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PGP-FM II-97-03
#1-Dividend Policy
• The Larger the RETENTION, the LESSER the dividends
• The choice between retention and payout would depend on the kind of effect each of these choices would have on the objective of maximizing the wealth of the shareholder
• Payout, if such a decision would lead to maximizing the Wealth of the shareholders: RETAIN otherwise
#2
• Irrelevance theory of Dividends• The theory rests on the BASIC premise that;
DIVIDEND DECISION IS A RESIDUAL DECISION
• The Dividend decision flows from out of the PRIMARY decision to RAISE CAPITAL or raise finances in a particular fashion than in another; funding is the primary decision and dividend decision that reflects SERVICING COSTS is only a PASSIVE RESIDUAL
#3
• If dividend policy is not a passive residual and instead is an “active Financing decision”, the decision to “retain or, payout” would rests on the premise—
• --Retain if the firm requires funds for Investment and CANNOT procure them AT A RATE CHEAPER THAN ITS EXISTING COST OF CAPITAL. It is of course assumed that the returns on the New Investment will be greater than the marginal cost of capital
#4
• Conversely, if Investment Opportunities are few and if the “returns on the New investments are LESS than the Cost of Capital, PAY OUT”
• Thus the GROUND RULE for the “adequacy of acceptable Investment opportunities” is the comparison of the ROI (r) with the Cost of Capital (Ke)
#5
• Theoretically, the following are the EXTREMES-
• --When adequate acceptable opportunities are available such that, ROI (r) is GREATER than the Cost of Capital (Key), the PAYOUT is ZERO
• -- OTHERWISE, the PAYOUT is 100%
#6• In all other cases, the payout will be between 0-
100%• The Passive Residual theory of Dividends is
thus based on the following parameters• 1) That financing decisions are primary and
dividend decisions are a mere “fallout of the Primary decision”
• 2)That, investors are INDIFFERENT between CAPITAL GAINS ( emanating from a possible BONUS ISSUE ) and Cash payout in form of dividends. So long as the firm is able to earn greater than its COC, investors would not mind if the profits are retained. In contrast, if ROR (r) is < Ke, investors would prefer to receive the Profits as DIVIDENDS
#7
• Modigliani & Miller Hypothesis• M&M contend that dividend decisions have
nothing to do with share prices and are of no consequence
• What matters is the Earning Capacity of the firm consequent to embarking on the New Project and the decision to split the earnings with the shareholders immediately or instead retain the profits is a matter of detail and holds no consequences on the share prices
#8
• If the Operational profits of a firm are to be determined by the level of EBIT then, the financing costs—whether as dividends or anything else-- would have no say on the Operational performance
• As already propounded by the two authors, if Method of Financing and the Capital structure were to be irrelevant in determining the Value of the Firm, the Costs of such financing that is a direct consequence of the above two Issues, can be no more relevant!
#9
• Assumptions of M&M Hypothesis• 1) Perfect Capital markets with Rational
Investors---- Securities are divisible, no transaction costs, information is free and available to all and there is No one Investor or a group of them to influence the markets
• 2)There are no taxes--- no difference between Revenue and Capital gains
#10
• 3) The Firm has a given Investment Policy that does not change—The implication of this parameter is that the firm could FINANCE New Investments from OUT of its retained earnings WITHOUT CHANGING ITS BUSINESS RISK (i.e. without undergoing a change in its required rate of return-(r)
#11
• 4) There is a perfect certainty in the estimation of the Future Profits and Dividends by ALL investors
• The Crux of M&M approach• The crux here again is the ARBITRAGE
ARGUMENT. If the Firm chooses to Payout and then raise Capital from the shareholders to meet its Capital Expenditure program, the effect of dividend payment on the shareholder’s wealth is EXACTLY OFFSET by the effect on the shareholder’s wealth to raise capital
#12
• When dividends are paid to the shareholder, the market value of the share decreases. This decrease is Identical to the extent of Dividends paid out. Essentially the gain in form of dividend receipt is offset exactly by a fall in the price of the stock. The Total market value PLUS Dividends of 2 firms which are identical except for their Pay outs must be the SAME
#13
• Proof of M&M Hypothesis• Step 1- The market price of a share at the
BEGINNING of a period must be equal to-the PRESENT VALUES OF THE SUMS of-a) The Dividends received at the END of the Year and b) The MARKET PRICE at the END of the Year
• __1__ (D1+P1)= Po• (1+Ke)
#14• Step 2 –Assuming that there is No external financing,( that
the Entire capital is Equity alone!) the total capitalized value of the firm would be the discounted value of
• no = __1__(nD1+nP1)--------(2)• (1+ Key)• Step 3 If the firm’s Internal sources( retained earnings) were to
fall short of the Investment outlay and delta n becomes the New shares to be issued at the end of period 1 at a price of P1. Equation 2 can be written as
• no= __1__(nD1+(n+delta n)P1-delta n *P1)--(3)• (1+Ke)• This is because (3) is the same as (2) , upon simplification!
15• It is easy to recognize that equation (2) and equation 3 are identical• Step 4 –If additional share Issue were to finance the additional
Investment,• Delta nP1= I-(E-nD1)= I –E + nD1-(4)• where delta nP1-> Amount obtained by the sale of New shares• I->Investment• E Earnings of the firm, nD1->Dividends paid and E-nD1 is Retained
earnings• I—(E-nD1) is nothing but-----’ the additional funds raised for
Investment’ ie Investment LESS what is generated as Internal Accruals’
16• Step 5– If we incorporate equation (3) with equation (4), • nPo= nD1+ (n+ delta n) P1- (I- E+nD1)***• --------------------------------------------• (1+Ke)• Thus nPo=(n+ delta n) P1- I + E• ------------------------------• (1+ Ke)• Since D ( dividends ) do not figure in the final equation they are
NOT relevant!• ***As delta nP1= I-E+nD1 as per (4)
17
• Problem --- A company belongs to a risk class for which the appropriate rate of capitalization is 10%. It currently has 25,000 shares outstanding, selling at Rs 100/- a piece. The firm is contemplating the declaration of a dividend @ Rs 5/ per share at the end of the current financial year. It expects to have a net income of Rs 2,50,000/ and has new Investments of Rs 5 lakhs on the block. Show under M&M hypothesis that the payment of a dividend does not affect the value of the firm
18
• Solution• A) VALUE of the Firm WHEN DIVIDENDS
ARE PAID• 1) Price per share at the END OF YEAR 1• Po=__1___(D1+P1)• (1+Ke)• 100=__1__(5+P1) or, P1= Rs 105/-• (1.10)
19• 2) Quantity of shares to be issued at Rs 105/ share to meet the
Investment shortfall• Additional amount = Investment outlay-{ Earnings- Payout} i.e. I-{E –
nD1 }• = 5,00,000-{ 2,50,000- 25,000* 5}• =Rs. 3,75,000• NO.of shares to be issued =375000/ 105• = 75,000/ 21 shares• Value of the firm=nPo=(n+delta n) P1- I+E• ----------------------------• ( 1+Ke)• 25000+ 75000/ 21} Rs 105-Rs 5,00,000+Rs 2,50,000• {old shares+ New shares} Issue}-Investment + Earnings• price} • 1.10• =Rs. 25,00,000/---Value of the firm when dividends are paid
20
• B) Value of the Firm when dividends are NOT paid
• 1) PRICE per share at the END OF THE YEAR • nPo=P1 / (1+Ke) • 100=P1 / (1.10) P1= Rs 110/-at year end
Recognize that when dividends were paid out in the earlier case, the market price was only Rs 105/( lesser than Rs. 110/ now, when dividends have not been paid) !
21
• 2) Amount required to be raised by Issue of New shares• Delta nP1= I – E= Rs. 5,00,000-2,50,000• =Rs. 2,50,000/-• 3) No. of additional shares to be issued• =Rs 2,50,000 /110=25,000/11• 4) Value of the firm• = 25,000+25,000/11}*Rs110-5,00,000+250000• 1.10• No.oldshrs+No. new shrs} Price/shr-Invst+Erngs• =Rs 25,00,000/- Same as in (A) earlier!
#22
• A critique of M& M • The CRITICAL observation in M&M Hypothesis rests on
the INDIFFERENCE of the Investors between gains by way of Capital gains and by way of Revenue gains. The balancing nature of this arbitrage can be looked at in 2 ways—
• 1) The fall in the price of the share MATCHING exactly the DIVIDENDS PAID OUT
• 2) The Indifference of the Investor towards receiving the profit share as either Dividends or as Bonus later on
• However the big question is “Is a balance always struck and that too exactly?”
23
• The arguments of M/M are appealing though of practically “no significance” The grey areas
• 1) Market imperfections—Taxes exist, floatation costs are present
• 2) Tax effect– Taxes are present. The Tax effect on a person’s Revenue gains would depend on the tax bracket he is in. Capital gains are a fixed percentage and do not vary with the Income earned There is a variance no doubt!
• 3) Statutory restrictions—Mutual funds MUST distribute 90% of their Income i.e. there cannot be indifference between Retention and Distribution
• 4) Informational content of Dividends, preference for current Income etc.
24• Relevance of Dividends theory• A) Walter’s model b) Gordon’s model• 1) Walter’s model—The Financing Policy, the
Investment Policy and the Dividend policy are all INTERRELATED. The choice of an appropriate dividend policy DOES AFFECT THE VALUE OF THE FIRM
• 2) The main premise behind this theory is the comparison between the Cost of Capital (Ke) and its Internal rate of return (r). Distribute if COC( or the reqd. rate of return) is greater than the Internal rate of return;(ROI) not otherwise
25
• The rationale is that if r>k, the firm is able to earn more than what the shareholders can if, the earnings were distributed to them as dividends. On the other hand, if r<k, the shareholders interests would be better served if the dividends were paid to them as they seemingly have better Investment opportunities.
• Walters model thus relates the distribution of dividends (retention of earnings) to available Investment Opportunities. If a firm has adequately profitable investment opportunities, it will be able to earn more than what investors expect so that, r>k. Such firms are called “GROWTH FIRMS” and for these firms, the Optimum dividend policy is ZERO
26
• Where r<k, the Optimum policy would be, for obvious reasons, a 100% payout!
• Where r=k, there is INDIFFERENCE as to the payout which can range from 0-100%!
• Assumptions of Walter’s model• 1) All financing is done through RETAINED
EARNINGS-no external Debt or Equity financing is envisaged
• 2)The Firm’s Business risk-r& k –do NOT change with Investments
27• 3) The firm has a perpetual life• Walter’s model on FIRM VALUATION • P= ___D_____ P- M.P of the share• Ke- g Ke-Cost of capital• D-Initial dividend• g-Expected Growth rate of Earnings i.e
b*r• b-Retention rate (E-D) /E• r-Expected rate of return on the firm’s Investment• br- measures the growth rate in dividends, which is the
product of—a) the Earnings retention percentage b) The profitability of the retained earnings (r)
• The bias in the model lies in the assumption that Ke is>g as otherwise P becomes an Imaginary figure
• The Assumption is that ‘Investor expectation is ALWAYS greater than Growth rate in Dividends’
#28• Substituting, • Ke=D /P+ g• Therefore, Ke=__D_ +__delta P__ ( as g=Increase in Prices ie deltaP)• P P • This is so since delta P is “ change in Prices” and therefore g= deltaP• Also since delta P=__r__ (E-D)• Ke• Substituting the value of delta P,• Ke=__D_+__r_ (E-D)• Ke• ------------------------ E EPS D DPS• P• Or P=D + r ( E-D)• Ke• ------------------• Ke
# 29
• Meaningfully,• P=--D+ r(E-D)• --- ---• Ke• ------------------------• Ke• D/Ke Capitalized value. of ALL DIVIDENDS• r/Ke (E-D)}• ----------- }-- Capitalized valueof All Capital Gains• Ke }• Thus the Price of a Security is related to its DIVIDENDS!
#30
• Walter’s model w.r.t the effect of Dividend/Retention Policy ON the Market value of shares ( under different assumptions of ‘r’)
• The following Info. Is available in respect of a firm
• Capitalization rate (ke)- 0.10• EPS-10 Rs Assumed rates of return(r)
i)15%ii)8% iii) 10%• The effect of Dividend Policy on Market Price of
a share
31
• Solution i) When r is 0.15, r>ke. • a) When D/P=0i.e DPS is zero• P=__D_+r__(E-D)• Ke• --------------- =0 + 0.15/0.10(10-0)• Ke -------------------------• 0.10• = Rs 150/-
32
• B) When D/P ratio is 100% i.e DPS is Rs10/-• P=_10_+__ {0.15}__ { 10-10}• {0.10 }• ____________________ =10/0.10• 0.10 =RS 100• Recognize that when r>ke, and DPS is ZERO,
the Market Price is Rs. 150. M.P decreases when dividend is paid out under these circumstances ( Rs 100/)
#33
• Assignment- calculate the market price when r is 15%, ke is 10%, EPS is 10Rs
• and DPS is a) Rs 2.50 b) Rs 7.50
• ---------------------------------------------------
• Case 2) The data remaining the same except for the following changes
• R=8% i.e r< Ke ( which is 10%)
#34
• A) When D/P is zero• P=_0_+ {0.08 } {10-10}• --------• {0.10} • -------------------------- = Rs. 80/• 0.10• Assignment; With the same data calculate the
M.P when a) DPS is Rs 5 b) When DPS is Rs 10/-
#35
• Assignment; If r=0.10 and Ke= 0.10 what would be the MP when a)DPS is Rs2.50 b) When DPS is Rs 10/ . What is your observation?
• --------------------------------------------------• Interpretations• A) When the firm is able to earn a rate of return
r, that is greater than its Cost of Capital Ke, a lower distribution increases the MP of the share The Optimum Payout is Zero-when the Price of the share is maximum. If Payout is 100% the MP is 0
#36
• 2) When r<ke, a 100% payout maximizes the MP of the share. Here the MP is positively correlated to the Payout
• 3) When r=Ke, the market Price is constant at all degrees of payout i.e. MP is INDIFFERENT to payout
• Limitations of Walter’s model• 1) Assumption of a constant Ke; Ke changes
with the risk complexion of the firm• 2) Assumption of a constant ‘r’. ‘r’ is rarely
constant!
#37
• GORDON”S MODEL• Like Walter’s model, Gordon’s model
emphasizes that dividend policy of a firm is related to the Market Price of a share and that Investors place a ‘positive premium’ on ‘current dividends’
• Investors are Risk- averse; they are rational• They place a premium on returns that are
‘certain’ and penalize returns that are not
• Chart. Y axis- Discount
Retention rate
38a
• The Discount on the market Price of the share is CONSTANT ‘but up to a certain percentage of RETENTION. Up to this percentage of retention, the market does not add DISCOUNT to the share price; beyond price, the discount increases the discount sharply.
• This is because the Market PENALISES a firm for uncertainty in its Dividend payment
39
• Gordon model
• P= ___E__(1-b)_ E-> EPS
• Ke- br b->% earnings retained
• 1-b-> D/P or % of earnings distributed
• Ke-> Capitalization rate or COC
• br=g=rate of return of an all equity firm.( It is retention rate* rate of return )
40-
From the following information in respect of the rate of return (r), the capitalization rate (ke) and EPS of a firm determine the value of the shares from the following data
r=12%E-> Rs 20 D-(D/P ratio) Retention ratio (b) Ke%1 10 90 20
70 30 14
41
• Solution• P= __E__(1-b)__ } D/P 10%->retention 90• Ke-br } i.e ‘b’ is 0.90• }br-> 0.90* 0.12=0.108• P= __20_(1-0.90)__ = Rs 21.74• 0.20- 0.108• 2) When D/Pis 70 and retention 30• P= __20_(1- 0.30)__ br=0.30* 12= 0.036• 0.14- 0.036 Ans= Rs 134.62• As can be seen, the better the payout, the better the MP
42
• Dividend Policy- factors determining it
• --Dividend Payout %
• --Stability of dividends
• ----Legal, contractual, Internal constraints
• --Owner’s requirement
• ---Capital market consideration
• ---Inflation etc.
43
• An Optimum Dividend Policy should strike a balance between a) Maximizing the wealth of the shareholders b) providing funds for growth
• These Objectives are NOT mutually exclusive but are ‘interrelated’
• Dividend Policy must not be viewed as a ‘Passive residual’ but must be a decision based on solid ground rules which could include—
• a) Earnings b) Earnings growth c) S/h preference for current dividends 4) Investment opportunities5) Cost of raising funds 6) The extent of ‘retention necessary’ and therefore the Payout
44
• Stability of Dividends
• 1) CONSTANT DPS
• 2) CONSTANT PAYOUT RATIO= DPS/EPS *100
• 3) CONSTANT DPS plus EXTRA DIVIDENDS
Risk & return
• For any Asset, the term ‘returns’ comprises of—
• a) Revenue Yield –like Dividend yield, Interest yield etc
• b) Capital gains
• R= ___Dt_+(Pt-Po)___
• Po
• Average return= __1__ sum of Ri
• n
#2
• Rate of return and Holding period
• Suppose you invest Re 1 in a Company for 5 years. The rates of return are 18% , 9% , 0% (10%) and 14%. What is the worth of your Investment?
• The Investment worth after 5years
• (1+0.18) * (1+.09) * (1+.00) *(1+ -0.10) *(1+0.14)=1.18* 1.09* 1*0.90*1.14= 1.32 Rs
3
• Since the Initial Investment is Re 1, the return is Rs 1.32-1= Rs 0.32 or 32%. This is over a 5 year period assuming that the Dividends received are re-invested in shares. The Compound Annual rate is
5 Root of ( 1.18*1.09* 1.0*0.90*1.14 –1)
=5th root of 1.3196- 1= 0.057 0r 5.78%
The Compound annual rate is called the Geometric Mean returns
4
• Suppose you have invested Re 1 in the shares of HLL in the beginning of the year 1993 and held it for 2 years. If the returns for 1993 is 16.52% and in 1994 is 22.71 %
• The Investment after 2 years is
• (1+0.1652) * ( 1+ 0.2271)= Rs 1.43= 43%
• The Compound Geometric mean return is
• 2 root of ( 1.1652* 1.2271) –1 =0.195 or 19.5%
5-Risks & Returns• Risk is a measure of variability. Standard
deviation and Variance reflect it
• Variance=__1_sum of ( Ri- R bar)^2
• n-1
• Rates of Return under the different Economic conditions
• Economy M.P Dividend Yield C/gain Rtns
• 1 2 3 4=3/** 5=(2)-** 6=4+5
• Growth 305.5 4.00 0.015 0.169 0.185
• Xpansion 285.5 3.25 0.012 0.093 0.105
•
6-contd.• Eco M. P Dividend Yield C/G Retns• Stagnation 261.25 2.50 0.010 00 0.010• Decline 243.50 2.00 0.008 -0.068 -0.060 • **- refers to Rs. 261.25 i.e current M.P• Ex. Yield-> 4/ 261.25=0.0153;
3.25/261.25=0.012; 2.50/261.25=0.010 2/ 261.25= 0.008
• Ex. C/G305.5-261.25 / 261.25= 0.169; 285.50- 261.25 / 261.25= 0.093; 261.25-261.25/ 261.25= 0; 243.50- 261.25/ 261.25=(0.068)
7-contd.• Ex. Retns. R1= __4+(305.5-261.25)_=0.185
• 261.25 18.5%
• R2=__3.25+_(285.5-261.25)_=0.105 10.5%
• 261.25
• R3= 2.5+__(261.25- 261.25)_=0.01 1%
• 261.25
• R4=2.00+__( 243.5- 261.25)_=(0.060) (6%)
• 261.25
• The Total return is a anticipated to vary between (6%) and 18.5%
8• Having worked the returns R1 to R4 you
can now work out the “Expected rate of Returns” by assigning PROBABILITIES to each outcome
• Expected ROR= rate of return under scenario1* probability of scenario 1+ rate of return under scenario 2* probability of scenario 2+ rate of return under scenario3* probability under scenario3 + rate of return under scenario4* probability under scenario4
9
• The “ Expected rate of return” is the “Average rate of return”. The DISPERSION is explained by the standard deviation. For ex. If the expected rate of return is 6% and the std. deviation is 8%, the investment in this asset could be a risky proposition as there is a 68% chance that the ‘returns’ could VARY between + 14% and –2% assuming the distribution to be a Normal one
10• The shares of M have the following returns
and the following probabilities associated with it.
• Return% -20 -10 10 15 20 25 30• Probability• 0.05 0.10 0.2 0.25 0.2 0.15 0.05• Exp. return= (20)* 0.05+ (10)*0.1+10*0.2+
15*0.25+20*0.2+25*0.15+30*0.05=14.5%
Variance=(-20-14.5)^2*0.05+(-10-14.5)^2*0.10+(10-14.5)^2*0.2 +contd.
+
11
• + (15-14.5)^2*0.25+ (20-14.5)^2*0.2+(25-14.5)^2*0.15 +(30-14.5)^2 * 0.05
• = 59.25+60.03+4.05+0.06+6.05+16.54+12.0=158
• Std. deviation= root of Rs.158= Rs.12.57
• There is a 68% chance that the returns will vary between Rs 27 and Rs 2/-
12
• Risk preference
• Investors normally prefer investments with HIGHER rate of returns and LOWER variability( dispersion). According to the Law of Diminishing marginal utility, as a person runs into more and more wealth, the utility he gets for the additional wealth acquired by him increases but at a declining rate
12-a• Attitudes towards risk• Assume you were a contestant in a game “let’s make a deal’. The
host Monty Hall explains that you get to keep whatever you find behind either Door #1 or, door #2. Behind one door is $10,000/ and behind the other ‘nothing’
• You choose to open door #1 and claim your prize. But before you can make a move, Monty says he will offer you a sum of money to call off the whole deal. {before reading any further , decide for yourself WHAT DOLLAR AMOUNT WOULD MAKE YOU INDIFFERENT BETWEEN –a) taking what is behind the door and b) accepting the ‘call off deal money Monty pays’} That is, determine an amount such tnat ONE DOLLAR MORE may prompt you to take the ‘deal money’ and ONE DOLLAR LESS makes you ‘keep the door’
• Now let’s assume that you decide that if Monty offers you $ 2999 or less, you will keep the door. At $ 3000 you cannot decide between taking the cash and keeping the door and at $ 3001 you would like to take the ‘deal money’ and give up the door
12-b• Monty now offers $3500/. So you take cash (accepting the ‘deal
money’ ) and give up the door ( Never mind that there could be $10,000/ behind it! )
• What has this example got to do with risks? Everything! We know for sure that an average Investor is ‘averse to risks’. Let’s see why. By keeping the door, you had a 50% chance of getting $ 10,000/ The Expected Value ‘in keeping the door’ was $ 5,000/ ( 10,000*0.5+0*0.5). In our example above you find yourself Indifferent between a RISKY ( uncertain) $5,000 return (expected) AND a CERTAIN(Definite) sum of Rs 3000/-. In other words, this certain or riskless amount of $3000/-- what one calls a CERTAINITY EQUIVALENT(C/E)– provided you the SAME utility or satisfaction as a RISKY gamble with expected value of $ 5000
• It would be AMAZING if your ACTUAL Certainty Equivalent in this situation was exactly $3000’ the ‘deal money’ offered by the host Monty! The Figure you wrote down is less than $ 5000/ and most people would react the same way as you did as, they are ‘Risk averse’
• We can possibly use the relationship of an Individual’s Certainty equivalent to the expected Value of a risky investment to DEFINE THEIR ATTITUDE TOWARDS RISK. In general, if a person’s –
• * Certainty Equivalent<Expected Value, he is AVERSE to risks
• *Certainty Equivalent= Expected Value, he is INDIFFERENT to Risks
• * Certainty Equivalent> Expected Value, he is a RISK SEEKER( prefers risks!)
12-D• In our example, any Certainty Equivalent LESS THAN THE
EXPECTED VALUE of $5000/ indicates ‘Risk Aversion’. • For RISK AVERSE Individuals, the difference between the
Certainty Equivalent AND the Expected Value represents ‘ RISK PREMIUM”
• Risk Premium is the ADDITIONAL RETURNS that RISKY INVESTMENTS must offer to the Individual for him to ACCEPT the RISK and not accept the DEAL offered
• Investors are generally ‘risk averse’. This implies that risky investments must offer HIGHER EXPECTED RETURNS than LESS RISKY INVESTMENTS in order that they BUY and HOLD it
• We talk of Expected returns here. And in order to have Low risks , one must be ready to accept ‘low expected returns’
• In short, the Business of Investment has ‘no free rides and no free lunches’ss
•
13Risk returns for Investors in different Risk
An Investor who is ‘Risk averse’ will choose-
i) From Investments with EQUAL rates of returns, the Investment which has the LOWEST risk. i.e. the Lowest std. deviation !
ii)From Investment with the SAME risks, that which gives the HIGHEST return
14• A ‘Risk seeking Investor” prefers
Investment with HIGH risk irrespective of the Returns it will provide!
• A ‘Risk Neutral Investor” looks NOT at the risks but at the ‘Returns’; he chooses investments that give him the HIGHEST RETURNS
• The next question
• How would a Risk averse Investor make choices when Investments have--- > risk & > Returns;;, < risk & < returns
15
• Portfolio theory and Asset Pricing model
• A Portfolio is a combination/ bundle of assets/ securities
• This Portfolio theory is based on the assumption that Investors are ‘Risk averse’ and that the Returns are ‘normally distributed”
16
• Portfolio returns– 2 asset case
• The Return of a Portfolio is equal to the Weighted average of the returns of INDIVIDUAL assets ( or securities) in the Portfolio with WEIGHTS being equal to the PROPORTION OF INVESTMENT VALUE in each asset
• Suppose you have an opportunity of investing in Asset X or in Asset Y.
17
• Eco Probability Returns %
• X Y
• A 0.10 -8 14
• B 0.2 10 -4
• C 0.4 8 6
• D 0.2 5 15
• E 0.1 -4 20
18
• Expected rate of Return---Share X
• (-8*0.1)+ (10* 0.2)+( 8*0.4)+( 5* 0.2)+
(-4*0.1) =5%
Similarly the Expected rate of return for Yis
= ( 14* 0.1) + (-4*0.2) +( 6*0.4) + ( 15* 0.2) +( 20*0.1)= 8%
Now suppose you decide to invest 50% of your wealth in X and 50% in Y What is the Expected return on a portfolio of X and Y?
19• I) This can be done in 2 ways
• a) Calculate the combined outcome under EACH state of the Economy
• b) Multiply each combined outcome by its Probability ( See the Table that follows)
• II) Direct Method First calculate the Expected Return on the Portfolio i.e. weighted average of the Expected returns of the 2 assets X and Y in the Portfolio
• E(r)= ( 0.5*5)+ ( 0.5*8)= 6.50%
20
• Table – see Slide # 19• Eco Prob. Comb. Rtn Exp.rtn• 1 2 3 4=2*3• A 0.1 (-8*0.5)+ (14*0.5)=3.0 3*.1=0.3• B 0.2 ( 10*.5)+(-4*.5)=3.0 3*0.2=0.6• C 0.4 (8*.5)+(6*.5)=7.0 7*0.4= 2.8• D 0.2 (5*0.5)+(15*0.5)=10 10*0.2=2.0• E 0.1 ( -4*0.5)+( 20*0.5)=8 8*0.1=0.8• -----------• 6.50
21
• Note that if ‘w’ is the proportion of Investment in Asset X, (1-w) is the Investment in Asset Y. Given the Expected Returns of Individual assets, the Portfolio return depends on the WEIGHTS ( Investment proportion) of assets X and Y. You will be able to CHANGE the Expected rate of Return on the Portfolio by changing the PROPORTION INVESTED in INDIVIDUAL ASSETS
22• How much would you earn if you invested
20% in Asset X and the remaining in Asset Y?
• E(Rp)= 0.2*5+ (1-0.2) * 8= 7.40%
• Thus the job of a Portfolio manager, is to –
• 1) select appropriate assets that form the Portfolio after aligning them to the ‘risk- return profile’ of the portfolio participants
• 2) assign ‘weights’ to individual assets in the portfolio
23
• Now, why invest in BOTH X and Y when Y yields much higher than X?
• Because of ensuing risks; under ‘unfavourable state’, Y may yield a negative return of 4%!
• The chances of incurring a negative return gets theoretically eliminated when X and Y combine into a Portfolio
25a-to be read with/after #25
• Does this mean that there is a 68.3% probability that returns will vary between 40% and 0%?
• This is true of Individual assets. In a Portfolio one needs to look at the Coeff. Of Correlation and Covariance as a consequence. While returns of the Portfolio could be more than the returns of one of the Assets comprising it, the Risks could be totally eliminated if the Coeff. Of Correlation of the 2 Assets were Perfectly negative
24• Portfolio risk -2 asset case• Returns on Individual assets fluctuate
MORE than their Port. returns. Why is this so?
• This is because a ‘scientific portfolio diversification’ could ELIMINATE ALL RISKS in a 2 Asset Portfolio
• Investments in A and B• ECO Prob. Retns.A% Retns.B%• Good 0.5 40 0• Bad 0.5 0 40
25• Recognize that the average returns for
Both A and B is 20%
• The expected returns
• E(R) = 0.5* 40 +0.5* 0= 20%
• Variance= 0.5( 40-20)^2+ 0.5(0-20)^2
• = 200 + 200= 400
• Std. deviation= root of 400= 20%
• So A has an Expected Value of 20% with a std. deviation of 20%. The same is true of asset B.{ Now go to #25a}
26• Ans: The Expected return of the
PORTFOLIO is the SAME as the Expected returns on INDIVIDUAL assets A and B
• E(R)= 0.5*20+ 0.5*20= 20%
• This is the same as the return on Individual Assets BUT the RISK is TOTALLY ELIMINATED. Why so?
• Because if the Economy were good A would yield 40% and B 0%. The Expected return would be 20%
27
• Similarly, in a Bad economy, A would yield 0% and B 40% with an Expected value of 20%!
• Thus there is NO RANGE, NO DISPERSION and NO RISK as a consequence!
• Caveat; In reality it is very difficult to locate 2 assets whose returns move in “absolutely opposite” directions for ALL States of the Economy!
28• Measuring Portfolio risks for 2 assets
• While the Portfolio Return is the Weighted Average of the returns on the 2 assets, the Portfolio Risk is NOT the weighted average of the Risks of the 2 assets!
• The Portfolio risks depends on the ‘co-movement” of the returns from the 2 assets
• Covariance- Measures the co-movement of 2 assets
29
• Covariance
• Step 1 ; Determine the Expected returns of each Asset
• Step 2; Determine the Deviation of the returns from the Expected returns
• Step 3 Determine the product of each deviation with its Probability
• Step 4 Sum up Step 3
30
• Step 1 E(Rx)= {0.1*(-8)}+{0.2*.1}+{0.4*8}+
{0.2*5}+ {0.1*(4)} = 5%
E(Ry)={0.1*14} +{0.2*(-4)}+{0.4*6}+{0.2*15}
+{0.1*20} = 8%
If Equal amounts are invested in X and Y the Expected returns on the PORTFOLIO is-
E(Rp)= 5*0.5+ 8* 0.5= 6.5%
31• The table below shows the calculation of variations
from the Expected returns and Covariance #17• Eco Prob. Return Deviation Product of• X Y from Ex.Rtn deviation &• Xbr Ybr Probability• A 0.1 -8 14 -13(-8-5) 6(14-8) -7.80• B 0.2 10 -4 5 (10-5) -12(-4-8) -12• C 0.4 8 6 3(8-5) -2(6-8) -2.4• D 0,2 5 15 0(5-5) 7(15-8) 0• E 0.1 -4 20 -9(-4-5) 12(-8-4) -10.8• sum=--33.0
32• In the calculations note that the value of
Xbar=5% and of Y bar= 8% as seen earlier in #30
• Alternatively, Cov x,y can be calculated thus
• Summation {Rx-E (Rx)} {Ry –E (R y)} Pi
• Rx- Xbar Ry- Y bar prob.
• ={0.1(-8-5)(14-8)}+ 0.2(10-5)(-4-8)+0.4(8-5) (6-8) +0.2(5-5) (15-8)+0.1(-4-5)(20-8)
• -7.8-12-2.40+0-10.8= (33)
32-a• A Note on the importance of the concept of COVARIANCE in
PORTFOLIO Management• Covariance; Is a Statistical measure of the degree to which two
variables ( Ex. Securities Returns) move together. A POSITIVE VALUE indicates that on an average,(need not always be so!) they move in THE SAME DIRECTION
• Unlike the Portfolio returns, the Portfolio risks( measured by the variance) is NOT the Weighted Average of the Std. deviations of Individual Assets. To take a weighted average of the Std. deviations of the Individual Assets would mean IGNORING a SPECIAL RELATIONSHIP of COVARIANCE that exists among INDIVIDUAL ASSETS
• Covariance does NOT affect Portfolio returns. It provides for the POSSIBILITY of ELIMINATING/REDUCING risks WITHOUT reducing POTENTIAL RETURNS The Startling revelation- The RISKINESS of a PORTFOLIO depends MUCH MORE on the Covariance of the paired securities THAN it does ,on the Std. deviation of Individual assets !
32-b• Thus a combination of INDIVIDUALLY RISKY securities could
STILL COMPRISE a ‘moderate to Low risk Portfolio’ SO LONG AS the Securities do not LOCK-STEP each other! In short, LOW COVARIANCES lead to low Portfolio risks!
• DIVERSIFICATION• Is the process of “spreading” risks ACROSS a NUMBER and
VARIETY of Assets / Investments . Concentration on Numbers by itself could make the “Diversification Process” at best, NAÏVE. This would imply that Investing $ 10,000 across 10 securities would MAKE a PORTFOLIO MORE DIVERSIFIED than $10,000/ being invested ACROSS 5 Securities. SILLY! The fact remains that NAÏVE DIVERSIFICATION IGNORES the COVARIANCE( Correlation) between / among Security’s returns.
• DIVERSIFICATION Is a process of risk reduction. If the Returns in an asset A is CYCLICAL-moves with the Economy in general, it would make sense to include Asset B ( which is Counter-cyclical) in building a 2 asset Portfolio
32-c
• The returns on these 2 assets are Negatively correlated. Equal amounts invested in these two assets will go to REDUCE the DISPERSION of returns from the portfolio because the risks of the 2 assets are ’off setting’
• Investing in the World markets can help achieve greater financial diversification than investing in ONE country abroad
33• What is the relationship between the Returns of
Security X and Security Y?• The following possibilities exist• 1) Positive covariance; Meaning that the returns of
BOTH X and Y are –• a) ABOVE the AVERAGE RETURNS of Portfolio• b) BELOW the AVERAGE RETURNS of Portfolio• 2) Negative covariance; The returns from ONE
asset is ABOVE the AVERAGE RETURNS while the returns from ANOTHER is BELOW its AVERAGE and vice-versa( av.returns is 6.5% while Xis 5% and Yis 8%. Also covariance was a negative 33%)
34
• 3) ZERO covariance; returns on X and Y could show NO pattern– that is, there could be NO relationship in their movements. This lack of relationship could be due to ‘randomness’
• In the example seen earlier, the minus sign denotes Negative relationship; the number 33 cannot however be explained
35• Correlation• Is the measure of LINEAR relationship between
2 variables. There is a relationship between Covariance and Correlation
• Covariance XY=Std. deviation X * Std. deviation Y *Correlation x, y
• Corrln. x,y= Covariance x,y / SD x, SD y• The value of the correlation known as
Correlation Co-efficient can be Positive, Negative or Zero. Here again the ‘sign’ of the Correlation coefficient depends on the ‘sign’ of the Covariance (as SD cannot be negative)
36• Using the earlier data for assets X and Y,
• Variance X= 0.1(-8-5)^2+ 0.2(10-5)^2+0.4(8-5)^2+0.2(5-5)^2+0(-4-5)^2
• =16.9+ 3.6+0+8.1 =33.6
• Std. dev.=root of 33.6=5.80%
• Variance Y= 0.1(14-8)^2+0.2(-4-8)^2+0.4(6-8)^2+0.2(15-8)^2+0.1(20-8)^2
• =3.6+ 28.8+1.6+9.8+14.4=58.2
• Std.dev.y=root of 58.2=7.63
37
• The Correlation of the 2 securities X and Y
• Corx,y= __Cov x,y_________
• Std.dev.x, Std. dev. Y
• = ____-33_ =___-33_ =-0.746
• 5.8* 7.63 44.25
• --0.746 represents a HIGH degree of Negative relationship
38• Variance and Std. deviation of a 2 asset
Portfolio
• The Variance
• =Sda^2 Wa^2+Sdb^2Wb^2+2Wa Wb Pab where,
• P ab=Cov a b=Sda.* Sdb *Corrln a b
• Therefore variance of a Portfolio
• =Sda^2 Wa^2+ Sdb^2 Wb^2+ 2 Wa Wb Sda. Sdb Corrln a, b
• s
39• It may be noted that the Variance of a Portfolio INCLUDES the Proportionate Variances of Individual Securities AND the Co variances of the securities put together
• The Risk of a Portfolio will be LESSER than the Risk of Individual assets for LOW or Negative correlation
• The Portfolio risk depends on the Correlation between the assets. The concept of a weighted average Risk, which is just an average of the risks of the 2 assets is SILLY -----contd.
40• This is because the concept of a risk for a
Portfolio must take cognizance of the correlation between the assets in the Portfolio
• The Weighted average (silly) of the SD (risk) of securities X and Y is 6.70 as is seen { 5.8 *0.5+ 7.63* 0.5= 6.70%}. However, at a Coeff. of Corrln. of 1, the Portfolio risk represented by its Sstd. deviation is the SAME as its weighted average i.e. 6.70%
41• So, unless the Coeff. Of correlation is
LESS than ONE, the risk in combining two assets is the SAME as the risk in NOT DOING SO!
• There is this concept of a MINIMUM VARIANCE PORTFOLIO the risk of the Portfolio is the LEAST.
• MVP= Wx*= ___SDy____ approximately
• SDx+ SDy
• Wy*= 1-Wx
42• In our earlier example we had the
Standard deviation of Y as 7.63, the standard dev. of X as 5.80 and the Covariance as –33. the MVP is
• W*x=___7.63* 7.63__--(-33)_____=0.578
• 7.63*7.63+5.8*5.8- 2(-33)
• W*y=1-0.578= 0.422 Accurate Minimum Variance Portfolio The Sd for this Portfolio is worked out in the next slide
• contd.
43• Var.p=Sd x ^2 * W x^2+ Sdy^2 *Wy^2+
2.0 Wx Wy Sd x Sdy. Cov x,y
• =33.6(0.578)^2 + 58.2(0.422)+ 2 (0.578) (0.422) (5.8)(7.63) (-0.746)
• = 11.23+ 10.36-16.11= 5.48(Port.Variance
• Std. deviation=root of 5.48= 2.34
• A portfolio comprising of an investment of 57.8% in asset x and 42.2% in Asset y would lead to the Lowest risk of 2.34% . Any other combination would only entail HIGHER risks
44• Suppose the Coeff. Of correlation was
+ve 0.25. What would the measure of risk be like? Equal weights
• Variance= 33.6(0.5)^2+ 58.2(0.5)^2+ 2(0.5)(0.5)(5.8)(7.63)(0.25)
• =8.4+14.55+5.53= 28.48
• Std. deviation= root of 28.48= 5.34
• This Portfolio risk though lower than the weighted average risk of 6.70% is FAR HIGHER than the Minimum Variance risk of 2.34
45• Illustration Securities M and N are ‘equally
risky’ but have DIFFERENT expected returns
• M N
• Expected returns(% ) 16% 24%
• Weight 0.50 0.50
• Stand.dev. 20 20
• What is the Portfolio risk (variance) if a) Corr m, n = +1 b) Corr = -1 c) Corr = 0
d) Corr= +0.1 e) Corr= -0.10?
45a• Var.P= (Wx*SDx +Wy*SDy)^2 When P=+1
• SDp =WxSDx+WySDy
• Var.P= (Wx*SDx-Wy*SDy)^2 When P=-1
• SDp = Wx*SDx- Wy*SDy
• Var.P=(Wx*SDx)^2+( Wy*SDy)^2} When P=0
• SDp=Square root of (Wx^2*Sdx^2)+(Wy^2*SDy^2)
46• When Correlation between 2 assets is +1.0, the Risk of the portfolio gets reduced to the Weighted Average risk of 2 Assets
• i.e SDp= Sdx Wx + Sdy Wy
• = 20*0.5+ 20* 0.5= 20%
• The Portfolio Sd when Corrl.= -1.0
• SDp= root of { 20^2* 0.5^2+ 20^2* 0.5^2
+2*0.5*0.5*20*20—1.0} or Wx*sdx-WySDy
= root of 100+100-200=0
At Corrln. -1.0 the risk is ELIMINATED!
47• Where the correlation is 0, the equation for
Sdp reduces to
• =root of Sdx^2* Wx^2+ Sdy^2* Wy^2
• =root of 20^2 *0.5^2+ 20^2*0.5^2
• = root of 200=14.4%
• D) the Portfolio Variance under a weak +corr of 0.1
• Variance=rt.of 20^2*0.5^2+20^2*0.5^2+ 2*0.5*0.5*20*20*0.1=rt.of 220=14.83%
48• Theoretically it is possible to largely reduce
or, completely eliminate risks in a 2 asset portfolio by having assets which are ‘perfectly negatively correlated’
• Portfolio risk- return analysis
• Positive correlation
• When one more invests in a HIGH Risk, HIGH yielding security (where the 2 securities are ‘perfectly positively correlated’), the portfolio Returns increases but so does the risk!
49• The relationship is perfectly ‘linear’.
B
/
/
C
A ‘risk averse Investor’ would prefer to invest at point C while, a ‘risk seeker would attempt to invest at B the others would attempt to invest at any point along BC
50• The relationship is perfectly ‘linear’.
B
/
/
C
A ‘risk averse Investor’ would prefer to invest at point C while, a ‘risk seeker would attempt to invest at B the others would attempt to invest at any point along BC
C
BA
51
• Both lines AC and AB represent lines -1.0 The Std. deviation
• =Std. deviation x* W x– Std. deviation y* Wy
• Also Wx* that is Minimum Variance Portfolio is = ____Std.deviation x___
• Std.deviation x+ Std. dev.Y
• Wy*= 1-Wx* {In Wx* SD is ZERO}
52**
• In slide #50, the diagram shows line AC superseding line AB. The risk keeps reducing on line AC till point A when it reaches zero.
• Limits to diversification
A C
B
53
• If we ADD up figures that we saw for Perfectly + Correlation and the figure on slide#50 for Perfectly –ve Correlation, we get the figure on slide #52. This triangle ACB represents the LIMITS of diversification which has a boundary between +1 and -1. It is only within this triangle that various risk and return combinations can lie in a 2 Asset portfolio
54
• 3) Zero correlation• As seen earlier, assets with Zero correlation help
REDUCE risks of the Portfolio( compared to the risk associated with Individual assets) without reducing( often increasing) returns in the process. You could increase the weight of a MORE RISKY ASSET in the Portfolio WITHOUT increasing the risks of the Portfolio possibly
54a-Limits of Diversification• An analysis of the Diagram on slide#52 • Lines AT and BT denote Lines formed by
Perfectly Negatively correlated assets while the Line AB represents the line formed by assets that are Perfectly positively correlated. Thus the triangle ABC represents the LIMITS OF DIVERSIFICATION as even assets not at all correlated-( zero correlation) must lie within +1 and -1!
• Let us Examine a clock-wise movement from point A along the pivots B and T
54-b• As you move from A to T in a clock wise
direction and inside the Triangle ABT, we find that as we move from perfectly positively correlated asset(+1.0) to perfectly negatively correlated asset and upto the extreme T where Std. deviation or risk is ZERO, we find that—
• A) For the same returns the risk actually reduces-- lines 1-6 on Xaxis
• B) In fact as the risk goes about decreasing, the returns actually are increasing-Line BT
• C) After the point T, along TA any further increase in returns comes only with increasing risks! Lines 61
55****• Systematic and Unsystematic risks
• We have seen that combining Assets that are NOT perfectly positively correlated, helps LESSEN the risk of a Portfolio. The Questions that then crop up are---
• 1) How much Risk reduction is reasonably possible?
• 2 ) How many DIFFERENT security holdings would be required in a Portfolio?
• The Figure would help. X axis No.of securities in a Portfolio Y Std. deviation
56• Research studies have looked at what
happens to ‘Portfolio risk” as RANDOMLY SELECTED STOCKS ARE COMBINED to form an Equally Weighted Portfolio
• When we look at a SINGLE STOCK the risk of the Portfolio is the Std. deviation of that stock. As the NUMBER ( and value) of RANDOMLY selected stocks held in the portfolio is increased, the TOTAL RISK of the Portfolio is reduced. Such a reduction is at a DECREASING RATE however!
57
• Thus a substantial amount of risk of a portfolio is ELIMINATED by ‘relatively moderate’ extent of DIVERSIFICATION say 15 to 20 RANDOMLY SELECTED stocks in EQUIDOLLAR amounts (not just numbers)
• Total risk=Systematic risk+ Unsystematic risk• Systematic risk Undiversifiable or unaviodable • Unsystematic risk- - Diversifiable/ avoidable
This is denoted by the Flattening of the Total Risk curve
58• Systematic risks affect the OVERALL MARKETS
by way of such changes as increase in Interest rates, Increase in Oil Prices etc. These CANNOT be ‘diversified away”. In other words even Investors who hold well- diversified portfolios are subject to these risks!
• Unsystematic risks are UNIQUE to a PARTICULAR INDUSTRY or a PARTICULAR COMPANY. They are unaffected by Economic, Political and other factors that that affect all other securities. A Wild cat strike can affect only one company or only one Industry, a technological break through cannot make all products obsolete!
59• For most stocks, Systematic risks account
for almost 50 % of the stock’s TOTAL RISK! By ‘considered diversification’ these risks are reduced or sometimes eliminated at least in a 2 asset Portfolio! Thus not all the Std. deviation in a stock is relevant as some portion pertaining to Unsystematic risk can be diversified away!
• Investors who bear Systematic risks need to be compensated for it. This logic is the essence of CAPM
60-CAPM• The Capital Asset Pricing Model
• Based on the behaviour of Risk averse investors, there is an IMPLIED EQUILIBRIUM RELATIONSHIP between Risk and Expected returns for each security. In MARKET EQUILIBRIUM, a security is supposed to provide an ‘Expected return’ COMMENSURATE’ with its SYSTEMATIC risk. The relationship
• Between a)Expected return b) Systematic risks is the basis of CAPM
61
• Developed by Nobel Laureate William Sharpe, the model helps us draw certain implications about RISKS and the SIZE OF RISK PREMIUMS necessary to compensate Risk bearing.
• Before trying to understand the concept of CAPM we need to understand the combination of assets in a Portfolio, the feasible region and the Efficient Frontier, The Indifference curves Optimal portfolio, the concept of Beta and the Characteristic and Security market Line
B
C
A
ZRETURNS
RISK
D
EFFICIENT FRONTIER.
62***• Feasible Region and Efficient Frontier• The collection of all possible portfolios
represents the feasible region.(the Portfolio Opportunity Set) The feasible region is the shaded region in the Portfolio combination
• Given the Feasible region, which portfolio should the Investor choose?
• The Investor should choose the portfolio that MAXIMISES his UTILITY FUNCTION
• The choice involves the 2 steps— #63
63• 1) Delineation of the set of Efficient Portfolios• 2) Selection of an OPTIMAL PORTFOLIO from
the set of Efficient Portfolios• A Portfolio is EFFICIENT if a) It gives the SAME
returns for a lower std.deviation• b) Gives higher returns for the same std.
deviation• Thus Portfolios lying along BC forms the
EFFICIENT portfolio. The boundary may be referred to as the ‘Efficient Frontier” All other Portfolios are “inefficient”. A portfolio Z is Inefficient as portfolio B dominates it. The Efficient frontier is the SAME for ALL investors as the Portfolio theory assumes that ALL investors have HOMOGENOUS expectations.
64
• How does one obtain an OPTIMAL Portfolio from a Efficient frontier? By graphical analysis for a 2 asset portfolio and with Quadratic Programming for a n asset portfolio
• Selection of Optimal portfolio from an Efficient Portfolio
• After selecting the efficient portfolio by Quadratic analysis, one will then have to select an Optimal portfolio
I4
I3
I2I1
X Y ZEXPECTED RETURN
RISK
INDIFFERENCE CURVES
65***• The selection of an Optimal Portfolio
starts with one ‘defining the risk- return preferences’ of an Investor. For this we start by plotting “Indifference curves”. Any point on ONE Indifference curve is as good as ANY OTHER POINT on the same Indifference curve. This is because the risk- return proportions at these two points make no difference!
• However this is NOT SO for points on 2 different Indifference curves!
B
C
A
RETURNS
RISK
D
YY3
Y2
Y1X
X3
X2
X1
OPTIMAL PORTFOLIO.
66***
• The Utility or the ‘level of satisfaction’ INCREASES as one moves LEFTWARDS I-2 gives a higher level of satisfaction than I-1 and so on. For ex. IF points X Y and Z represented an expected return of RS. 5 on I/C curves I-3, I-2 and I-1 respectively, it is clear that the SAME RETURN is being achieved at LESSER RISKS as the Std. deviation at I-1 is lesser than at I-2 which again is lesser than at I-3; all for the SAME returns! This gives the Investor Higher satisfaction as he achieves the SAME returns with Lesser risk as he moves fromI-1 to I-3!
67• OPTIMAL PORTFOLIO• GIVEN the Efficient frontier(step1) and the Risk-
return indifference curves (step 2), the OPTIMAL PORTFOLIO is located at the’ point of Tangency BETWEEN the EFFICIENT FRONTIER and the INDIFFERENCE curves’
• 2 investors X and Y, confronted with DIFFERENT Indifference curves X1,X2and X3 AND Y1,Y2 and Y3 locate their OPTIMAL PORTFOLIOS at X* and Y*-the points of tangency- as in the diagrams
68• The Point of Tangency locates the OPTIMAL
PORTFOLIO as it is the meeting point ‘of what best the stock can offer (Efficient Portfolio- best returns at that level of risk) AND what the Investor best expects( Indifference curves)
• Optimal Portfolio with LENDING and Borrowing at Risk less rate
• Suppose Investors can LEND and BORROW at ‘risk less rate’. Per se this looks trivial but is NOT SO as, the risk less Asset has some SPECIAL CHARACTERISTICS. To get this we look at the equation for the Std. deviation of a Portfolio of 2 assets
B
C
A
RETURNS
RISK
D
.Q.U. . .
V
G
M
LENDING BORROWING
SHIFTING THE EFFICIENT FRONTIER.
Rf
69**• Std, deviation of a Portfolio, SDp• = ROOT (W1^2* SD 1^2) +( W2^2* SD2^ 2) + (2
W1W2*P12*SD1*SD2)• If one of the Assets, say Asset2 is a RISK FREE
ASSET, it means that SD2=0 • Therefore SDp= W1* SD1- What a
transformation!• If an Investor LENDS a portion of his funds at Rf
and invests the remaining in asset M a risky asset( M is the Optimal Portfolio), the Efficient Frontier which was BC all the while, gets shifted to any point he prefers along the line Rf M
70• For obvious reasons, a point U on RfM
DOMINATES the point B on the Efficient frontier BC
• Further if he BORROWS money at risk free rate and invests it in M ( he will be called an ‘Aggressive Investor”) he can , if he wishes, reach the point G, which is EVEN BEYOND M (beyond even the Optimal Portfolio)!!
• Since RfMG dominates BC, every investor would do well to choose some combination of Rf and M. A conservative investor may choose a point U, an Aggressive Investor may choose a point V. A conservative investor includes and weighs more of Rf in his Portfolio while an Aggressor weighs more of M in his Portfolio!
71
• At the point V discussed earlier, the Weight of Rf turns NEGATIVE as money is borrowed and invested into Asset M
• Wf+ Wm=1. Now by Borrowing,
• Wf+ Wm + Wb=1. Since Wb is another form of Wm, to keep the equation intact, Wf should be less than than 1! ( because we started off saying Wf + Wm =1
72
• The task of a Portfolio manager can be separated into—
• 1)Location of M, the Optimal Portfolio of risky assets• 2) Choice of a Combination of Rf and M depending
on one’s tolerance for risks• Ex. I have located Tisco and Reliance as the assets
constituting an Optimal Portfolio. Now for my client S’ how much should he Borrow at the treasury rate at 6% and invest even that amount into Tisco and Reliance( portfolio M)?
• This is the Separation Theorem propounded by James Tobin for which he was awarded a Nobel Prize
• RV= ___{ E(Rm)- Rf}______________• Std.deviation of Portfolio M• RV Reward Variability Ratio
73• The Concept of a Market Portfolio• In CAPM there are 2 types of Investment opportunities
which concern Investors• 1) A RISK FREE Security whose holdings over the
Holding Period is KNOWN WITH CERTAINTY-(Risk free Treasury Bonds of a Sovereign State , for example)
• 2)A MARKET PORTFOLIO of Common stocks• Frequently , the rate on Short term to Intermediate term
Treasury securities is used as a SURROGATE for a Risk Free Security. And a MARKET PORTFOLIO is a portfolio of ALL COMMON STOCKS and Weighted according to the Aggregate Market values outstanding. As a Market Portfolio is unwieldy to work with, people use a S&P500/ NIFTY/ SENSEX as a surrogate
• A Market Portfolio represents a Limit to ‘attainable diversification” as one CANNOT hold a Portfolio that is more diversified than a Market Portfolio. Thus ALL risks associated with a Market Portfolio are Systematic / Unavoidable risks
• THE CHARACTERISTIC LINE*******• We are now in a position to compare the Expected
Returns for an Individual Stock WITH the Expected returns for a Market Portfolio( S&P 500, proxy). In our comparison it is useful to deal with ‘returns in excess of risk- free rate’– a BENCH MARK against which the returns from the risky assets are contrasted. The Excess return is simply the EXPECTED RETURN LESS the RISK FREE RETURN. The Chart shows the comparison of the Expected excess for a specific stock with the returns
from the Market Portfolio
CHARACTERISTIC LIN
E
EXCESS RETURN
ON STOCK
EXCESS RETURN ON MARKET
75****• The DARK red line is the Characteristic line ; it
depicts the expected relationship between EXCESS RETURNS for a stock AND the Excess returns for a Market Portfolio.
• The Expected relationship may be based on a past experience in which case the Actual excess return for the stock and the Market Portfolio would be plotted on a graph and a REGRESSION LINE best characterizing the relationship is drawn. Each point represents ‘excess returns for a stock and the excess return of S&P 500, say in the past given month and in the last 60 months in total
• The Monthly returns are- Returns=• __ Dividends+_ ( Ending price-Opening Price)_____• Beginning price
76• The Narrower the spread, the GREATER the CORRELATION;
alternatively, the wider the spread, the GREATER the Unsystematic risks and LESSER the Correlation with the excess return on the Market Portfolio
• The concept of BETA-- an Index of Systematic risks Beta is simply the SLOPE of the Characteristic Line i.e. the change in the Excess returns on the Stock OVER the change in the Excess returns for the Market Portfolio __ Y2-Y1___
• X2- X1 If the slope is 1, it means that the Excess returns on the Stock varies PROPORTIONATELY with the excess returns on the Market Portfolio. In other words, the stock has the same Systematic risks as the Market as a whole! If the market goes up 5% on the whole, we could expect the Stock to go up around 5% as well. If the slope is>1, it would mean that the excess returns on the stock would be more than proportionate to the Excess returns on the Market portfolio –in other words the stock is riskier than the WH OLE MARKET !
76a
• The wider the relative distance of the points FROM the characteristic line, the greater the Unsystematic risks of the stock. This means that the Excess return from the stock shows increasingly LOWER correlation WITH the Excess return on the Market Portfolio.
• We know that Unsystematic risks can be reduced / eliminated through Efficient diversification. For example for a Portfolio of say 20 carefully selected stocks, the data points will hover close to the characteristic line
B>1
B=1
B<1
EXCESS RETURN
ON STOCK
EXCESS RETURN ON MARKET
77***• A stock with a Beta of >1, is an Aggressive stock while the one with
a Beta of <1 is a Defensive stock• The greater the slope of the Characteristic line for a stock, as
depicted by Beta>1.0, the GREATER THE SYSTEMATIC RISK• This means that for BOTH upward and downward movements
in the Market’s Excess returns, movements in the ‘excess returns in Individual stocks ‘ are greater or lesser, depending on its beta
• With the Beta of a Market Portfolio equal to 1 by definition, Beta thus becomes an INDEX of the Systematic or Unavoidable risks relative to that of the Market Portfolio.
• In addition a Portfolio’s Beta is simply a weighted average of the Individual’s stock betas with, the weights being assigned in proportion of total portfolio’s market value, represented by each stock. Thus the Beta of a stock is—
• A stock’s contribution to the RISK of a HIGHLY diversified portfolio of stocks
78• Required Rate of Return and the Security
Market line ( SML)• Unsystematic risks can be diversified away. The
major risk associated with a stock is therefore the Systematic risk. The greater the Beta of a stock, the greater the Systematic risks and greater the Required rate of return. The required rate of return is in case of Inefficient Portfolio--
• Rj=Rf+ { ( Rm- Rf) Bj}• Where Rj is the Reqd. rate of return• Rm-> Is Expected return for Market Portfolio• Rf -> Risk free return, Bj-> Beta coefficient for
the stock j.
79
• Put another way, the Required rate of Return for a stock is equal to the return required by the market for riskless Investment PLUS a Risk Premium
• The Risk Premium is a function of—• 1) The Expected market return LESS the Risk
free rate of return, which represents the risk premium required for that stock in the market AND is a function once again of its--
• 2) Its Beta coefficient
80• Suppose the Expected return on Treasury
security is 8%, the Expected return on the Market Portfolio is 13 % and the Beta of A Corp. is 1.30.
• The Beta indicates that A Corp. has MORE Systematic risks than a typical stock index.
• Rj= 0.08+ {(0.13-0.08) 1.30}= 14.5%• This means that the Market expects A Corp. to
return 14.50% as against the Expectation of 13% on the Market Portfolio. This is because Acorp. has a higher Systematic risks(30% more than on the Market security) and therefore Investors in it demand a higher rate of return for investing in it
SMLRISK
PREMIUM
RISK FREE RETURN
BETA
EX
PE
CT
ED
RE
TU
RN
1
Rf
SECURITY MARKET LINE.
81***• Security Market Line• The Equation Rj= Rf + {( Rm- Rf) B} describes
the relationship between an individual Security’s Expected returns and its Systematic risk as denoted by Beta. This linear relationship is known as Security Market Line and is illustrated in the figure. The expected one year return is shown on the Y axis. Beta, the Index of Systematic risks is on the Horizontal axis.At zero risk, the SML has an intercept equal to the Risk free rate of return as, investors expect to be compensated for the Time value of money. Of course as risk increases, the Required rate of return also increases
82• CAPM provides us the means to estimate
the ‘REQUIRED rate of return’ on a security. The RROR can then be used as a ‘Discounting rate’ in a Dividend Valuation model. Recall the Intrinsic value of a share can be expressed as the Present Value of a stream of ‘Expected future dividends’ Po= __-sum of Dt____
• (1+ Ke)
• Where ‘growth’ is seen P= Dt____
• Ke-g
83• If A ltd.declares a dividend of Rs 2/ share,
the expected annual growth in dividends per share is 10% and if the Required rate of return for A ltd. is 14.5%
• P= ___2.0___ = Rs. 44.44
• 0.145- 0.10
• If this value of Rs. 44.50 is the SAME as the Current Market Price, the ‘ Actual Return on the stock and the “required rate of return would be Equal, denoting a state of Equilibrium.
84• In the Previous Example, the Required
rate of returns---based on the Investors expectations regarding such factors as 1) returns on riskless assets 2) Company performance expected 3) The state of the economy as such etc--- matches the market price. If the match does not remain the state of equilibrium is lost and recognizable price changes are the result!
85• Suppose Inflation in the Economy has come down and a state of
relatively stable growth has set in. Interest rates have declined and the risk aversion of the Investors have decreased with a decrease in the growth in dividends( Dividends Rs 2/-)s
• Before After• Risk free rate 0.08 0.07• Expected market return Rm 0.13 0.11• Beta of A ltd. 1.30 1.20• Dividend growth A ltd 0.10 0.09• The RROR for A ltd Rj(or Ke) = 0.07+ { ( 0.11-0.07)* 1.20} =
11.80%• P= __D____= ___2.0_______= Rs 71.43 ( See #83)• Ke-g 0.118- 0.09• Thus, with an improvement in the economic and company specific
factors, there is an appreciation in the price of the company’s stock. If these expectations represented ‘market consensus’ the price of Rs 71.50 Rs would be the Equilibrium price
STOCK X (under priced)
STOCK Y (Over Priced)
RE
QU
IRE
D R
AT
E O
F R
ET
UR
N
BETA
Rf
BETA AND R R O R.
SML
86***• Underpriced and Overpriced stocks• At market equilibrium, the Required rate of return on a stock is equal
to its Expected rate of return ( We generally do not use Actual rate of return in place of Expected rate of return as, markets generally discount the future and at any point in time it is the future price that is to be looked at). What happens when the asset is not in Equilibrium?
• Let us say that for some reason stocks X and Y are ‘improperly priced’. Stock X is underpriced relative to SML and Stock Y is overpriced relative to SML
• A look at the Chart indicates that Y axis represents the Required rate of return. This is akin to the DISCOUNT the stock is trading at compared to its relevant fundamentals
• Stock X, some investors DEMAND, should provide a ‘rate of return’ GREATER than that required based on its relevant fundamentals and consequently a return GREATER than that required, based on its SYSTEMATIC RISKS
87• Some Investors seeing an OPPORTUNITY would go about buying
the asset. This would push the Market prices up, and the required rate of return down. The process would continue till the RROR hit the SML.
• Similarly in the case of Stock Y, investors holding the stock would sell it recognizing that they would obtain a ‘higher return for the SAME AMOUNT of Systematic risks( should they invest in other stocks with ‘ comparable systematic risks’) the selling pressure would drive the prices of the stocks down increasing the RROR in that process, till such RROR ( the Discount to the price based on fundamentals) meets with SML
• Prices adjust quickly to new Information. • The SML concept becomes a useful means of determining the
Expected RROR for an asset. The RROR can then be used as a DISCOUNTING rate while valuing the asset.
• P= ___D_____ P= ___Dt____ where RROR=Ke• (1+ Ke) Ke- g
88• THE CAPM after all!• Portfolio theory is a normative which PRESCRIBES how ‘utility
maximizing’ investors should behave; rationally though!• The CAPM was developed later to examine “ the
RELATIONSHIP between Risk and Returns in the Capital markets IF INVESTORS BEHAVED IN CONFORMITY WITH THE PRESCRIPTIONS OF THE PORTFOLIO THEORY”. CAPM is thus an extension of the Portfolio theory and basically concerns itself with 2 sets of Questions---
• 1) What is the appropriate measure of Risk for an EFFICIENT portfolio?
• 2) What is the relationship between Risk and Returns for an Efficient Portfolio?
• 3) What is the appropriate measure of risk for an ‘Inefficient Portfolio’?
• 4) What is the relationship between Risk and Returns for an Individual security or an Inefficient portfolio?
89• ASSUMPTIONS of CAPM• 1) Individuals are risk averse• 2) Individuals seek to maximize the ‘expected utility’ of their
Portfolios over a single period planning Horizon• 3) Individuals have HOMOGENOUS EXPECTATIONS. This means
that they have IDENTICAL subjective estimates of the means, variances and co variances among returns
• 4) Individuals can BORROW and LEND freely at Risk free rate of return
• 5) The markets are Perfect with no taxes, transaction costs etc.• 6) Securities are completely divisible and the markets are
competitive
• THE CAPITAL MARKET LINE• An important assumption of the CAPM is that Investors can
BORROW or LEND freely at RISK FREE rate of interest
90***• This means that investors would be interested ONLY in that
Portfolio of Risky investments at which the line from the point representing the risk free investment is TANGENTIAL to the FEASIBLE REGION of the risky portfolios This was explained in the graph adjunct to slide # 69. The shaded region represents the FEASIBLE region of Risky assets, Rf represents Risk free investments, and Point B represents the point at which the straight line from Rf is TANGENTIAL to the FEASIBLE region of Risky portfolios. By a suitable combination of Rf and M investors can reach any point along RfMG. Since RfBGs DOMINATES the EFFICIENT FRONTIER BC, ALL investors would hold a portfolio of Rf and M in some combination or the other
• The chart for slide #90 indicates what investors with DIFFERING RISK ATTITUDES WOULD DO! An investor with a risk-return preference denoted by INDIFFERENCE CURVES A1 to A4 would LEND a part of his funds at the risk free rate of interest and invest the REMAINING in Portfolio M since this LEADS to the position he desires at A*
91• On the other hand an Investor with with Indifference curves C1 to C4
would BORROW some funds at the RISK FREE rate of Interest and put HIS OWN FUNDS PLUS THE FUNDS HE HAS BORROWED INTO THE PORTFOLIO M to reach the position desired by him at C* An Investor confronted with Indifference curves B1 to B4 would put ALL his funds into Portfolio M so that he arrives at B*
• Summarizing,• At A*- An investor LENDS a PART OF HIS FUNDS at RISK
FREE rate and INVESTS the REMAINING in Portfolio M• At B*- An Investor invests ALL his money in M• At C*- An Investor BORROWS at risk free rate and puts BOTH
THE BORROWED FUNDS AS WELL AS HIS OWN FUNDS into Portfolio M
• The Straight line passing through Rf and M is the Capital Market line. It can be expressed as
• E(Ri)= Rf+ Lambda * Std.deviation• Where E(Ri) is the Expected returns on a portfolio held by investo
92-r.w 94a• RfRisk free rate of Interest• Lambda Slope of the Capital Market line• Lambda, the slope of the Capital Market line is obtained as Follows• =___E(Rm-Rf)_ This is essentially a RISK-REWARD ratio• Std.deviation• The slope of the Capital market line represents the Price risk in the
markets • SECURITY MARKET LINE• The Capital Market Line reflects the relationship between Risk and
Returns for an EFFICIENT PORTFOLIO but does NOT spell out the relationship between risk and return for an INEFFICIENT ASSETS—either Individual security OR (Inefficient )Portfolio
• The relationship between risk and return for inefficient portfolio or single security is expressed by the Security Market Line
• E(Rf)= Rf+ { (E ( Rm)-Rf)} _COV .i,m__ m Market PortfolioM• Variance m• COVi,m covariance of returns between M and Portfolio i
93• Inefficient asset can be—• A) a single asset or• B) A Dominated Portfolio• Capital allocation line- connects a Rf asset to ANY
Portfolio on the Efficient frontier• Capital market Line connects a Rf asset to
Portfolio M of Risky assets through the feasible region of Risky portfolios and with the connecting line being tangential to the feasible region of risky portfolios
• The Expected return and Standard deviation values associated with SINGLE security will normally lie BELOW the Capital Market Line as, undiversified holdings are usually ‘inefficient’- Therefore SML would be dominated by CML!
94***
• CAPM-SML• Rq= Risk free rate+ { Expected return on
Market portfolio- Risk free rate}* Security Beta
• = Risk free rate+ {Market risk premium} Beta of the Security
• Market risk premium is calculated as the difference between the Average return on Index and the Risk free rate
94a
• Get this clear
• CAPM- SML + CML
• ( Exp.retns vs } (Exp.rtns vs Std.
• Beta of Individual Sec) } {dev.of Portfolio}
• R= Rf+ B( Rm-Rf) Rp=Rf+Rm-Rf*SDp
• B=SDp/ SDm *Coeff SDm
• of Corrln.
95
96
97-BETA
98
99
• Diversification• When there are just 2 assets in a Portfolio, there are equal
number of Variance and covariance terms. As the number of securities increases, the number of variance and covariance terms increase much faster. In a portfolio of N securities, there are N variance terms but N^2-N covariance terms. If the securities in a portfolio have equal weights,
• Variance=_1_*Av. Variance+(1—1/N)Av.Variance• N • As N increases, the portfolio variance steadily approaches
Average variance. This is the limit to which Portfolio risk can be reduced through Naïve diversification
100
• Portfolio selection• Deals with portfolio selection based on Mean variance
Model developed by Harry Markowitz. The model procedure has 2 parts 1) Technical; Determination of a set of “ efficient portfolios from out of an ‘available feasible set’
• 2) personal- choosing the best risk-return opportunity CONSISTENT with the Investors attitude to risks
• The Technical part can be looked at in• A) One step Optimization• B) Two step optimization
101
• One step optimization• This approach BEGINS with the DELIMITATION
( identification) of Efficient Portfolio having one or more RISKY asset and culminates with the Capital market line. The capital Market line is a straight line that represents the EFFICIENT portfolio that can be formed by combining a RISKY asset with Risk free borrowing as also Lending opportunities
• In one step optimization one is not going from asset classes to individual Securities. Rather the move is straight to Individual securities – like ACC, Tisco etc
• Two or Three step optimization• Is also called ‘Top Down approach’, It is more
STRUCTURED and preferred by Institutional Investors
102
• The process here begins with
• I) Capital allocation Decision
• This involves APPORTIONMENT of the TOTAL INVESTIBLE FUNDS between 1)Risk free assets 2) OPTIMAL PORTFOLIO of Risky assets
• II) Asset Allocation decision
• This refers to the Construction of an Optimal Portfolio of risky assets referred to above(stage I) contd.
103
• The Risky Investments are distributed ACROSS ASSET CLASSES like shares, bonds, bullion, Real estate etc.
• III) The Final stage is the SECURITY SELECTION DECISION i.e selecting securities WITHIN each asset class
• To summarize, in a 3 step optimization,• Step 1) Capital Allocation Into a) Risk Free assets ( say Rs.
1crore) b) Optimal portfolio of risky assets ( Rs. 4 crores)• Step 2) Asset allocation ---Risky asset portfolio only ( Rs. 4 crores)
split into say –Stocks;Rs. 2 crores, Bullion Rs. 1 crore and Oil; Rs 1 crore
• Step 3) MRF Rs 50 lakhs, ACC RS 50 lakhs, Tisco Rs. 50 lakhs and Reliance Rs. 50 lakhs
• The 3 step optimization is called “ top down optimization” as the focus of the ‘top management’ is always on INDEPENDENT OPTIMIZATION of Risky portfolios i.e. independent optimization of both Asset class And Security portfolios WITHIN EACH asset class The Investment Manager looks to adjust Portfolio weights to take advantage of forecasted changes
104******• The ONE step optimization is explained below• Locating an Efficient portfolio• The first step to the Technical aspect of Optimal Portfolio selection
is to determine ‘risk-return’ opportunities available to an investor.this is also referred to as the ‘determination of a FEASIBLE SETOF PORTFOLIOS or ‘determination of the Portfolio OPPORTUNITY SET’ or the ‘MINIMUM VARIANCE PORTFOLIO OPPORTUNITY SET’
• Graphically, these are summarized by the MINIMUM VARIANCE FRONTIER OF RISKY ASSETS’.
• Each point along the MINIMUM VARIANCE FRONTIER of risky assets represents the lowest possible variance (not necessarily highest returns) that can be obtained for a GIVEN return of a GIVEN portfolio. The point to the EXTREME LEFT on the Minimum variance Frontier –Point C– represents the GLOBAL MINIMUM VARIANCE portfolio. Similarly the HIGHEST point represents the GLOBAL MAXIMUM RETURN portfolio F
105
• The line segment FC represents the ‘Efficient frontier’ of risky assets. It is so called as, it is a ‘Dominating Portfolio”; it dominates other portfolios in the sense that it ---
• a) Offers Maximum Returns for a given level of Risks or
• b) Offers the Lowest risks for a GIVEN Return
• The Efficient Frontier is CONVEX as, ALL Assets have a Correlation between +1 and -1
106
• Delineation of Efficient frontier through Markowitz Diversification rests on 4 assumptions—
• 1) The Rate of return is the most important outcome.
• 2) Investors are Averse to Risks. They seek HIGHEST returns for a GIVEN level of Risks
• 3) Investors estimate risks as a ‘variability of Expected returns’
106
• d) Investors base their decisions solely on Risks( variance) and Returns
• Investors who conform to these rules are known as “Markowitz Diversifiers” who prefer an Efficient Portfolio to any other!
• To illustrate the Concept of “dominance” and “Efficient Frontier” let us take a simple example with 2 Assets, X (return 10% and Std. dev.15%) and Y (return returns 20% and Std. dev. 26%). Low positive correlation between their returns- contd.
107
• permits ‘diversification gains’.
• A large number of Asset Portfolios can be formed by blending these assets in DIFFERENT PROPORTIONS. The table in Slide # 108 presents such portfolios with their respective risks and returns. A graph of this was already discussed( slide# 104)
• The Line AF depicts the Minimum variance Frontier. Points A&F represents PURE HOLDINGS of X and Y. The Inflexion point C represents the ‘Global Minimum Variance Portfolio”
108
• CF is the Efficient Frontier. Portfolios A and B are Dominated portfolios and hence Inefficient. Put up a charts
• Port E(r) Std. dev. Dominated? Eff?
• A 10 15 Yes-B,C NO
• B 12 13 Yes-C No
• C 13 12 No Yes
• D 15 16 No Yes
• E 18 22 NO Yes
• F 20 26 No Yes
109*****
• Efficient Frontier with ‘Margined short sales”
• A ‘Short Sale’ comes into being when a person sells to another an asset BORROWED from yet another! Margin is a deposit of a small percentage of Market price. Edward Dyl has pointed out that when Margined Short sales are possible, one can construct a Portfolio that the SAME expected return for ‘ an even lower variance’ Thus the Efficient frontier—contd.
110*******
with Margined short sales DOMINATES the Efficient frontier without it!
Efficient Frontier with ONE Risk free asset
A Risk free Security is one that has ‘Zero variance” or ‘Zero Std. deviation”. James Tobin has pointed out that---;
a) A portfolio made up of ‘Risky assets” and ONE Risk FREE asset generates Investment opportunities( Portfolio Opportunity set) with LINEAR relationship
111****
• between Expected returns and risks• b) ONE SUCH (point M- tangential) Portfolio at
M dominates the Portfolio formed by mixing ONLY RISKY ASSETS
• If C is the ‘ Complete Portfolio” of a combination of a Risk free asset AND Risky assets, Rf represents Risk free rate, and M represents the Risky portfolio, then the slope of the Capital allocation Line that connects the Risk free rate to all portfolios-
• Slope= __E(Rc)-Rf_// SD m• •
112
• The Slope of the CAL is essentially REWARD to RISK ratio
• Looking at the charts for CAL, we find that there are 3 CALS –all originating at F (Rf) and connecting portfolios A,M and Z. The Line FM which is at a tangent to the Minimum variance Frontier at M, is the BEST Capital Allocation Line. In other words, combination of Portfolio M(of risky assets) with Risk free asset F on the point M representing a certain combo of M and F is the BEST combo
113• Point M represents the PURE portfolio of
100% Risky assets with returns E(Rm) and SDm. This is because M is the Tip of the Efficient Capital Asset line FM. The investor can invest along the line FM at any point and with any combination of Risk and return that suits him! Portfolios represented by the Line Segment FM are known as LENDING PORTFOLIOS ( because you are investing at ‘risk free
114
• only with OWN funds. With Own Funds, the Efficient frontier of a Portfolio would end at M. FM, if extended beyond M, opens up ‘further opportunities” for HIGHER RETURNS.
• These are REAL opportunities and an Investor can exploit them by’borrowing funds at Risk- free rate Rf and investing them in ‘Risky Asset M”! This is known as creating a Leveraged/ margined/ Borrowing Portfolio.
115• With borrowing, the weight of the Risky
asset M in the Portfolio exceeds 1. Negative weight for the Risk free asset ensures that the ‘total weight” is 1
• For example, an investor has Rs. 2,00,000/. He borrows an additional sum of Rs. 1,00,000/ and invests in M. The weight of the Risky asset M in the Portfolio is Rs. 3,00,000/ 2,00,000=1.50. the weight of the Risk Free asset in the Portfolio is therefore, -0.50 which means borrowings are 50%of Owned Funds
116*****
• It may be noted that the ‘steepest CAL’ ( i.e. is at TANGENT at M) with BORROWING/LENDING dominates ALL OTHER PORTFOLIOS! These Lending or Borrowing Portfolios dominate the ‘Efficient frontier of risky assets”
• This CAL is now the ‘NEW Efficient frontier” of Risky assets with one Risk Free Asset”. This forms the Optimal Portfolio for all Investors irrespective of the Investor’s risk preferences!
117
• This is because Risks are taken care of by the presence of an Investment in a risk free asset, the returns are ensured by Investment in High yielding Risky assets, the spread is created by borrowing at risk free rate and investing those funds on Risky assets etc. This essentially is the Capital market line in CAPM
• Also the Capital Market Line reflects the line from the risk free rate joining the PEAK of the Efficient Frontier. The Portfolio contains a risk free asset along
118****
• The best portfolio the Efficient Frontier contains( the risk free asset line touches the Efficient Frontier exactly at the point it turns down for worse!) Add lending/ borrowing Portfolio to this and what better can an Investor expect beyond this?
• The CAPM goes about working out the ‘expected rate of return’ or RROR as—
• E(Rt)= Rf+ {Rm- Rf} B where Rf is the Risk free rate of return, {Rm-Rf} is the Market risk Premium and B Beta
119****• The Chart explains in words what the Equation depicts. The equation
says that the Expected return on an asset varies DIRECTLY with 1) The SYSTEMATIC RISKS –represented by Beta
• 2)the Market Premium of the Market Portfolio. The Risk Premium for an asset or a Portfolio is a function of its Beta (B). The Risk premium of a Market Portfolio also referred as REWARD depends on the LEVEL OF RISK FREE RETURN AND the return on the Market Portfolio
• In short 3 information are required to work out the RROR according to the CAPM
• 1) Risk free Rate
• 2) Risk Premium on Market Portfolio
• 3) beta
• The risk free Rate- is the rate of return available on assets like T-Bills, Money Market funds etc. are taken as a Proxy for Risk Free rate. They carry NO( or very low) DEFAULT RISKS and negligible interest rate risk. During Inflationary conditions the REAL interest rates could fall to zero or go negative however!
120
• Risk Premium on the Market Portfolio
• Is the DIFFERENCE between • A) the Expected returns on the Market Portfolio AND• B) the Risk Free Rate Of Return• The CAPM holds that, IN EQUILIBRIUM, the MARKET PORTFOLIO
is the unanimously desirable risky portfolio. The Market Portfolio consists of ALL securities held in EXACT PROPORTION to their MARKET VALUES( market capitalization) It is an EFFICIENT PORTFOLIO but entails NO LENDING OR BORROWING however( this is the kind of Market premium that CAPM allows. Lending and Borrowing could create huge EXPECTED RETURNS jacking up and unduly so, the Market premium and the Expected returns in that process!). The Risk Premium on the Market Portfolio is -- 1) PROPORTIONAL to its risk ( i.e. Variance) 2) the degree of Risk Aversion of the investors.
• BETA: It measures the RISK ( Volatility) of an INDIVIDUAL ASSET RELATIVE to the Market portfolio ( or a proxy for it like Nifty!)
121• Statistically, Beta is the Covariance of the Asset’s returns with
respect to the returns of the Market Portfolio DIVIDED by the Variance of the Market Portfolio
• B= ___Covar j,m_____ It may be recalled that the Covariance
• Variance m of 2 assets is the product of the
Coeff. Of Correlation of 2 assets AND their respective Standard deviation. Covar j, m= Coeff. of corrln. j, m * SDj SDm
Nifty/ Sensex are often taken as a proxy for Market Portfolio. The deficiencies in such a move are---1) A Market Portfolio comprises of ALL ASSETS held in the form that is ‘proportionate to the Quantity supplied” While Nifty/ Sensex comprise of just 50/30 of the several thousands of Stocks Listed. Not all Issues floated are LISTED and N/S are concerned only with scrips that are ‘listed’. Besides even among Listed securities, there is this concept of a ‘free float’ which is not 100% for all listed securities at ALL POINTS IN TIME!
The Covariance of the Market Portfolio with itself is the VARIANCE of the Portfolio. This is because the Coeff. Of Corrln. of the Market
122• ---Portfolio with itself is the VARIANCE of the Portfolio. This is
because Covar m,m=Coeff. Of Corrln. m,m *SDm SDm. Since Coeff. Of Corrln. Of a Market Portfolio with itself is 1, the Covar. Is 1* Sdm SDm= SDm^2= Variance m
• As a consequence BETA which is___ Cov. M,m___=__Var. m_=1• Var. m Var.m• Beta of a Market Portfolio with itself is ONE• This classifies ALL PORTFOLIOS into 2 categories----• 1) Assets with B<1 called DEFENSIVE ASSETS• 2) Assets with B>1 called AGGRESSIVE ASSETS ( see charts for
slides 118/119)• Risk Free assets have a BETA equal to ZERO• It may be noted that the BETA of a Portfolio is the weighted
average of the Betas of the assets included in the Portfolio. The weights are the Relative share of the assets in the Portfolio
• Ex 2 Assets with Beta values 0.8 and 1.20 have been combined in the proportion of 3:1. The Betas of the Portfolios will be ---
123
• 3/4 * 0.8 + 1/4 * 1.20 = 0.90 If the SD of the Market Portfolio is 30%, the SD of the COMBINED PORTFOLIO is 0.9 * 30%=27% This shows that the Portfolio Risk i.e. Portfolio SD, is driven by the BETA of each Security
• ------------------------------------------------------------------------------------------
BOND CONCEPTS• A BOND is a Debt Instrument requiring the Issuer ( also called the
Debtor or the Borrower) to repay the LENDER or the Investor---• a) the amount borrowed b) Interest over a specified period of time• The Date on which the Bond is to be repaid is called ‘Maturity Date’• Assuming that the Issuer does not default or redeem the Issue
PRIOR to the Maturity Date, an Investor holding this Bond till the Maturity date is assured of a known cash flow pattern
• ISSUERS of Bonds• 1 The Federal Govt. & its Agencies• 2 Municipal Govts.• 3 Corporate Sector• Term to Maturity of a Bond• Is the number of years over which, the Issuer has promised to meet
his ‘obligations towards the Bond’• Maturity of the bond ; Is the date on which the Bond will cease to
exist i.e. the Borrower will redeem the debt
2• 1-5 years maturity--- Short term Bonds• 5-12 years-- Medium term Bonds• > 12 years- Long term bonds• Importance of the concept ‘Bond Maturity’• 1) Indicates the number of years over which the Bond Holder can
expect to receive his payments of COUPONS and the number of years over which the Bond Principal will be ‘repaid’
• 2) The Yield on a Bond depends on its ‘Term to maturity’• 3) the Price of a Bond will fluctuate over its life, as yields in the
Market change!• Greater the TERM TO MATURITY with other factors being
constant, the GREATER the VOLATILITY (in the Price) resulting from a change in Market Yields
• The PRINCIPAL is the amount that the Issuer agrees to repay the Bond holder on the Maturity Date. This is also referred to as Redemption value/Maturity value/Par/Face Value of the Bond
3• The Interest paid to the Holders of the Bond is called COUPON• In USA and Japan coupon is Semiannual. Zero Coupon Bonds carry
‘no coupon’ They are issued at a ‘huge discount’ to the FV and are redeemed at FV
• Floating Rate Bonds• Here coupon rates are RESET in tune with a pre-determined
BENCH MARK generally linked to a FINANCIAL INDEX• In some cases the coupon rate INCREASES with a FALL in the
INDEX and FALLS with an INCREASE in INDEX. Rates designed on these parameters are called ‘Inverse Floaters’. Institutional Investors hold them as ‘HEDGE VEHICLES”
• Junk Bonds• Are HIGH YIELD BONDS issued generally by Corporates, very
often not enjoying a high financial standing • A Leveraged Buy out(LBO) or a Re-capitalization financed by
‘high yield bonds’ with consequent heavy interest payment burdens, can place severe cash flow constraints on the Issuer
4• To reduce this burden, firms involved in Leveraged Buy Outs and
re-capitalization, issue ‘deferred coupon bonds’ that let the Issuer AVOID making CASH payments for some ‘specified period of time’ ( Ballooning Interest Payments) thereby reducing the cash strain on the acquirer.
• There are 3 types of ‘deferred coupon structures”---• 1) Deferred Interest Bonds• 2)Step-up bonds• 3) Payment-in –kind Bonds• In STEP-UP Bonds, the coupon the coupon rate for the first few
years is low; it increases thereafter, giving the required return on an average
• In Payment –in-kind bonds, the coupon is satisfied by issue of Baby bonds
• There is another High Yield bond structure which helps the Issuer RESET the coupon rate so that the bond trades at a predetermined price
5• In addition to indicating the coupon payments that an Investor
should expect to receive over the term of the bond, the COUPON RATE also indicates the degree to which a BOND’S PRICE is expected to be affected by changes in INTEREST RATES
• As will be illustrated later, with ALL OTHER FACTORS REMAINING CONSTANT, the HIGHER the coupon rate, the LESSER the PRICE CHANGE, in response to a change in INTEREST RATES. Consequently, the coupon rate and the Term to Maturity have OPPOSITE EFFECTS on the ‘PRICE VOLATILITY” of a bond, when interest rates change.
• AMORTIZATION FEATURE• The repayment of the PRINCIPAL of a bond can be either a) By
way of the Principal being repaid IN FULL at MATURITY b) The Principal being repaid over the life of the bond( staggered repayments over the life of the bond)—called AMORTIZATION . Investors do not talk of BOND MATURITY while referring to Amortizing securities; they’d rather compute a ‘weighted average life’ while dealing with Bond maturity
7• EMBEDDED OPTIONS• It is common for a Bond Issue to include a PROVISION in the
INDENTURE ( AGREEMENT)-- a clause that gives the BOND HOLDER or the ISSUER-- an OPTION to initiate the stated action mentioned in the bond which could affect the interests of the counter party The most common feature is the call feature . The provision grants the ISSUER the RIGHT ( optionally exercisable) to RETIRE the debt, fully or partially, BEFORE THE SCHEDULED MATURITY DATE
• Inclusion of a CALL FEATURE benefits ‘bond Issuers’ by allowing them to replace the ‘old bond’ by a new bond at a ‘lower rate of interest’ This call feature could be ‘detrimental’ to the interests of the Bond holder
• The right to call an Obligation is included in in MOST LOANS and therefore in all securities created ‘from out of all such Loans. This is because the Borrower typically should enjoy the right to repay/payoff the loan anytime he pleases. The borrower has the right to alter the amortization schedule if he is so pleased !
8• An Issue may also include a provision that allows a BOND HOLDER
to effect a change in the MATURITY of the bond. An Issue with a PUT provision included in the Indenture , grants the BOND HOLDER the RIGHT to sell the Issue BACK TO THE ISSUER at PAR VALUE on the DESIGNATED DATES Here the advantage to the Investor is that if Interest rates rise after the Issue date, thereby reducing a BOND’S PRICE , the investor can force the Issuer to redeem the bond at PAR VALUE He can then invest this ‘PUT’ amount in new bonds at HIGHER rates of interest!
• A Convertible bond is an Issue giving the Bond Holder the RIGHT to EXCHANGE the bond for a specified number of Common Stock. Such a feature helps the Bond Holder to take advantage of favorable movement in stock prices
• Some Issues allow either the Issuer or, the Bond Holder, the RIGHT to SELECT THE CURRENCY in which the cash flows will be paid.
• Multi feature Embedded options are ‘difficult to value’
9• Risks associated with investing in Bonds• 1) Interest rate Risk 2) Reinvestment risk 3) Call risk 4) Default risk
5) Inflation Risk 6) Exchange rate risk 7)Liquidity risks 8)Volatility risks 9) RISK RISK
• Interest rate risk The Price of a typical bond will vary inversely with change in interest rates. If an investor has to sell a bond PRIOR to the MATURITY DATE, an INCREASE in INTEREST rates at that time, will mean REALIZATION OF CAPITAL LOSSES ( selling the bond BELOW the Purchase Price). This risk is referred to as Interest Rate Risk/ Market risk. This risk is by far the BIGGEST risk faced by the Investors in bonds
• The actual degree of sensitivity of a Bond’s Price to CHANGES in Market interest rates depends on various characteristics of an issue such as Coupon rates, maturity, the options embedded in the issue
• Reinvestment Risks Calculation of the yield of a bond assumes that all cash flows received are ‘reinvested’ at the YTM / IRRi.e. interest received from A is reinvested in B The additional income from such re-investment called ‘interest on Interest’ would depend on the Interest rate prevailing at the time of REINVESTMENT
10• Variability in the rates of interest at the time of Reinvestment
( reinvestment rate variability) is called REINVESTMENT RISK. This risk is that Interest rate at which the cash Inflows received interim, are re-invested, could FALL
• Reinvestment risk is GREATER for• 1) LONGER HOLDING PERIODS and• 2) HIGH COUPON BONDS with LARGE, EARLY cash flows• It is to be noted that Interest rate risk and Re-investment risk
have ‘OFF SETTING EFFECTS’. The reason; Interest rate risk is that risk that interest rates will INCREASE ( thereby, reducing the Bond Price) and ‘Reinvestment Risk’ is the risk that interest rates WILL FALL!
• A strategy based on ‘offsetting effects’ is called IMMUNIZATION.• CALL RISK• Many Bonds include a provision that allows the Issuer to CALL, all
or any part of the bond BEFORE THE Maturity date. The Issuer often as a part of Indenture, appropriates this right for himself so that, he can refinance the bond in future if market rates fall below Coupon rates
11• From the Investor’s perspective, there are 3 disadvantages to a
call provision i.e Bonds with embedded optiond.• 1) The cash flow pattern of a callable bond is NOT known with
certainty• 2) Since it is highly likely that the ISSUER will CALL the bond
when Interest rates DROP, the investor gets exposed to REINVESTMENT RISKS
• 3) CAPITAL APPRECIATION of the bond stands reduced as the price of the callable bond, will only VERY VERY rarely RISE above the price at which the Issuer is entitled to call!
• Even though, the investor is compensated usually for taking a ‘call risk’ by means of a LOWER PRICE or a HIGHER YIELD, it is NOT easy to determine if this compensation is sufficient! In any case, the returns from a callable bond is very different from an ‘otherwise similar non callable bond’
• Call risk is so pervasive in Bond Portfolio Management that many market participants consider it SECOND ONLY TO INTEREST RATE RISK in importance
12• DEFAULT RISK• Also referred to as Credit risk, it refers to the probability of default by the
Issuer of the Bond ( inability of the Issuer to make Timely coupon payments as also the return of Principal). Default risks is gauged normally by the ‘credit standing ‘ assigned to the issuer by ‘credit rating agencies’
• Bonds carrying ‘credit risks” trade in the market at prices LOWER than comparable government securities, no matter that they offer higher rates of interest
• Except in the case of JUNK BONDS, an investor is normally more concerned with changes in PERCEIVED DEFAULT RISKS than with ACTUAL DEFAULT! Even though the actual default of the Issuing Corporation may be ‘highly unlikely’, they reason that the impact of a change in the ‘perceived default risk’ or, the ‘spread’ demanded by the market, for a given level of default risk, can have an IMMEDIATE IMPACT on the price of the Bond
• INFLATION RISK• Also called ‘Purchasing power risk’ arises because of the Variation in the
value of the Cash flows measured in terms of ‘what money can buy’
13• The Real rate of return could reduce drastically and in extreme
circumstances even turn ‘negative’ under ‘galloping Inflation’ To counter this risk—to the extent possible– Floating rate Bonds have been structured
• FRB’s have a lower rate of Inflation risk; especially Inverse Floaters• EXCHANGE/ CURRENCY RATE RISK• An investor who invests across the border attracts this risk• LIQUIDITY RISK• Also known as ‘marketability risk’ it represents the EASE with which an
Issue can be SOLD at or near its value. The Primary measure of Liquidity is the “size of the spread” between the ‘Bid price” and the ‘Ask price” quoted by the dealer. The WIDER the ‘dealer spread’, the HIGHER the Liquidity risk! For a person who is clear about holding the security till Maturity, the Liquidity risk is Very low!
• VOLATILITY RISK• Typically, the VALUE of an option RISES when ‘expected interest volatility’
INCREASES. -------- CONTINUED-----
14• In the case of a CALLABLE BOND or in the case of a MORTGAGE
BACKED SECURITY, in which the investor has granted the ISSUER an option, the price of the Security FALLS as, the Investor has given away a very valuable option. The risk that a CHANGE in VOLATILITY will affect the PRICE of a Bond is called ‘Volatility risk’
• RISK, RISK• Is NOT KNOWING the RISK OF A SECURITY; this could happen when
the Bond gets saddled with several embedded features not all of which subject themselves to to financial evaluation! While the future is not very predictable, there is no reason why the outcome of an Investment strategy cannot be known in advance
• There are 2 ways to MITIGATE these risks• 1)Extensive use of research models to evaluate these securities• 2) Generally trying to AVOID investing in Securities that CANNOT be
understood!
15• Pricing of Bonds• Future Value Pn= Po (1+r)^n when interest is
paid once a year• Interest paid more than once a year• r = ___Annual interest rate_________• no. of times interest is paid per year• If interest is 9.20 Rs/ annum, period 6years and
if payment are semi-annual• r =0.092 /2 =0.046 n=2*6=12• If Rs 10m is invested, • P12= 100,00,000 (1+0.046)^12 =$171,54,600
•
16• Future Value of an Ordinary Annuity• F.V.= A{ (1+r)^n-1}• r • Suppose a Portfolio Manager purchases $ 20 million par value 15 yr. 10% bond. Interest is paid
yearly. How much will the Portfolio Manager have if a) If the bond is held to maturity? B) annual payments are reinvested at 8%p.a?
• The Portfolio manager will have—• A) $ 20 million when the Bond matures• B)15 annual interest payments of $ 2 million• C) The RE-investment returns @8%
• P15=20,00,000{(1.08)^15- 1 = $ 543,04,250/- This figure answers• 0.08 all the 3 questions ABC above• Because $ 300,00,000/- represents the total future $ amount of Interest earned(15*
200,00,000), earned by the recipient, the balance of $ 243,04,250/- that is ( $ 543,04,250 – 300,00,000) is the interest earned by reinvesting these annual payments. Total future dollars
• 1 Interest payments- 300,00,000• 2 Interest on reinvestment--- 243,04,250• 543,04,250• 3 Par value on maturity 200,00,000• TOTAL------------------------- 743,04,250
•
17• If we rework assuming that the Bond is
being paid every 6 months ( based on Annual rate) with immediate reinvestment @ 8%p.a.
• Interest 6 monthly 200,00,000*6/12*10/100 = $ 10,00,000=A
• R=0.08/2 = 0.04, n= 15*2=30 periods• P30= 10,00,000{( 1.04)^30-1}=560,85,000• 0.04• Maturity value-> $ 200,00,000 Interest-> $
300,00,000 Therefore, interest on re-investment of interest is $ 260,85,000/-
18
• Present Value of any inflow/outflow=• Po= Pn{1/ (1+r)^n}• “If the Present Value of a $ 50,00,000/ bond
@10% over 7years is $ 25,65,791/-” means-this sum of $ 25,65,791/ - at 10% p.a. over a period of 7 years will grow to $ 50,00,000/ If this Financial instrument is trading for MORE THAN $ 25,65,791/-now, a person investing in it now will get less than 10% when it matures 7 years from now
19
• There are 2 properties of a present value-• A) For a GIVEN FUTURE VALUE, at a specified time in
the future, the HIGHER the interest rate or the Discount rate, the LOWER the Present Value AND
• The HIGHER THE INTEREST RATE, on any sum to be invested today, the LESSER the Investment to realize a specified future sum
• B) For a given Interest rate, (discount rate) the FURTHER into the Future the future value is received, the GREATER the period for the interest to accumulate and GREATER the sum. To accumulate a certain sum, greater the interest rate and greater the period it t is allowed to accumulate, the LESSER the Principal required to accumulate it!
20
• Present Value of an Ordinary Annuity• P.V.a= A{ (1+r)^n-1}• {r(1+r)^n }• The terms in the brackets denote the Value of an
Ordinary Annuity of $1 for n periods• Suppose the Investor expects to receive $100 at
the end of each year for the next 8 years, and if the cost of capital is 8%, the PV of the Annuity-
• PVa= 100{ __(1.09)^8-1)} = 553| 48 $• { .09(1.09)^8 }
21
• P.V. annuity- payments occur >1/year
• 1) Annual interest rate is to be divided by 2 if semiannual; by 4 if Quarterly etc
• 2)The annual periods are to be adjusted by multiplying by 2 for semi-annual payments and by 4 for Quarterly payments
• --------------------------------------------------
• BOND PRICING
22-BOND PRICING
• The Price of any Financial Instrument is equal to the Present Value of the Expected cash flows from the instrument. Therefore determining the prices requires
• a) An ESTIMATE of the EXPECTED CASH FLOWS
• b) An ESTIMATE of the APPROPRIATE REQUIRED YIELD
• The “required yield” reflects the yield for financial instruments with COMPARABLE RISK or, ALTERNATE/ IDENTICAL/ SUBSTITUTE Instruments
23• The Cash flows of a non-callable bond would be—• 1) Periodic coupon interest payments to the Maturity
date• 2) the Maturity Value• Our assumptions in calculating cash flows• a) Coupon payments are semi-annual• b) The coupon interest is FIXED for the term of a Bond• For a 20 year bond with a 10% coupon rate and a par
value $1000/- has the foll. Cash flows• Annual coupon interest= $1000* 0.10= $100/-• Semi-annual interest= $100/2= $50 • Or, r/2=5% 5%*1000$=50$• There are 40 semiannual cash flows of $50 and a final
1000$ flow ---contd--
24
• The ‘Required yields “ is an ‘opportunity concept” and is determined by checking the yields offered on ‘Comparable bonds in the Market”. By ‘comparable” we mean ‘Non- callable” bonds of the same CREDIT QUALITY, same Maturity etc. The Required Yield is expressed typically as an “Annual Interest rate’ When the cash flows occur semi-annually, the market convention is to use ONE-Half the annual Interest Rate as the periodic interest rate with which to DISCOUNT the Cash Flows
25
• Price P= C + C2 + Cn +M• (1+r) (1+r)^2 (1+r)^n (1+r)^n• Where n no.of periods, C=peiodical
coupon payments; M maturity value• Consider a 20year bond 10% coupon
bond of par value $1000/ If the Required yield is 11% and the coupon are paid semi-annually, the Price of the Bond can be calculated thus---
26
• The semi-annual coupon payments are equivalent to an Ordinary annuity, the PV is = C{( 1+r)^n-1}
• (1+r)^n*r• There are 1) 40 semi-annual coupon payments of $50
each 2) $1,000? Is to be received 40 six months from now
• 50{ (__1.055)^40-1}__ = $ 802.31• (1.055)^40 (0.055)• PV of Maturity value= 1000/ (1.055)^40= $117.46• PRICE of the Bond=802.31+ 117.46= 919.77• Suppose instead of 11% yield,(5.50% for 6 months, the
Required yield is 6.80%• CONTD.
27
• Price of the bond• 50{(1.034)^40-1} = 50(21.69029)• {.034) (1.034)}• = 1084|50• The PV of the Maturity amount=
1000/(1.034)^40 = 262.53• PRICE= 1084|50 + 262|53= 1347|04• If the Required yield were EQUAL to the
coupon rate of 10%, the price of the Bond would be equal to its Maturity Value of Rs. 1,000/- ( Verify this)
28
• Pricing Zero coupon Bonds• Some bonds DO NOT make any “periodic
coupon payments”. Instead, the Investor realizes the interest as the ‘difference between the Maturity value and Purchase price” These bonds are called “Zero coupon bonds’
• The Price of a zero Coupon bond=• Po= M/ ( 1+r)^n ! ( because all C’s are zero)• Note In PV calculations, it is the number of
half- years and NOT the no. of years that are used!
29• The Price of a $1,000/- zero coupon bond which
matures 15 years from now with a required yield • r=0.094/2= 0.047 n=30• P= $1000 / (1.047)^30= $252|12• -----------------------------------------------• Price- yield relationship• The fundamental property of a bond is that “its
price moves in the opposite direction to its required yield” REASON; the Price of a Bond is the PV of its cash Flows. A higher required return leads to a Lower ‘discounted value” and vice-versa!
• The Price-yield relationship is a Convex curve
30
• Coupon rate, Required yield, Price relationship• As Yields in the Market place change, the only variable
that can change to ‘compensate’ for the NEW required rate of return is the PRICE of the bond
• When COUPON rate is EQUAL to the REQUIRED RATE , the Price of the bond will be equal to its PAR VALUE
• When YIELDS in the Market place RISE above the Coupon rate, at a given point in time, the PRICE of the Bond ADJUSTS itself so that investors contemplating the purchase of the bond can realize some ADDITIONAL INTEREST! If it did not, the Investor would NOT buy this Bond! A lack of Demand would push the prices down- making this bond attractive too!
31
• The Capital appreciation realized by “holding the bond to Maturity” represents a form of ‘interest’ to an investor that compensates for the coupon rate being LOWER than the REQUIRED YIELD
• When a Bond SELLS BELOW its Par Value, it is said to be selling at a ‘discount’ If the REQUIRED YIELD is < coupon rate, the Bond sells at a PREMIUM. If the required yield is > coupon rate the Bond sells at a discount
• Relationship between Bond price and TIME if interest rates are ‘unchanged’
• If the required yield DOES NOT change between• a) The TIME the Bond is PURCHASED AND• b) The TIME it MATURES• the coupon rate will be the ‘required yield’
32• As the Bond moves closer to the Maturity date, the bond attempts
the ‘par value’ and remains at, or, close to it• Broad reasons for ‘change in the price of the bond’• a) There is a change in the ‘required rate of yield’ owing to
changes in the Credit Quality of the Borrower• b) The Bond is moving close to its Maturity date• c) Bonds of ‘comparable yields’ have undergone a ‘price
change’—prices here follow!• -----------------------------------------------------------------------------------• Caveats in pricing a bond—• 1) the next coupon payment is just 6 months away• 2) the Cash flows are known• 3) The appropriate required yield can be determined• 4) One rate is used to discount ALL cash flows
33• A) When the NEXT payment is due in <6 months• P= sum of _C___________ + __M__________________• ( 1+r) ^v (1+r)^ n-1 ( 1+r)^v (1+r) ^ n-1• Where v = days between settlement period and Next coupon• days in a six month’s period• Where v=1, the equation reduces to the form that we already know• B) When the Cash flows MAY NOT BE KNOWN• For non-callable bonds, assuming that the Issuer DOES NOT
DEFAULT, the cash flows are KNOWN. For other bonds where there is a likelihood of the Issuer Calling the Bond, the Cash flows may not be known with certainty
• With CALLABLE BONDS, the Cash flows will, in fact, depend on the level of Current interest rates RELATIVE TO the Coupon rates ( Because the Issuer would call up only if the Current interest rates are ‘lower’ than the coupon rate! ). Thus for Callable BONDS, the Current Interest rates are CRUCIAL
34• Pricing Floating rate and Inverse Floating Rate Notes• The Cash flow is not known for either a floating rate or an Inverse
Floating rate Security; it depends on the REFERENCE RATE in the future
• Price of a FLOATER• The Price of a Floater hinges on 2 factors• 1) The SPREAD over the REFERENCE RATE• 2) Any restrictions that have been imposed on the re-setting of
the Coupon rate!• The Coupon rate of a Floater is equal to—• a) a Reference rate PLUS• b) Some spread / Margin• For ex. A Floater can be set at • a) The rate on a 3 month Treasury Bill ( Reference rate) plus • b) 50 BASIS POINTS
35• For example; A Floater may have a MAXIMUM Coupon rate called a
CAP; or a Minimum called a FLOOR• The Price of a Floater trades CLOSE to its PAR VALUE so long as • 1) the SPREAD that the market requires remains unchanged• 2) Neither the CAP nor the FLOOR is breached• If the market requires a BIGGER SPREAD--the FRN will trade
ABOVE PAR• If the market is happy with a SMALLER SPREAD-- the FRN will
trade BELOW PAR• IF the Coupon rate is, by a restriction such as a CAP, barred
from changing to ‘reference plus spread’---- the FRN recognizing the restriction, will trade BELOW PAR
• ----------------------------------------------------------------------------------------• Price of an Inverse Floater• In general, an INVERSE FLOATER is created from a ‘Fixed rate
security” The Security from which it is created is called a “COLLATERAL”
36• From the Collateral, 2 Bonds are created—a) Floater b) Inverse
floater The bonds are created such that • A) the Total coupon Interest paid to the two bonds, in each
period is LESS THAN or EQUAL TO the COLLATERAL’s coupon interest in each period and
• B) the TOTAL par value of the 2 bonds is LESS THAN or EQUAL TO the COLLATERAL”S par value
• Evidently, the Floater and the Inverse Floater are structured so that the Cash Flow from the Collateral is sufficient to satisfy the obligations of the 2 Bonds. For ex. Consider a 10 yr, 7.50% coupon semi-annual pay bond. Suppose that $ 100 million of the Bond is used as a collateral to create a Floater with a par value of $ 50 million and an Inverse Floater of par value $50 million. Suppose the coupon rate is re-set every 6 months based on the following—
• Floater coupon Reference rate + 1% } 15% cap for Floater• Inverse Floater- (14%--- reference rate) }s 0% Floor for
Inv/Fltr
37• Ans; Notice that the TOTAL PAR VALUE of the Floater and the Inverse
Floater equals the par value of the Collateral $ 100 million. The WEIGHTED AVERAGE of the Coupon rate of this combo, is—
• 0.50(reference rate+1%) + 0.50( 14%-- reference rate)• =0.50RR+ 0.50+ 7.00%-0.50RR == 7.50%• Thus REGARDLESS of the level of the reference rate, the combined
coupon rate of the 2 bonds is EQUAL TO the coupon rate of the Collateral, 7.50%( see # 36)
• There is only one ISSUE with the Coupon formula given here. Suppose the reference rate is >14%. Then the formula for the coupon rate for INVERSE FLOATER turns NEGATIVE. To prevent this from happening, a FLOOR is placed on the coupon ratefor an Inverse floater.
• Typically the floor is set at Zero. Because of the floor, the coupon rate on the Floater must be restricted so that, the coupon interest paid on the 2 bonds DOES NOT EXCEED THE COUPON INTEREST on the Collateral. In our hypothetical structure, the MAXIMUM coupon rate that must be imposed on the Floater is 15%.
38• Inflation and Yields• When Inflation rates shoot up –as happened in the USA in 1978-79
when it got past 10%--SHORT_TERM Bonds maturing in less than a Year could yield more than a 10 year maturity bond. The Govt. under such circumstances fights Inflation by mopping up money supplies by floating Treasury bonds. This mop up increases the supply of Bonds decreasing the Prices of Bonds and resulting in Higher Yields. A negatively sloped Inverted Yield curve comes into existence! When the Inflation subsides the Curve turns normal
• -----------------------------------------------------------------• Accrued Interest
• Book-------------------Purchase ---------------------Next B/c• Closure date• ----Accrued interest ---Interest belongs------• Belongs to Seller to new Buyer
39• The Interest that has accrued belongs to the Seller and is paid by
the Buyer ADDING IT ON TO THE BUYING PRICE• --------------------------------------------------------------------------------• The Yield on any Investment is the interest rate that will make the
Present Value of the Cash Flow from the Investment EQUAL to the Price/ cost of Investment
• P=CF1/ (1+y) + CF2 /(1+y)^2 +CFn /(1+y)^n +M /(1+y)^n• P- Price of the Bond n- no.of years y yield to maturity• Where only one single Cash flow is concerned• P= CFn /( 1+r)^n• A 20yr 8% Bond makes many coupon payments, ALMOST ALL
of them coming BEFORE the Bond’s Maturity date. Each of these payments may be considered to have its OWN MATURITY DATE. In this case a Coupon Bond is a PORTFOLIO OF COUPON PAYMENTS. The Effective Maturity of a Bond is therefore some sort of an AVERAGE of the maturities of ALL cash flows paid by the Bond. The Zero Coupon Bond has a CLEAR MATURITY—end of Bond life
40• The Effective YIELD, on the other hand is
• = (1+ periodic interest rate)^m-1
• Where m is the frequency of payments per year
• Periodic interest rate= (1+Effective Annual Yield)^1/m-1
• Suppose the Periodic interest is 4% and the frequency of payments is twice a year
• Effective annual yield= (1.04)^2-1= 0.816
• Or 8.16%
41• If interest is paid quarterly, the periodic
interest rate is 2% (i.e. 8%/4) and the Effective Annual Yield is 8.24% as follows
• = (1.02)^4-1= 8.24%
• We can also determine the periodic interest rate that will produce a given Annual Interest rate by solving for it
• Periodic interest rate= (1+effective yield)^1/m -1
• (1+.12)^1/4-1=1.0287-1=.0287=2.87%
42• Yield to Maturity• Is the interest rate/IRR that will equate the sum of the PV of the Inflows to the
Current Price of a Bond• YTM for a semi-annual Bond• Step1 Compute the ‘periodic interest rate “y” that satisfies the relationship• P=__C____ + ___C___ + +___M_____• (1+y) ( 1 + y)^2 (1+y)^n• n= No. of periods*2 • Step2 For a semi-annual pay bond DOUBLING the periodic interest rate OR the
discount rate (y) gives the yield to maturity . This YTM is called ‘Bond equivalent Yield’
• Consider a bond with Cash flows1) 30 coupon payments of $35 each every 6 months 2) $1000/- to be paid 6 months from now 3) Price $ 769|42
• The PV of the Cash flows is as follows
43Annual Semi PV of 30 PV of $1000 PV of Cash
Interest annual payments of 30 periods from flows
Rate rate (y%) $35 (a) now (b)
9.0 4.50 $ 570.11 $267 $ 837.11
9.50 4.75 $ 553|71 248|53 $802.24
10.00 5.00 $ 538.04 231.28 $ 769.42
10.50 5.25 $ 532.04 215.45 $738.49
11.0 5.50 $508.68 200.64 $ 709.32
(a) The Present value of Coupon payments
____ $ 35{ (1+r)^n-1 }____ =___35{(1.045)^30-1}_ _ = 570.11
r(1+r)^n (1.045)^30 (0.0450)
(b) The present value of Maturity value is found using the formula
$ 1000{ __1__} = 1000 / (1+y)^30 =1000/3.7453 = 267$
(1+y)^30
(c) When a 5% semi-annual interest rate is used, the PV of the Cash flow is $ 769.42 Therefore y is 5% and the YTM on a BOND EQUIVALENT BASIS IS 10%
44• The YTM of a say, 10 year zero coupon Bond with maturity value $1,000/
selling for $439.18 is---• 439.18= ___1000__ at 8.40% the equation is satisfied• (1+r)^n• The relationship among Coupon rate, current yield and YTM is as follows---• Bond selling Relationship• At Par Coupon rate=current yield= ytm• At Discount Coupon rate < current yield< ytm• At Premium Coupon rate> current yield> ytm
• Yield to call• In respect of callable bonds, the Price at which the Bonds may be called is
called the ‘call price’. For some issues, the CALL PRICE IS THE SAME REGARDLESS OF WHEN THE BOND IS CALLED. For other callable issues, the call price depends on when the Issue is called. That is there is a CALL SCHEDULE that specifies a CALL PRICE for EACH CALL DATE!
45• For Callable Issues, the practice has been to calculate BOTH a Yield to call
and a Yield to Maturity. The YTC assumes that the Issuer will call the Bond at some CALL DATE. Typically investors calculate a YIELD TO FIRST CALL and a YIELD TO PAR CALL; yield to par value is when the Issuer can call the Bond at Par Value
• For ANY YTC one needs to look at the Cash flows till the Call date as well as the amount receivable as Capital sum on the assumed date of call(M)
• To illustrate, consider an 18 yr 11% coupon Bond with a Maturity value of $ 1000/ selling for $1,169/- suppose the FIRST call is 8 years from now and that the call price is $1,055/- The Cash flows for the bond if called in 13 years are---
• 1) 26 coupon payments of $55/ (i.e. 11/100*1000=110/2=55)• 2) $1,055 due in 16 six months periods from now• Date of_____________ issuer eligibility_______________ CALL Made• Issue Year 8 at $1,055 13th year
• 1.1.1995 31.12.2003 31.12.2008
46• By trial & error let us try 8% (A) (B)
• Annual Interest Semi annual PV of 16 PV of $1055 PV of Cash
Rate rate y % payments 16 periods Flows
of $ 55 each from now
8.000 4.000 640.88 + 563.27 = 1204.15
8.2500 4.1250 635.01 + 552.55 = 1,187.56
8.500 4.2500 629.22 + 542.05 = 1,171. 26
8.535 4.2675 628.41 + 540.50 =1169
8.600 4.300 626.92 + 537.50 = 1164.83
(A)The PV of the Coupon rate is found using the equation _55{ (1+r)^n-1}___ =55{ ( 1.04)^16 -1} = 640.88
{r ( 1+r) ^n } (.04) (1.04)^16
(B) The PV of the call price
$1,055 { __1____} = ___1055___ = 563.27
{1+ y } ^16 !.04 ^16
47• The periodic interest rate is of course, 4.2675% at which the PV of
Cash Flows equals the market Price of $ 1169. therefore the Yield to First call on a ‘BOND EQUIVALENT BASIS’ is 8.535 %
• Suppose the FIRST PAR CALL date for the Bond is 13 years from now. Then the yield to first par call is the interest rate that will make the Present value of $55, every six months for the next 26 months PLUS the PAR value of $1,000/, 26 months from now, equal to a price of $1,169/- ( assignment)
• YIELD TO PUT As explained earlier, an Issue can be PUTABLE. This means that the Bond Holder can force the ISSUER to BUY BACK HIS BONDS at the PRICE SPECIFIED! As with a callable issue, a PUTABLE ISSUE can have a PUT SCHEDULE . The Schedule specifies when the Issue can be PUT and the Price thereof called the PUT price
• The Yield to Put is that interest rate that makes the Present Value of Cash flows to the ASSUMED PUT DATE PLUS the PUT price on the date set forth in the PUT SCHEDULE, equal to the Bond’s Price
•
48• For example consider again the 11%
coupon 18 years Issue selling for $1,169/ assume the Issue is PUTABLE at par ($1,000) in 5 years. The yield to PUT is the interest rate that makes the PV of $55/period for 10 six months period PLUS the PUT price of $1,000/ equal to $1,169
• Calculate—(Assignment)
49• Yield To Worst ( YTW)• Is the MINIMUM of YTM, YTC, YTP every day• YIELD ( Internal rate of return) for a Portfolio• The Yield for a Portfolio of Bonds is NOT simply the average/weighted
average of the ‘yield to maturity’ of Individual bonds in the Portfolio• It is computed by 1) Determining the Cash Flows and• 2) Determining the INTEREST RATE that will make the PRESENT
VALUE of the Cash flows EQUAL TO the MARKET VALUE of the Portfolio
• Yield of a portfolio• Bond Coupon Maturity Par value Price YTM• A 7.000 5yrs-10halfyrs $10,000,000 9,209,000 9.0 %• B 10.500 7yrs- 14halfyrs $ 20,000,000 20,000,000 10.5%• C 6.000 3 yrs- 6half yrs. $30,000,000 _ 28,050,000_ 8.50%• 57,25,9000
50• It is assumed that the coupon payment date is the same for all the 3 bonds• Period cash Bond A(1/2yr) Bond B(1/2yr) Bond C(1/2yr) Portfolio• 1 $3,50,000 $1,050,000 900,000 $2,300,000• 2 $3,50,000 $1,050,000 900,000 $2,300,000• 3 $3,50,000 $1,050,000 900,000 $2,300,000• 4 $3,50,000 $1,050,000 900,000 $2,300,000• 5 30,900,000 $32,300,000• 6 $ 1,400,000• 7 $ 1,400,000
• 8• 9• 10 $10,35,000 $11,400,000• 11 $1,050,000• 12• 13• 14 $21,050,000 21,050,000
51• To find the yield ( internal rate of return) for the 3 bond Portfolio, we need to
find such INTEREST RATE that EQUALIZES the sum of the cash flows in the last column to a sum of $ 57,259,000/-( the total value of the portfolio). At 4.77% this turns out so! Doubling this rate we get a figure of 9,54% which is the Yield on the Portfolio
• Yield Measure for FLOATERS• The rate the Coupon gives out on Floaters depends on the REFERENCE
RATE to which the Security is linked. Because the value of the Reference rate, in the future is NOT known WITH CERTAINTY, it is not possible to determine the cash flows. This means that a Yield to maturity cannot be calculated for a FLOATER
• A conventional measure used to estimate the Potential return for a FLOATER is the Security’s EFFECTIVE MARGIN . This measure estimates the AVERAGE SPREAD / MARGIN over the REFERENCE RATE and over the life of the security.
52• Procedure for calculating EFFECTIVE MARGIN• Step 1 Determine the CASH FLOWS assuming that the REFERENCE
RATE DOES NOT CHANGE over the life of the security• Step 2 Select a MARGIN (spread)• Step 3 DISCOUNT the Cash Flows found in Step 1 by the Current Value
of the REFERENCE Rate PLUS the Margin (spread) selected in Step 2• Step 4 Compare the PV of the Cash Flows as calculated in Step 3 WITH
the PRICE. If the PV is NOT EQUAL to the Security’s PRICE,go back to step 2 and iterate!
• For a security selling AT PAR, the EFFECTIVE MARGIN is simply “ the SPREAD over the REFERENCE RATE”
-----------------------------------------------------------------------------------------------
Potential sources of a Bond’s Dollar return
The Potential Income----a) Periodic Coupon
b) Capital gain/ capital loss on SALE of the Instrument
c) Interest income generated from REINVESTMENT OF PERIODIC CASH flow
53• For a Standard Bond that makes ONLY Coupon Payments and NO re-
payments of Capital prior to Maturity, the Interim cash flows are the only receipts/ inflows and the re-investment of these amounts leads to ‘Interest on Interest Income’. For AMORTIZING SECURITIES, the Re-investment Income is “Income earned on BOTH Coupon receipts AND Principal repayments re-invested’
• Any measure of a Bond’s potential Yield should take into account ALL these potential sources of return The Current Yield takes into a/c ONLY the COUPON payments( No consideration is given to any Capital Gain/ Loss OR FOR ‘Interest on interest payments”. The YTM takes into a/c the Coupon Receipts PLUS CAPITAL GAINS (loss) PLUS Interest on Interest component. However as will be demonstrated later, IMPLICIT in the assumption of YTM is the HIDDEN NOTION that all receipts are re-invested at the YTM rates computed! The YTM is only a PROMISED YIELD that will be realized ONLY IF 1) The Bond is HELD TO MATURITY 2) The Coupon and the ‘amortized amounts’ of the PRINCIPAL re-payments can be ‘reinvested’ at the ‘computed YTM rates If either of the 2 conditions FAIL, YTM can be DIFFERENT from the
53a• Cash flow yield
• Here we look at ‘amortizing securities’ where—
• 1) Principal re-payments are made on schedule
• 2) Interest payments are made on schedule
• 3) Principal can be PRE-PAID i.e Principal can be paid over and above the schedule
54-contd. from#53• ---can be different from the realized yield.
• The YTC suffers from the same limitations as YTM
• The Cash flow yield also takes into a/c all the 3 sources as does the YTM. It makes 2 more assumptions– 1) It assumes that periodic Principal repayments( receipts) are re-invested at Cash Flow yield rate 2) It assumes that pre-payments projected to obtain the Cash Flows are actually realized
55• Determining the ‘Interest on Interest’ Dollar
return• If ‘r’ denotes the semi-annual re-investment rate,
the Interest on Interest PLUS the total coupon payments can be found from the following Equation
• Coupon interest ] =C{(1+r)^n-1} (A)• + Interest on interest] r• The above is the ‘future value of Annuity’• The total dollar amount of coupon interest is
found by multiplying the semi-annual coupon interest BY the no. of periods =C* n-(B)
56• Interest on Interest• Is (A)—(B) in the previous slide• i.e• C{ (1+r)^n-1} -- C*n• r • That is the Future value of the amount of Coupon interest(including
the Interest on Interest portion) LESS Total coupon interest• Ex Let us consider the 15year 7% Bond that we have used to
illustrate how to compute Current yield & YTM. If the price of the bond of par value is $ 1,000/ is $ 769.40, the YTM of this Bond is 10% as seen earlier. Assuming an Annual reinvestment rate of 10%(i.e. 5% semiannual), the interest on interest PLUS total coupon payments are—
• Coupon Interest ]=____35 [(1+.05)^30-1}____ =$2325.36• + Interest on Interest] .05• 35=1000*.07/2• Total Interest=C*n=35*30= $1050
57• Therefore, Interest on Interest= $2325.36-30(35)=$1275.36
• YTM AND REINVESTMENT RISKS• Let’s look at the potential total $ return from holding this bond(#56)
to Maturity. The sources of Inflows—• 1) Total coupon interest of $1,050i.e. 35*30 every 6 months for
15years• 2) Interest on Interest of $1,275.36 earned from re-investing the
semi-annual coupon interest payments @5% every 6months• 3) A capital gain of $230.60 (i.e. $1,000-769.40)• The Total potential $ return if the coupon is reinvested at YTM of
10% then is 1050+1275.36+230.60= 2555.96 $• This is the SAME as $2325.36 representing Coupon interest PLUS
Interest on Interest AND Capital gains of $230.60• Notice that if an Investor places the Money that would have been
used to purchase this Bond, $ 769.40 in a savings Bank a/c earning 5% semi-annually for 15 years, the savings would be--
58• The savings 769.40 (1.05)^30= $3,325.30• For an Initial $ Investment of $769.40, the $ returns = 3,325.30-
769.40=$2555.90• This is what we found by breaking down the $ return on the bond
assuming a re-investment rate equal to YTM! Thus it can be seen that for the Bond to yield 10%, the investor MUST GENERATE $1,275.36 by re-investing the coupon receipts. This also means that, to generate a YTM of 10% APPROXIMATELY HALF i.e. 1275.36/2555.96=49.80%, must come from re-investment of coupon payments
• The investor will realize the YTM (that he anticipated at the time of purchase) ONLY IF he held the Bond to Maturity and he is able to realize the returns for the period of Holding at the YTM rates. The Risk that the Investor faces is that the future RE_INVESTMENT rates will be LESS than the YTM rates that existed ( ‘on the calculator’) at the time the investment was made. This is called “REINVESTMENT RISKS”
• There are 2 characteristics of a bond that determine the importance of ‘interest to interest component’ and therefore the re-investment risks’ 1) MATURITY 2) COUPON
59• For a GIVEN YTM and a GIVEN COUPON RATE, the longer the maturity,
the more DEPENDENT the bond’s ‘total dollar return’ on the ‘interest to interest component’, to realize the YTM at the time of purchase! i.e. the longer the maturity, the GREATER the RE-INVESTMENT Risk!( because the market interest rates may change unfavorably compared to the YTM and there is enough time for this to happen—especially if the maturity period is long)
• Cash Flow Yield and RE-Investment risk• For AMORTIZING SECURITIES, the re-investment risk is greater than it
is for Non-amortizing securities. The reason is that the investor has to re-invest NOT only the periodic interest repayments, but ALSO the PERIODIC CAPITAL REPAYMENTS at the Yield rate! In Mortgage/ asset backed securities, the CASH FLOWS are the MONTHLY and NOT semi-annually ( as with non-amortizing securities). Thus in Amortizing Securities which are Asset based / Mortgage backed, the REINVESTMENT RISK is high! There is only one aspect in Non-amortizing securities that increase REINVESTMENT RISK- the Borrower can PRE-PAY! He does this when interest rates decline! This is the reinvestment risk for a Non Amortizing Security
• Yield curves• Yield curves are a graphic representation of the relationship between Yield and
Time( Time to Maturity). The Y/C captures the relationship at a certain point in time. The shape of the Y/C changes over time.
• To be representative, the Instruments plotted on the Y/c must have common characteristics – same credit risks, same tax treatment etc Of course the Maturity Dates of the Yield of the Instruments being compared are DIFFERENT. Usually the Yields of a 3 month Treasury Bond is compared with the Yields of a 5 Year and 30 year US Treasury Bonds.
• Types of Yield curves• 1 ) Ascending yield curves• Shows yield increases for Longer maturities. Common feature of Bonds in Developed
countries.• 2) Descending Y/C• Short term interest rates are much greater than Long term interest rates. This could
indicate a ‘possible recession”• 3) Flat Yield curves• Short Term rates are the same as the Long term rates.• 4) Humped Y/c • The Hump comes as a result of ‘expectations’ that the short term interest will increase
first and then Fall or, interest rates will Fall first and then Increase!
59a
59 b• Y/c are usually ‘upward sloping’– the Longer the maturity, the Higher the Yield
but with “DIMINISHING MARGINAL GROWTH” The reasons A) Investors need to be compensated for the “Time Value of Money” B) Need to be compensated for ‘uncertainties’ that increase with duration
• Also if the Market expects more VOLATILITY in the FUTURE , notwithstanding a Possible Decline in Interest rates, a HIKE IN RISK PREMIUM as a result of ‘uncertainty” can influence a “spread” and cause Yields to INCREASE. This is known as “Liquidity Spread”
• In the OPPOSITE SITUATION– Low activity regime—the Short term Yields could “GO PAST” Long term yields resulting in an “ Inverted Yield Curve”. The market’s anticipation of a Falling Interest rate regime causes a ‘Inverted yield Curve”. Strongly inverted Yield Curves have historically preceded ‘Economic Depressions”
• The MOST IMPORTANT FACTOR determining the shape of the Yield curve is the CURRENCY in which it is denominated. The ECONOMIC SITUATION IN COUNTRIES AS WELL AS the Corporates using such CURRENCY determine the shape of the Yield Curve. For ex. The sluggish Economic Growth in Japan through out 1990-2000 meant that the “Yen Yield curve ”traded “ LOW—a 3 month Y/C is virtually zero while a 30 year Y/C is just around 2%. Compare this with the British Pound that averaged 4-5% during the same period!
• Different Credit institutions borrow at different rates based on their ‘Credit Worthiness”. The Yield Curve pertaining to the bonds issued by GOVT. ----
59c• -----In the country’s currency is known as “GOVT. Bond Yield Curve”• Banks and Institutions with HIGH CREDIT RATING borrow money from each
other at LIBOR! These yield Curves are only slightly higher than Govt. Yield curves. These LIBOR curves at which Banks borrow/ lend to one another are known as SWAP CURVES
• On the other hand, Corporate Y/C for Bonds issued by Corporates are HIGHER than LIBOR and GOVT. Y/C, as the credit worthiness of the Corporates is LOWER. The Corporate Y/C are reflected in terms of their” credit spread over the relevant Swap Curve! i.e. say, LIBOR + 0.25
• NORMAL YIELD CURVE• From Post Depressions to now, Y/C have generally been ‘normal’ i.e. Yields
Increase, as Time to Maturity increases! A +ve sloping Y/c reflects expectations of Growth and Inflation both! On HIGHER INFLATION, the expectations are that the Central Bank would hike Interest rates to contain Inflation!
• Y/c are also inverted for some periods of DEFLATION; deflation makes ‘CURRENT CASH FLOWS LESS VALUABLE THAN FUTURE CASH FLOWS!
• STEEP YIELD CURVE• Historically, the 20 year Treasury Bond Yield has averaged approximately 2 %
points ABOVE 3 months Treasury Bills. In situations where this gap INCREASES, the Economy is expected to improve in the future! This can be seen by way of the onset of Economic Expansion AFTER Recession!!
59dREASON-- Yield on the 3 month Bond could be could be FALLING consequent to
a rising bond prices, in the course of an economic recovery.
THEORIES regarding YIELD
There are 5 theories that try to explain why Yields vary with Maturity. Two of these theories are extreme positions while the third attempts a middle ground between the two.
1 ) Market Expectations theory
( 1 + I k.t) = (1+ iyear1 s,t) ( 1+ I year 2 s,t) +-----( 1 + I year n s,t)
The Hypothesis assumes that that the various maturities are PERFECT SUBSTITUTES and the SLOPE of the Y/C depends on the Expectations of the Market Participants about the future). These expected rates along with an assumption that ‘arbitrage opportunities” will be MINIMAL is enough information to construct a Yield Curve For Example, if Investors have an expectation of what 1 year interest rates will be NEXT YEAR, the 2 year interest rates can be calculated as the COMPOUNDING of this year’s interest rate BY the next years’ interest rates. More generally, rates on a LONG TERM INSTRUMENT are equal to the GEOMETRIC MEAN of the Yield on a series of short term Instruments
2) LIQUIDITY PREFERENCE THEORY
Investors need to be compensated for holding LONG TERM BONDs by way of a LIQUIDITY PREMIUM. ------contd.---
59-e• Time value compensation plus compensation for uncertainties are built into the
Liquidity premium. Thus the curve of LT Bonds is Steeper and slopes more upwards than the Y/C of a short term Bond.
• 3) Market Segmentation Theory• This is also called SEGMENTED MARKET HYPOTHESIS. Financial Instruments
of different kinds / terms are NOT SUBSTITUTABLE!. As a consequence, the demand for Long and Short term Instruments are determined INDEPENDENTLY!. Investors, generally Risk averse, prefer short term Instruments. Demand for this pushes up the PRICE of these Bonds , REDUCING their YIELDS. This explains the fact that ‘short term yields” are lesser than Long Term Yields!
• 4) Preferred Habitat theory• in addition to Interest Rate Expectations, investors have DISTINCT
INVESTMENT HORIZONS and require sufficient financial motivations to buy instruments NOT IN THEIR “PREFERRED HABITAT
60• TOTAL RETURN• The YTM is a PROMISED YIELD. The Investor , AT THE TIME OF
PURCHASE, is promised a yield as calculated by YTM if---• A) The Bond is Held to Maturity AND • B) ALL coupon interest payments are ‘reinvested’ at YTM rates• While point A) is simple, point B) can make the situation tricky! If
Market interest rates continue to be lower than the YTM for extended periods of time, the ‘realized total returns’ would turn out to be much lower than the ‘Purchase point YTM’
• So, rather than make an assumption that the reinvestment rate will be at YTM, one could very well make an EXPLICIT ASSUMPTION that the reinvestment would be at the ‘explicitly stated re-investment rates” . The Yield of a Bond calculated on this basis is called ‘ the Total return’ The Total return is a measure of Yield that makes an EXPLICIT ASSUMPTION about the ‘reinvestment rates’
• Let us examine point a) above, by way of an example.
61• Suppose an Investor who has a 5year Investment Horizon is looking at the
Foll. 4 Bonds• Bond Coupon% Maturity(yrs) YTM(%)• A 5 3 9.0• B 6 20 8.60• C 11 15 9,20• D 8 5 8.00• Assuming that all the 4 Bonds are of Equal credit Quality, which Bond is the
MOST attractive to the Investor?• You might first want to look at Bond C as the YTM is the Highest
@9.20%But the Instrument has a maturity period of 15 years whereas, the investor has an Investment horizon of only 5 years. This calls for selling the Bond after 5 years AT A PRICE that depends on the YIELD REQUIRED in the market for a 10 years 11% Coupon Bond at that time!
• Bond A offers the 2nd highest YTM @ 9%. How does this look?ss
62• On the surface of it, it looks alright as it eliminates the possibility of realizing a
CAPITAL loss when the Bond is sold PRIOR to its maturity date! • The Maturity period is the lowest. The Problem is that it is LOWER than the
investors Investment HORIZON of 5 years! So, the re-investment risk exists nevertheless! The Yield that the Investor realizes depends on the
INTEREST RATES three years from now on a 2 year bond on to which the proceeds of the 3 year old bond must be rolled over!
the YTM does not help identify the Bond that suits the Investor ‘the best’
IT is the Expectations of the Investor that matters! Specifically, it depends
on the RATE AT WHICH THE RE_INVESTMENT CAN BE MADE(expectation
Of the Investor). Consequently any of these bonds could end up as being
“best suited to the Expectations of the Investor” for the differential period!
The YTC is subject to the same issues as the YTM. First it assumes that the Bond will be held until the first call date; next that the interest will be reinvested
At YTC rates etc. There are Re-investment risks arising out of this concept
So we might have to look at TOTAL RETURNS!
63
• Computing the TOTAL RETURNS for a bond• The idea is simple. The Objective is to first compute the Total Future
Dollars that will result from investing in a bond assuming a PARTICULAR RE-INVESTMENT rate. The Total return is then computed as the interest rate that will make the initial investment in the Bond GROW to the computed TOTAL FUTURE DOLLARS
• The procedure for computing the total return for a Bond held over some investment horizon can be summarized as follows.—
• For an ASSUMED re-investment rate , the total dollar return for a bond held over some re-investment horizon can be computed for BOTH the Coupon interest payment and interest on interest components. In addition at the end of the period the Investor will receive some Par value or some other value
64• The Total Return is then that Interest rate that will make the
amount invested in the bond ( current Market Price PLUS accrued interest ) GROW to the FUTURE DOLLARS available at the end of the Planned Investment Horizon
• STEP 1 Compute the total Coupon Payments PLUS interest on Interest , based on an ‘assumed reinvestment rate’ ( the re-investment rate is ONE HALF of the Annual Interest rate that the Investor assumes can be earned on the re-investment of Coupon interest payment)
• STEP 2 Determine the ‘projected Sale Price” at the END of the Planned Investment horizon . The “Projected sale price’ will depend on the ‘Projected Required Yield’ at the end of the planned Investment horizon THE PROJECTED SALES PRICE is EQUAL TO the PRESENT VALUE of the REMAINING CASH FLOWS of the bond( the value at which the bond can be sold/surrendered) DISCOUNTED AT the PROJECTED REQUIRED YIELD
65• Step 3 Sum the values computed in step 1 and in step 2. the sum is
the TOTAL FUTURE DOLLARS THAT CAN BE RECEIVED FROM THE INVESTMENT , GIVEN the assumed re-investment rate and the Projected Required yield at the end of the Investment horizon
• Step 4 To obtain the Semi-annual Total Return use the formula—• { Total Future DOLLARS } ^ 1/ n -1 • {Purchase price of Bond }• n no. of half yearly periods in the Investment horizon• Step 5 as interest is assumed to be paid semi-annually, DOUBLE
the Investment rate got in step 4 The resulting figure is the TOTAL RETURN
• Ex. Suppose an investor with a 3-yr.Investment Horizon is considering purchasing a 20 yr. 8% coupon Bond for $ 828.40 The YTM for this bond is 10% The Investor expects to re-invest the coupon interest payments at an Annual Interest rate of 6% and at the end of the Planned Investment Horizon their 17 year bond would be selling at a price to offer a YTM of 7%. The TOTAL RETURN--
66• Step 1 Coupon payments are $40 ( i.e 1000*8%*6/12) every 6 months
for 3 years (6 periods—investors horizon)• • A—3yrs--------------B----------17 yrs----------------------C• What he receives from the Bond in 3 years• Coupon+ Interest on Interest• =$40 {(1.03)^6 -1} = $ 258.74• 0.03• Step 2 Determining the ‘projected sale price’at the end of 3 yrs, at YTM
@7% for 17yr bonds. This is the PV of the 34 coupon payments {i.e. 20yrs*2- (3*2)} of $40 PLUS the PV of Maturity value of $1000/ discounted at 3.50%. The Projected sale Price is $ 1089.51 as under—
• FUTURE $= 34{( 1.0+ 0.0350)^ 34- 1} = 788.25
(1.035)^34 (0.035)
PLUS 1000/ (1.035) ^34 =310.47 TOTAL=1098.67
67• Note; The TOTAL Future Dollars computed here DIFFERS from
Total Dollar Returns that we have used in showing the importance of ‘interest on Interest’ earlier. The Total Dollar return there includes the Capital gains/ capital Loss and NOT the Purchase price, which is used in calculating the ‘Total Future Dollars’
• Step 3
• Yr0----3Yr---------| |-------------------------------Yr 20• $258.74 1098.67• PV of 34 coupons+ PV of $1000• Total= $ 1357.25 • Step 4 To obtain semi-annual return• {1,357.25} ^1/6 -1 = .0858= 8.58% • { 828.40 }• Step 5 Double 8.58% for a total return of 17.16%
68• APPLICATION OF THE CONCEPT OF TOTAL RETURNS• Horizon Analysis allows a Portfolio manager to project the
performance of a bond on the basis of Planned Investment Horizon and ‘expectations’ concerning reinvestment rates and ‘future market yields’. This permits the Portfolio manager to evaluate which of the several potential bonds considered for acquisition will perform the best over the planned Investment horizon.
• This job cannot be done using YTM– hence the need for ‘total return concept’/ ‘Horizon Analysis’
• Horizon Analysis can also be used for ‘Bond swaps’. One evaluates the total returns of a bond in a Portfolio WITH another Bond outside the Portfolio with an idea of a SWAP
• Limitations• 1) Reinvestment rates 2) Investment Horizon #) Future yields
and Projected sales price
69• Bond Price Volatility
• To employ effective Bond Portfolio Strategies, it is essential that one understands ‘volatility in the prices of Bonds, consequent to a change in the rate of Interest in the Economy.
• A fundamental principle of an OPTION FREE BOND ( i.e. a Bond without an Option embedded in it) is that the Price of the Bond CHANGES in the direction OPPOSITE to that of a change in the REQUIRED YIELD of the bond.
• This Principle flows from the fact that the price of a Bond is equal to the PV of the Expected Cash Flows from it. An Increase/ (Decrease) in the REQUIRED YIELD decreases( increases) the PV of its expected cash Flows and therefore decreases( increases) the Bond’s Price
•
70
• The Sensitivity of Bond Prices to change in interest rates is measured by the ‘Duration of a bond’. There is an Inverse relationship between bond prices and yields, as we have seen already. As interest rates move up or down, bond holders experience Capital losses or Capital gains. These gains or losses make Fixed income investments ‘risky’ even if the Coupon & Principal payments are guaranteed!
• Why do Bond Prices respond to changes in Interest rates?• In a competitive market all securities must offer the Expected rate of
return . If interest rates increase from 8% to 9%, the 8% bond loses ground so that the YIELD, as a consequence, is 9%! Likewise the fall in interest rates from 8% to 7% makes the 8% Bond attractive. The Yield falls to around 7%, pushing the Price up in the process
•
71• It is interesting to note that ‘decreases in yields ‘ have a
BIGGER impact on prices THAN increases in yields of EQUAL MAGNITUDE have on prices! An increase in the bond’s ‘yield to maturity’ results in a SMALLER PRICE CHANGE( downwards) than a decrease in the bond yield of ‘equal magnitude’
• The other interesting observations—• ---Prices of LONG TERM Bonds tend to be MORE SENSITIVE to
interest rate changes THAN prices of short term bonds• It is plain that longer the period, greater the uncertainty and
greater the sensitivity• ---The sensitivity of the Bond Prices to CHANGES IN YIELDS
increases at a DECREASING RATE as Maturity increases. • In other words, INTEREST RATE RISK is less than proportional
to ‘period to maturity’
72
• --Interest rate risk is INVERSELY related to the BOND’s Coupon rate
• Prices of ‘high coupon Bonds’ are LESS SENSITIVE to changes in Interest rates THAN the PRICES of LOW COUPON BONDS
• ---The Sensitivity of a Bond’s Price to a CHANGE IN YIELD is INVERSELY RELATED to the YTM at which the Bond is currently selling
If Bonds C and D are identical EXCEPT for their YTM’s, Bond C with a HIGHER YTM, is LESS SENSITIVE to changes in Yields than Bond D
The first five rules are called “ Malkeil’s Bond Pricing Relationship’ while the last was demonstrated by Homer
73• The 6 propositions confirm that Maturity is
a MAJOR determinant of Interest rates. However they also show that MATURITY ALONE is NOT sufficient to measure interest rate sensitivity . For ex. Bonds B and C could have the same maturity (30 yrs.) but a higher coupon rate in Bond B gives it lower sensitivity in bond PRICES to interest rate changes. Obviously we need to know MORE THAN a Bond’s maturity to quantify its interest rate risk!
74• To see why Bond characteristics such as
Coupon rates or YTM affects interest rate sensitivity, let us start with a simple example. The table gives Bond Prices for 8% semiannual coupon bonds at different a) Yields to maturity and b) Time to Maturity T
• YTM T=1yr T=10yr T=30 yrs• 8% 1000 1000 1000• 9% 990.84 934.96 907.99• 0.94% 6.50% 9.20%
75
• Prices of zero coupon Bond (semi annual compounding)
• YTM T=1yr T=10 yrs T=20 yrs
• 8% 924.56 456.39 208.19
• 9% 915.73 414.64 171.93
• Change 0.96% 9.15% 17.46 %
• In price %
76
• The interest rates are expressed as an Annual Percentage rates (APR) meaning that 6 months yield is doubled to obtain Annual yields.
• The shortest term bond (1yr) falls by <1% when interest rate increases from8% to 9% ie 1%). The 10 yr bond falls by 6.50%!! And the 20 yr by >9%!!! The SAME computation in a Zero coupon bond above shows that for EACH MATURITY, the PRICE of a zero coupon bond falls by a greater proportional amount than the price of a 8%(non-zero bond)!
77• We know that LONG TERM BONDS are MORE sensitive to interest
rate changes than short term bonds. The 20 yr,8% bond makes many coupon payments ALMOST ALL OF THEM coming obviously BEFORE the BOND’S MATURITY DATE. EACH of these payments may be considered to have its own Maturity date In this sense, a Coupon Bond is a Portfolio of coupon payments. The EFFECTIVE MATURITY of a Bond is therefore some sort of an AVERAGE of the maturities of ALL cash flows paid by the bond. The zero coupon has a well defined maturity concept—at the end of its life!
• Higher coupon bonds have a HIGHER fraction of value tied to coupons RATHER THAN to the (closing) or PAR VALUE . So the ‘portfolio of coupons’ is more heavily weighted towards the EARLIER SHORT TERM PAYMENTS which gives it LOWER EFFECTIVE MATURIRT! This explains the fifth rule that ‘price sensitivity falls with coupon rate!
78
• Similar logic explains the 6th rule that price sensitivity falls with yield to maturity. A higher yield REDUCES the PV of ALL of the bond’s payments; but more so, for the the MORE DISTANT PAYMENTS. Therefore, at Higher yields, a HIGHER fraction of the Bond’s value is accounted for by EARLIER receipts and hence have LOWER EFFECTIVE MATURITY and INTEREST RATE SENSITIVITY. The overall sensitivity of the bond’s price to changes in yield is thus LOWER
79-Duration• We need a measure of AVERAGE MATURITY of a Bond’s promised
cash Flows to serve as a useful summary statistic of the EFFECTIVE MATURITY OF THE BOND. We would also like to use the measure as a guide to the Sensitivity of a Bond reacting to the changes in interest rates, because we have noted that the PRICE SENSITIVITY tends to increase with the TIME TO MATURITY
• Fredrick Maculay termed this “Effective maturity concept”, the DURATION of the Bond ( basically an AVERAGE maturity concept)
• Maculay’s Duration is computed as the “weighted average” of the TIMES of each coupon or Principal payment made by the bond. The weight associated with EACH payment time clearly, should be related to the importance of that payment TO the VALUE OF THE BONDthat is accounted for by THAT PAYMENT This PROPORTION is just the PRESENT VALUE of the payment DIVIDED BY THE BOND PRICE!
•
80• Characteristics of a bond that affect its Price Volatility• There are 2 characteristics of an Option Free Bond that
determine its Price Volatility 1) Coupon rate 2) Term to Maturity• Characteristic 1; For a given TIME TO MATURITY and INITIAL
YIELD, the Lower the Coupon rate, GREATER the PRICE VOLATILITY
• Characteristic 2 ; for a given COUPON RATE , the LONGER the TIME TO MATURITY, the GREATER the PRICE VOLATILITY
• An application of the 2nd characteristic is that investors who want to increase a Portfolio’s Price Volatility BECAUSE THEY EXPECT THE INTEREST RATES TO FALL, (other factors held constant), SHOULD HOLD BONDS WITH LONG MATURITIES IN THE PORTFOLIO. Just the same way, in order to subject oneself to a LOWER PRICE VOLATILITY in a RISING RATE REGIME, one should hold Portfolios of bonds WITH SHORT TIME TO MATURITIES
81• Effects of YTM• We cannot ignore the fact that CREDIT CONSIDERATIONS cause
different bonds to trade at different yields, EVEN IF THEY HAVE THE SAME COUPONS AND THE SAME MATURITY It is seen that Higher the Initial Yield, LOWER the Price Volatility , when a change in the Yield takes place
• Measures of Bond Volatility• There are 3 measures commonly employed to measure the
‘Price Volatility’ of a Bond• 1) Price value of a Basis Point• Also referred to as a Dollar Value of an 01 it is the CHANGE in
the PRICE of a Bond if the REQUIRED YIELD CHANGES BY 01 Basis Point. This exhibits “Dollar Price Volatility” as opposed to the Percentage Price Volatility( Price change as a % of Initial Price).
82• Price Value of a Basis Point• Bond Initial Price Price @ 9.01% Price value• @9% yield yield of a Basis Point• 5yr 9% coupon 100.000 99.9604@ 0.0396 ( 100-99.9604)• 25yr 9% coupon 100.000 99.9013 0.0987• 5Yr 6% coupon *** 88.1309 88.0945 0.0364• 25 yr 6% coupon 70.3570 70.2824 0.0746• 5 yr zero coupon++ 64.3928 64.3620 0.0308• 25 yr zero coupon 11.0710 11.0445 0.0265 • @ 9/1.0901+9/1.0901^2+---109/1.0901^5=99.9604 • ***6/1.0901+ 6/1.0901^2+----6/1.0901^5+ 100/1.0901^5• ++ 100/1.0901^5• Note; Some small differences seem to be cropping up in the
calculations—check why
83
• Yield Value of Price change• Another measure of the Price Volatility of a Bond used by
investors is the change in the yield for a SPECIFIED PRICE CHANGE. This is estimated by FIRST CALCULATING the Bond’s YTM If the Price decreases by X dollars. The difference between the INITIAL YIELD and the NEW YIELD is the YIELD VALUE of an “X Dollar Price Change” The SMALLER this value, the GREATER the Price Volatility—the reason; it would take a SMALLER CHANGE IN YIELD, to produce a PRICE CHANGE OF X Dollars!
• Treasury Notes and Bonds Quoted in 32nds of a Percentage point of Par. Consequently, in the Treasury Market, investors compute the Yield value of a 32nd!
84
• BOND Initial Price Yield at Initial Yield Yield Value• minus a 32nd New Price of a 32nd
• 5yr,9% 99.96875# 9.008** 9.000 0.008• 25yr,9% 99.96875# 9.003 9.000 0.003
• # 100-(1/32*1)• ** 99.96875= 9/(1+x) + 9/(1+x)^2+------109/(1+x)^5• By trial & Error, x=1.09008• So the New Yield= 0.09008 or 9.008• By similar process the Value of an 8th can also be calculated•
85-Bond Duration• The Price of an ‘option free bond’ can be expressed as • P=C/(1+y) + C/ (1+y)^2+---C/(1+y)^n+ M/ (1+y) ^n-----(1)• Where ‘n’ is the no. of Semi-annual periods (yrs*2)• Now if the required Yield is changed by a very small amount,
what happens to PRICE is the question
dP/ dY = -- 1/(1+y){ __1C_ + __2C__ ------ nC___ +nM___} ----(2)
{(1+y) (1+y)^2 (1+y)^n (1+y)^n}
The terms in the Brackets indicate the WEIGHTED AVERAGE “term to Maturity” of the CASH FLOWS from the Bond where the weights are the PRESENT VALUES of the CASH FLOWS
Dividing Both the sides by P, we get the approximate PERCENTAGE PRICE CHANGE
dP/dY*1/P = -- 1/(1+y){ 1C_+__2C__+-----nC___- + __nM} _1__ -(3)
{ (1+y) (1+y)^2 (1+y)^n (1+y)^n} P
The Expression in the Brackets Divided (or Multipled by the reciprocal) of the Price is called MACAULAY DURATION
86
• Macaulay Duration• _1C___ + __2C___ +___3C-_+ nC-__ +__nM__• (1+y) (1+y) ^2 (1+y)^3 (1+y)^n (1+y)^n• __________________________________________ -----(4)• P• Which is SUM __tC__ + __nM__• (1+y)^t (1+y)^n• ---------------------------- ----(5)• P• Substituting (5) in Equation (3) above’• dP/ dY *1/p = -- 1/(1+y) * Macaulay Duration---(6)• This is called MODIFIED DURATION
87• Modified Duration= Macaulay Duration / (1+y)-----(7)• Substituting Equation (7) into (6)• dP/dy * 1/p = -- Modified duration -(8)• Equation (8) states that MODIFIED DURATION is related to the
approximate PERCENTAGE CHANGE in PRICE for a GIVEN CHANGE IN YIELD
• Because for all OPTION FREE BONDS, MODIFIED DURATION is POSITIVE, Equation (8) states that” there is an INVERSE RELATIONSHIP between MODIFIED DURATION __and_ “the approximate PERCENTAGE CHANGE IN PRICE __for_ a GIVEN CHANGE IN YIELD” ( this flows from the principle that ‘Bond prices move Opposite to the direction of change in Interest rates)
• Duration in YEARS= Duration in ‘m’ periods per year• m •
88• Fredrick Macaulay coined this term and used this measure as a PROXY
FOR THE AVERAGE LENGTH OF TIME , a Bond is OUTSTANDING
• To deal with the AMBIGUITY of the ‘Maturity of a Bond making several payments”, we need a measure of AVERAGE MATURITY of the Bond’s promised cash flows to serve as a useful statistic of the “EFFECTIVE MATURITY OF A BOND”
• This measure serves as a GUIDE to the SENSITIVITY of a Bond to INTEREST RATE CHANGES, because we have noted that’the Price sensitivity tends to INCREASE WITH INCREASE IN TIME TO MATURITY”
• Macaulay Duration is computed as the ‘weighted average’ of the TIMES to EACH PAYMENT ( whether coupon or Principal payment) made by the Bond. Therefore, the weight associated with Each PAYMENT TIME should be related to the importance of THAT payment TO the PRICE of the BOND. Therefore the WEIGHT applied to EACH “payment time” should be “ the PROPORTION of the TOTAL Value of the Bond that is ACCOUNTED for, BY THAT PAYMENT” . THIS PROPORTION IS THE “Present value of the payment DIVIDED BY the Bond Price!
89Macaulay’s duration and Modified duration for 6 hypothetical Bonds –Bond Macaulay Duration Modified Duration9% 5Yr 4.13 3.969% 25 yrs 10.33 9.886% 5yr 4.35 4.136% 25 yr 11.10 10.620% 5yr 5.00 4.780% 25yr 25.00 23.92Rather than use Equation (5) and then Eqn. (7) to obtain Modified
duration, we derive an Alternate Formula that Does NOT require Extensive calculations required by Eqn.(5). This is done by re-writing the PRICE of a BOND in terms of 2 Components--;
1) The PV of the ANNUITY where the annuity is the Sum of the Coupon payments
2) The PV of the PAR VALUE
90• The Price of the Bond can be written as• P= C { {1- __1__ }• {___{ (1+y)^n }__ + ___100____• { y } (1+y)^n• Macaulay & modified duration of a 5Yr 65 Bond selling to yield 9%• Coupon rate 6%, 5Yr, Par value 100$• Period Cash flow PV of [email protected]% PV of CF t*PV C/F• 1 2 3 4=2*3 5=4*1• 1 $3.00 0.956937 2.870813 2.870813• 2 do 0.915729 2.747190 5.49437• 3 do 0.872696 2.628890 7.8866• 4 do 0.838561 2.515684 10.06273• 5 do 0.802451 2.407353 12.03676• -• 10 103 0.643927 66.324551 663.24551• 88.130923 765.8952
91• Macaulay Duration in Half-Years = 765.8950 / 88.130923= 8.69
• ____do___ in years= 8.69/2 =4.35
• Modified Duration 4.35/(1+.0450) = 4.16
• NOTE Column 2= 6$/2 =3$, Column 3= 1/(1.0450)^n The Period in Column is Half years ANOTHER METHOD 8% coupon Bond
Period Time to Cash flow PV of Weight Column (B)
• payment Cash Flow times Column (E)
• A B C D E F=B*E
• 1 0.50 yr 40 38.095 0.0395** 0.0197
• 2 1.0 40 36.281^^ 0.0376 0.0376
• 3 1.50 40 34.554 0.0358 0.0537
• 4 2.00 1040 855.611 0.8871 1.7741
• sum 964.54 1.000 1.8852(Duration)
• Weight=PV of Each Payment /Bond Price= D column / D Total
• =38.095 /964,54= 0.0395**
• 40/ (1.05)^n = 40/1.05^2 = 36.281^^Yield assumed is10%or5% for half year
92• For a Zero coupon bond since cash flows are zero till period of
maturity( say 2 yrs) the Duration is Obviously 2 yrs( the full weight 1.00 comes in year 2 ; till then it is 0 all the way!) So the Duration of a zero coupon bond is EQUAL to its TIME TO MATURITY
• The coupon payments being made before maturity, the Effective Maturity( weighted average maturity) is LESS than Actual time to maturity, for all coupon bonds
• Duration is a Key concept in FIXED INCOME PORTFOLIO MANAGEMENT for at least 3 reasons—
• 1) It is a simple statistic of the EFFECTIVE AVERAGEMATURITY of the Portfolio
• 2)It is an essential tool in IMMUNIZING a Portfolio from Interest rate risk
• 3) It is a measure of the Interest rate sensitivity of a Portfolio
93• It can be shown that Delta P/ P= --D* delta Y --(1)• Where D* is called MODIFIED DURATION. This in turn is
something like the “Present value of Duration’• Delta y=delta (1+y)=change in the YTM• It can be shown that when interest rates change, the
PROPORTIONAL CHANGE in a Bond’s Price can be related to a “change in its YTM” i.e “y”
• Referring to the Equation(1), we can say that a percentage change in a Bond’s Price is nothing but “the Product of Modified Duration and a change in the Bond’s Yield to maturity
• Because the Percentage change in a Bond’s Price is PROPORTIONAL to the MODIFIED DURATION, the Modified duration is a Natural measure of the Bond’s Response to changes in interest rates
94• Consider a 2 year maturity 8% coupon bond making semi-annual payments
and selling at $964.54, for a YTM of 10%. The Duration of the Bond is 1.8852 years (seen earlier) or 1.8852* 2=3.7704 periods (half-years) with a per period interest rate of 5%. The Modified duration of the bond is 3.7704/ (1+r)= 3.7704 /1.05=3.591 periods
• Suppose the semi annual interest rate increases from 5.0 to 5.01%. The Bond Prices should FALL BY delta P/ P=-D*delta y= (3.591)*0.01=--0.3591%
• Computing the Price change of the bond DIRECTLY, the Bond PRICE will Fall by 964.54* .03591/100 = 0.3464
• Price= 964.54-0.3464=964.1936$
• The Zero Coupon 2year Bond initially sells for 1000/ (1.05)^3.7704= 831.9704. At HIGHER YIELDS the Bond sells for1000/(1.051)^3.7704= 831.6717$ This Price ALSO FALLS BY 0.0359%
• We Conclude that Bonds with EQUAL DURATION do in fact have EQUAL INTEREST RATE SENSITIVITY! And that (atleast for small changes in Yields) the PERCENTAGE PRICE CHANGE is the “Modified Duration TIMES the Change in Yield delta P/P= -D*delta y
95
• What is it that determines “DURATION”?• Malkeil’s Bond Price relations seen earlier characterizes the
DETERMINANTS of INTEREST RATE SENSITIVITY. Duration helps us Quantify that sensitivity This helps us in devising Investment Strategies!
• Put up the Chart Pg 8 here
96• Rule 1 for Duration• The Duration of a Zero coupon Bond is Equal to its Maturity-
already seen• Rule 2 for Duration• Holding MATURITY CONSTANT, a Bond’s Duration is HIGHER
when the COUPON RATE IS LOWER!• Let’s look at a Bond which offers HIGHER Cash flows in the
Initial Years and a Lower Cash Flow in the LATER years. The bond duration for this Bond will be LOWER than the Duration of another Bond that HAS THE SAME ‘time to maturity” but whose initial cash flows are lower and later cash flows higher. This is because Higher weights are attached to higher cash flows in the initial periods in the earlier bonds.
• This is corroborated by the Chart in #95 where the plot of the 15% coupon lies below the plot of the 3% coupon( though both have the same YTM)!!
97• Rule 3 for Duration• Holding Coupon rate Constant, a Bond’s Duration
GENERALLY increases with it’s TIME TO MATURITY• Duration ALWAYS increases for Bond’s selling at PAR or at
PREMIUM compared to the one selling at discount• This Property corresponds to Malkiel’s 3rd relationship. What is
however surprising is that the Duration NEED NOT necessarily INCREASE with Time to Maturity! For some Deep Discount Bonds THE DURATION MAY ACTUALLY FALL WITH INCREASE IN MATURITY!!!. Generally however—excepting these discount Bonds– it is safe to assume that Bond duration INCREASES with increase in Time to maturity!
98
99
100
101- Working capital
• C/A refers to those Assets that get converted to Cash within 1 year, WITHOUT--;
• A) undergoing a diminution in value• B) disrupting the Operations of the firm• Current liabilities are those liabilities which are
intended, at their inception, to be paid 1) in the Ordinary course of the business within 1 year 2) from out of the current assets 3) or, from out of the Earnings of the firm
• Gross working capital= Total current assets• Net W/c= C/A- C/L
102• Sometimes, the term ‘Working capital’ is referred to as
that PORTION of the Current assets that are financed from out of LONG TERM FUNDS ( because the ‘differential amount of the ( excess) of the C/A over the C/l is a ‘PERMANENT FEATURE’ of any going concern!
• The 3 basic features reflecting the LIQUIDITY in a firm are– 1) Current Ratio 2) Acid test ratio 3) NWC
• NWC helps in comparing the Liquidity of the SAME firm “over time” The Goal of a Finance manager is to maintain current assets and Current Liabilities in such a way that an a acceptable level of NWC is maintained Greater the NWC or, the Excess of C/A over C/L, the Greater the ability of the firm to pay up obligations when they become due. It is the non-synchronous nature of cash flows that makes NWC necessary! Cash ‘outflows’ especially payments of current liabilities, are quite predictable—contd.
103• It is only the Current assets—especially the ‘receipts
from the Debtors” that is quite ‘unpredictable’! NWC funding also becomes necessary as cash outflows and cash inflows do not match due in 1) quantum 2) Timings
• Trade off between Profitability and Risks
• NWC has a bearing on Profitability and risks. Risk here refers to the ‘risks of solvency” i.e. the probability of technically ‘turning insolvent’ In evaluating the profitability- risk trade-off, the 3 assumptions that are generally true are– 1) We are referring to a manufacturing firm 2) That C/A are “ less profitable ‘ than Fixed assets 3) That short term funds are LESS expensive than long term funds
104
• Effect of the level of Current assets on the PROFITABILTY – RISK Trade- off
• Total assets= C/A + F/A; an increase in C/A means a decrease in F/A
• Total liabilities= C/L + Long term liabilities; an increase in C/l means a decrease in LTL!
• The Quantum of Total assets as also the Quantum of Total Liabilities are FIXED!
105
• Liquidity Profitability Risks• > C/a/T/a > < <• < C/a/T/a < > >• > C/l /T/l* < > >• < C/l / T/l > < <• * If Current liabilities increases, the long
term Liabilities decrease, making the firm Less Liquid; lesser the liquidity, greater the profitability!
106
• Balance sheet• C/L Rs 3200 C/A Rs 5400• LTL Rs 4800 F/A Rs 8600• Equity _Rs 6000_ _______• Rs 14000 Rs 14000• If the co. earns around 2% on C/A and
12%on F/A, it earns Rs 1140/ in all.The NWC currently is around Rs 2200/ and the C/A/ T/A is 5400/14000= 0.386
107• Assuming that the co. increases its
investment by Rs 600/ in C/A ( and thus Rs 600/ less in F/A), the ratio of CA/TA will be 0.429(i.e 6000/ 14000). The Profits on total assets will be Rs 1080( 0.02* Rs 6000 + 0.12* 8000). Thus CA/TA ratio has increased from 0.386 to 0.429, the total profits decreased from Rs 1140 to Rs 1080/. The Risk measured by NWC decreases since the C/A have increased leading to better liquidity
108• Effect of change in C/L on Profitability- Risk-
return trade off• A CL reflects a short term liabilities/funding. An
increase in CL indicates that outflows of cash in the short run has been curtailed. This results in interest savings-( assuming that there is no charge on late payments or even on normal credits). This enhances profitability
• Any increase in C/L assuming ‘no change’ in C/A will adversely affect ‘Net Working capital’. This leads to an increase in risks( liquidity risk) but to an increase in profitability( due to interest savings, consequent to a hike in C/L)
109• Sources of C/A Financing
• Short term sources– out of C/L
• Long term sources—Share capital, retained earnings, debentures, LT loans, Pref. S/C etc.
• The Question –How much of C/A should be funded out of LTS and How much out of C/L?
Pwc- long term funding
fwc
110• There are 3 basic approaches to determining
Working Capital funding• 1) Hedging / matching approach Hedging refers
to a Risk Reduction approach where 2 opposing transactions are carried out SIMULTANEOUSLY so that the adverse effect of one is likely to be ‘counter balanced’ by the other, Here we hedge the ‘maturities of debts’ (liabilities) WITH the ‘matching maturities of Financial Needs’ {towards (financing) Current assets}. The maturities of the sources of funds should match the Assets to be financed
111
• For the purposes of Matching approach, C/A can be classified broadly into—
• A) Those that are required in certain quantities at ‘ virtually ALL points in time’, and hence undergo virtually no change at all!—Core current assets
• B) those that FLUCTUATE over time• i) needs to be financed by long term sources
like Equity, LT debts, Retained earnings, etc.• ii) financed out of C/L
112• 2) Conservative approach• Is a Strategy where the firm finances ALL its C/A
through its Long term sources only and uses its Short term funds only to meet ‘unexpected outflows’
• 3) Trade off plan that arranges long term funds to a certain fixed percentage of ‘total working capital requirement’
• Matching approach is RISKIER than Conservative approach as, in conservative approach ALL C/Assets are financed from out of Long term Sources. Long term sources are costlier; hence Conservative Plan is a Costlier one though it is much less risky than Matching !
113-Planning W/C
• Cash-------------------Inventory• Inventory------------- Receivables• Receivables--------- Cash• _A firm that does not sell on credit does
not have the 2nd phase of the Operating cycle ________________________________
• Public utilities-------Mfg---------------Trading• The ratio of CA/TA is ‘highest’ in Trading
114• The Operating Cycle seen in#113 creates the
need for W/C. It does not however, mean that the W/C becomes Zero as soon as an Operating cycle is completed. This we’ll understand when we get the concept of Permanent & Temporary W/C.
• A certain MINIMUM LEVEL of W/C is required to keep the business going on an uninterrupted basis. This quantum of W/C is called the Core/ Permanent W/C
• Requirement beyond the Core– Temporary W/C
115• In a Growth oriented/ Expanding firm, the
Permanent/ Core W/c may not be parallel to the X axis; it will be an ‘upward sloping line’
• Determinants of W/C• I) General nature of the business• A) Cash/ credit sales? B) Mfg. / service
industry? C) Trading Concern?• Mfg Spare parts inventory; Services no
inventory! Trading-High inventory but, scope for cash rotation.
• In Hotels CA/TA is the Lowest; everything gets converted to cash fast!
116• 2) Production Cycle• Is basically the time span between the procurement of Raw materials and
Completion of Finished goods. The Longer the time span, the higher the working capital required. Distilleries in contradistinction to Hotels involve heavy investment in Inventory and thus in Working capital. Heavy industries and Projects involve jobs that lock huge sums of money for extended periods of time; they generally insist on ‘advances’ from the Customers, to tide over this situation
• 3) Business cycle• Business fluctuations lead to 1) Cyclical changes in W/C 2) Seasonal changes
in W/c—Sugar Industry The variation may be in 2 directions 1) Upward phase Leading to boom in credit sales and a consequent lock up of heavy funds in W/C 2) Business Downswing; Decline in Business leads to a release of money locked up in W/C with Unfavorable consequences on Profits
• 4) Production policy; For Seasonal goods , what is the production policy that one should adopt?
• A) Produce only seasonally OR Produce throughout the year even though at lower capacity utilization
• B)Can one MATCH Production with Demand at various points of time ?• contd.
117• 4) Credit policy--- a) Credit from Suppliers –Quantum, Timing and the period
of credit vis a vis b) Credit Sales regarding the above three dimensions• 30days }• Supplier credit}------Prodn. 20 days---F/G------10 days credit-----Receivables• Matching of Credit period received with the credit period given out!• Decision variable; The contribution lost on sales vs Interest savings• 5) Growth & Expansion > Growth --- > Inventory• > Expansion, > greater the Stock of Inventory• 6) Vagaries in availability of Raw materials. Sugarcane---Only 3 months;
Evaporation losses in some stocked materials could be high• 7) Profit levels High profits could become ‘easy source of funds ‘ for W/c .
They could get locked up easily!• 8) Dividend policy Higher the dividend, lesser the retention and lesser the
Source for W/C• 9) Depreciation policy Depreciation could go into Sinking fund Investments ---
in which case they are Not available for working Capital except by way of a borrowing using these Investments as ‘collaterals’ Otherwise Depreciation holds back profits that can be used as a means for financing Current assets
• ----------End. GOOD LUCK FOR YOUR EXAMS>----------------
WORKING CAPITAL MANAGEMENT
