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Bond Analytics
MANISH BANSAL
Jeetay Investment Pvt. Ltd.
Email: [email protected]
Phone: +91 98924 86751
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Important terms linked to debtinstruments
Face value/ Par value
Issue price (at face value or at premium/discount
to the face value) Redemption value (at face value or at
premium/discount to the face value)
Rate of interest (Coupon) and frequency Maturity of the instrument
Terms of redemption
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Risks in Bonds
Interest rate risk - Does not exist, if instrument is held tillmaturity.
Reinvestment risk - Does not exist in zeros.
Call risk
Credit risk
Liquidity risk
Event risk
Risks in international \Cross Border Bonds
Currency risk
Political risk
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Price of a Bonds
In finance, price of any financial
instrument is present value of all future
cash flows
Hence, we need following to value bonds:
Stream of cash flows and their timings
Discount rate
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Time Value of Money
One Rupee received today is worth more
than one Rupee received tomorrow
The reasons for this phenomenon are
Opportunity cost
Loss in purchasing power or Inflation
Risk of lending the money
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Concept of Compounding
Compounding means that the interest
received on a sum of money is reinvested at
the same rate and this interest also earns
interest
Let us compare two situations
Bank A pays interest @10% compounded
annually
Bank B pays interest @10% compounded
semi-annually
Which of the two offers better return???6
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Concept of Compounding
Suppose an investor invests Rs.100 in both the banks After 1 year, Bank A pays back Rs.110 being the sum of
principal (Rs.100) and interest (Rs.10)
And, Bank B pays Rs. 110.25 being the sum of:
Principal (Rs.100)
Interest for two six month periods (Rs.5+5 = 10)
Interest for six months on the first interest of Rs.5
(Rs.5 X 0.05= 0.25)
Hence, Bank B offers better return. This happens
because in compounding, interest also starts earning
interest 7
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Concept of Compounding
Compounding frequency means the number oftimes interest is deemed to be paid out in a year
Higher the compounding frequency, higher the
return for same nominal rate since, the interestis paid out faster and starts earning interest
earlier
Nominal rate (i.e., the qouted rate) along withcompounding frequency determines the effective
rate of return on an instrument
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Conversion from Nominal toEffective rate
Nominal rates are quoted in the market along with theircompounding frequency, e.g., 10% quarterly
To convert nominal rate in to effective annualized rate,we use the following formula:
re= (1+rn/k*100)^k - 1
where
re
= Effective annualized yield
rn = Nominal yield
k = Compounding frequency
For example: 12% quarterly is 12.55% annualised
12% semi-annual is 12.36% annualised 9
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Effective rate computation
Formulae for various frequencies Semi - Annual Compounding
r effective = (1 + r/(2 * 100))^2 - 1
Monthly Compoundingr effective = (1 + r/(12 * 100))^12 - 1
Daily Compounding
r effective = (1 + r/(365 * 100))^365 - 1 Continuos Compounding
r effective = exp(r/100) - 1
where r = nominal annual rate of interest 10
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Difference betweenCompounded and payable
10% compounded quarterly is not equal to 10% payable
quarterly.
Compounding assumes that the interest payable is
reinvested at the same rate for remaining life of the bond. But, in case of payable, interest or coupon is actually
paid out and this may or may not be invested at the same
rate because interest rates at the time of payment could
be different from original rate.
Hence, we can not definitely calculate the total return in
case of payable, whereas, in case of compounding, we
can find the exact total return promised. 11
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Calculating Present Value
Present Value (Single or Bullet Cash Flow)Present value is the amount that must beinvested today in order to a get a given amountat a future date.
Computation of Present Value
Cash Flow (at time t) = Ct
Rate (per period) = r
No. of periods = t
Present value = Ct /(1+r)^t
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Calculating Present Value
Present Value (Multiple Cash Flows)Present value of multiple cash flows is the sum of presentvalues of individual cash flows calculated as explainedearlier.
Computation of Present Value
Cash flow at time t) = Ct
Rate of interest (per period) = r
No. of preriods = n
Present value = Ct /(1+ r)^(t)
(t = 0 to n)
(Please note that the term 1/(1+r)^t is also called thediscount factor or PV factor for maturity t) 13
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Calculating Present ValuePeriod (years Cash Flow Discount Factor Present Value
0 0 1.0000 0.00
0.5 7.5 0.9434 7.08
1 7.5 0.8900 6.67
1.5 7.5 0.8396 6.30
2 7.5 0.7921 5.942.5 7.5 0.7473 5.60
3 7.5 0.7050 5.29
3.5 7.5 0.6651 4.99
4 7.5 0.6274 4.71
4.5 7.5 0.5919 4.445 107.5 0.5584 60.03
Total Present Value = 111.04
Coupon= 15%
Yield = 12% 14
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Calculating Future Value
Future Value (Single or Bullet Cash Flow)Future value of a sum of money is theamount an investor would get on investing
the sum for a fixed period of time at afixed rate.
Computation of Future Value
Principal (Cash flow at time=0) = PRate of interest (per period) = r
No. of periods = n
Future value = P(1+r)^n 15
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Calculating Future Value
An investor invests a sum of Rs. 100,000in a financial instrument that promises to
pay 15% per year for 5 years. Interest is
compounded semiannually.
The future value of the investment would
be:
FV = 100,000*(1+0.075)^(5*2) =
206,103.2
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Calculating Future Value Future Value (Multiple Cash Flows)
Future value of multiple cash flows is the sum of futurevalues of individual cash flows calculated as explainedearlier.
Computation of Future Value
Cash flow at time t) = CtRate of interest (per period) = r
No. of periods = n
Future value = Ct (1+ r)^(n-t)
(t = 0 to n)
(the future value factor is raised to power (n-t) since acash flow after t years will be invested for the remaining(n-t) years.)
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Calculating Future ValuePeriod
(years)
Cash Flow Future Value
Factor
Future Value at
the end of 5
years
0 0 1.7908 0.00
0.5 7.5 1.6895 12.67
1 7.5 1.5938 11.95
1.5 7.5 1.5036 11.282 7.5 1.4185 10.64
2.5 7.5 1.3382 10.04
3 7.5 1.2625 9.47
3.5 7.5 1.1910 8.93
4 7.5 1.1236 8.43
4.5 7.5 1.0600 7.955 107.5 1.0000 107.50
Total Future Value = 198.86
Coupon= 15%
Yield = 12% 18
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Calculating Bond Price
Price of a bond is equal to the present value of expected
cash flows.
Therefore, to price a bond, we need:
1. Periodic cash flows - Cash flows for a typical coupon
bearing bond would be periodic coupon and redemption
value
2. Yield (discount rate)
Required yield depends on the yield offered on
comparable securities in the market. Instruments are
compared on the basis of maturity and credit risk.
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Calculating Bond Price
Compute the price of a 16 % coupon bond,interest payable semi-annually, with 3 yearsto maturity and a par value of Rs. 1,000.
Applicable discount rate for a bond of similarcredit rating is 16.5% payable semiannually.
If this Bond is issued at an upfront discountof 5%, would you buy?
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Calculating Bond Price
No. of periods Maturity Cash flows PV factor PV1 0.5 80 0.92379 73.902 1 80 0.85338 68.273 1.5 80 0.78834 63.074 2 80 0.72826 58.265 2.5 80 0.67276 53.826 3 1080 0.62149 671.21
Price 988.5321
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Relationship BetweenCoupon, Yield and price
From pricing methodology, we can infer therelationship between Coupon Rate,Required Yield and Price:
Keeping the coupon rate constant, anincrease in yield will lead to a decrease inthe price and vice versa
Keeping the required yield constant, anincrease in coupon rate will lead to anincrease in the price and vice versa
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Relative Value AnalysisVarious yield measures are used to compare
different bonds. These are:
Current Yield: The current yield relates to the
annual coupon interest to the market price of
financial instrument.
Current yield = Annual rupee coupon interest /
Market price
Yield-to-Maturity: Yield-to-maturity is the interestrate (internal rate of return) that will make
present value of cash flows equal to price of the
financial instrument. 23
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Relative Value Analysis
Yield for a Portfolio: Yield for a portfolio of bonds
is computed by determining the cash flows for
the portfolio and interest that will make the
present value of cash flows equal to the market
value of portfolio.
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Calculation of Yield to maturity
(YTM)YTM on any investment is computed bydetermining the interest rate that will make
present value of the cash flows from theinvestment equal to the price of investment.
Yield to maturity on a bond is also calledthe Internal Rate of Return (IRR)
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Calculation of Yield to maturity
Solving for the yield (y) using the bond pricing formularequires a trial and error procedure. The objective is to find
interest rate that will make the present value of cash flows
equal to the price. The following illustration will demonstrate
the procedure.
Illustration
A financial instrument offers Rs. 2,000 in the first 2 years,Rs. 2,500 in the third year and Rs. 4,000 in the fourth year.
The price of the instrument is Rs. 7,702. What is the yield
(Internal Rate of Return) offered by this instrument ?
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Calculation of Yield to maturity
Years Cash Flow DiscountingFactor
1/(1+ r)^ n
Present Valueat 14% Yield
1 2,000 0.8772 1,754.39
2 2,000 0.7695 1,538.94
3 2,500 0.6750 1,687.43
4 4,000 0.5921 2,368.32
Total Present Value 7,349.07
Present value of the bond computed here is less than givenprice of the bond, hence the YTM should be lesser than 14%.So, we try YTM = 12%
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Calculation of Yield to maturity
Years Cash Flow DiscountingFactor1/(1+ r)^ n
Present Valueat 12% Yield
1 2,000 0.8929 1,785.71
2 2,000 0.7972 1,594.39
3 2,500 0.7118 1,779.45
4 4,000 0.6355 2,542.07
Total Present Value 7,701.62
Present value computed here is very close to given price ofthe bond, hence the YTM is slightly lesser than 12%. We canget the exact value by more trials.
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Coupon, YTM and Current Yield
For a par bond, coupon rate, YTM andCurrent yield are equal.
For a premium bondCoupon > Current yield > YTM
For a discount bondCoupon < Current Yield
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Shortcomings of Yield-to-
MaturityYield-to-Maturity (YTM) is not a good
effective return measure for an investor
because it assumes:
Intermediate cash flows to the
investor are reinvested at a rate equal tothe yield-to-maturity, and
Investor holds the bond till maturity.
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Shortcomings of Yield-to-
MaturityAs a result of the above shortfalls, investor is exposed tothe following risks:
Interest Rate Risk : If investor does not hold bond till
maturity, an increase in future interest rates could lead to
a capital loss when the bond is sold in the secondary
market.
Reinvestment Risk : Assumption of the intermediatecashflows being reinvested at the yield-to-maturity,
exposes investor to the risk that the future reinvestment
rates would be less than the yield-to-maturity.31
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Problems with YTM
YTM is a convenient summary. However,
You can only calculate it after you know a
bond's price. It only applies to a single bond.
Ideally, one should not use yield tomaturity to value coupon paying bonds.
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Correct Way to Value Bonds
Dont use YTM Use Zero Coupon Spot Rate for each cash flow
Spot Rate for a maturity is defined as the
interest on a zero coupon bond of thatmaturity
Gives a correct picture of the value of eachcash flow by eliminating intermediate cash
flows and hence eliminating the interest rateand reinvestment risk
How do we calculate Zero Coupon Spot Rates?
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Volatility in Prices of Bonds
A fundamental property of a bond is that the price ofthe bond changes inversely to the change in the yieldof the bond. The graph of the price yield relationshipfor a typical bond is given below.
Yield
Price
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Volatility in Prices of Bonds
We study the volatility in bond prices to understand theirbehavior with changes in yield.
This is very important for the risk management of bond
portfolio.
We need to measure by how much the price of a bondwill change for a given change in yield.
We have already seen that the risk of investing in
coupon paying bonds can be divided as the reinvestmentrisk and the interest rate risk.
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Volatility in Prices of Bonds
To study the bond price volatility characteristics,
consider the following illustration.
Consider the following four bonds (face value 100)where the yield is 15%:
Coupon Maturity Price
Zero 5 years 49.72
Zero 25 years 3.0415% 5 years 100.00
15% 25 years 100.00
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Volatility in Prices of Bonds(Contd.)
Yield 0%-5 yr 0%-25 yr 15%-5 yr 15%-25 yr
14.99% 0.04% 0.22% 0.03% 0.06%
15.01% -0.04% -0.22% -0.03% -0.06%
14.00% 4.46% 24.40% 3.43% 6.87%
16.00% -4.24% -19.46% -3.27% -6.10%
Look at the following data carefully. It showsthe change on prices of bonds with changes inyield
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Volatility in Prices of Bonds
(Contd.)From the data it can be seen that:
Longer the maturity, higher the moves
Lower the coupon, higher the moves (note thatlower coupon means higher average maturitysince lesser proportion of present value is paidout before maturity)
For small change in interest rate, increase anddecrease in price are of almost same magnitudebut for large change in interest rate, increase inprice is more than decrease in price.
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Pull to Par Effect
Change in the price of a bond with time For any bond, selling at premium or at discount the price
moves toward the par value as the bond approaches
maturity date.
The explanation for Pull to Par Effect derives from the
bond pricing formula.
The difference in the prices of two bonds having equal
face value arises due to the difference in the couponrates.
As the bonds move toward maturity, the present value
of the coupon payments forms a lesser proportion of
bond price. 39
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Pull to Par Effect
As the bonds move toward maturity, the present valueof the face value forms a greater proportion of the bondprice.
Hence, the discount or premium bonds will converge topar value at maturity as shown below:
Premium Bond
Discount Bond
Time
Maturity
Price
Par
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Macaulay Duration
The weighted average time to maturity of a Bond is
called Macaulay Duration.
The weights are the present value of the cashflows
Larger cash flows get more weight than smaller cash
flows.
Since their present value is lower, distant cash flows
get less weight than more immediate cash flows.
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Macaulay Duration
Macaulay duration is defined for a bond with annualcashflow Ct, yield to maturity y, and maturity T as :
Dmac ={ t*PV(Ct)}/{ PV(Ct)}
Note that the denominator is simply the sum of presentvalue of all future cash flows of the bond and hence isequal to the price (inclusive of the accrued interest).
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Macaulay Duration - An Example
Consider a 2 year, 15% s.a. bondselling at par. The duration is calculatedas follows:
Time CashFlow
DiscountingFactor
PV/Price (PV/Price)*Time
1 7.5 0.9302 0.0698 0.06982 7.5 0.8653 0.0649 0.1298
3 7.5 0.8050 0.0604 0.18114 107.5 0.7488 0.8050 3.2198
Duration (in halfyears)
3.6005
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Macaulay Duration - ZeroCoupon Bond
Consider a zero coupon bond paying $1 at time T.
Its Macaulay duration is
T/(1 + y)^T
Dmac = ----------------
1/(1 + y)^T
= T
Macaulay duration equals maturity for a zero couponbond.
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Modified Duration
Macaulays Duration (Dmac) can be modified slightly togive a better risk measure called Modified Duration (MD)
Modified Duration (MD) = Dmac/(1+y/k)*k
where k = frequency of compounding
y = yield to maturity of the bond
MD directly gives the percentage change in price with aunit change in yield.
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Factors Affecting Duration Time to maturity
Higher the time to maturity higher the durationand hence higher the interest rate risk of the bond
Coupon rate
Lower the coupon rate higher the duration and
hence higher the interest rate risk of the bond
Current level of interest rates (yield)
Lower the yield higher is the duration and hence
higher the interest rate risk of the bond Thus, Modified Duration is a very convenient interest
rate risk measure for bonds. Higher the durationhigher the interest rate risk of the bond
46
d f d
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Modified Duration as InterestRate Sensitivity
Let the initial price of a bond be P0
If the yield moves by (y-y0), the new price P1 isapproximately given by
P1 ~ P0 + (-MD)*P*(y-y0)
Using the formula for MD as derived earlier.
(~ means approximately equal to)
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Example
At a yield of 10%, a 5 year 5% annual coupon bondhas a value of 81.05 and a Modified Duration of 4.08year.
Assume the bond's yield increases from 10% to10.01%.
Use its duration to calculate the bond's change in
value.
Calculate the new value longhand, using the newyield.
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Example
Using the duration
D P = (-MD) x P x D y
= - 4.08 x 81.05 x 0.0001
= - 0.033
Thus the changed price is = 81.05 - 0.033
= 81.017
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Dollar Duration
Modified Duration can also be expressed as thechange in dollar value of the bond for a unit changein yield. This is called Dollar duration.
Dollar duration is defined by
$D = -(dP/dy) = -MD*P
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Dollar Duration
From its definition, the change in price of a bonddue to a change in yield dy is given by
dP ~ - $ D x dy
Dollar duration gives the dollar change in value for a100 basis point change in interest rates.
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Price Value of a Basis Point
The Price Value of a Basis Point (PVBP) is the pricechange of a security for a one basis point change inyield.
It is equal to Dollar Duration divided by 100.
PVBP = $D/100
For example, the 5 year bond we looked at earlier, has
Dollar duration of 3.31
PVBP of 3.31 / 100 = 0.0331
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Duration and Immunisation
Duration of a 5 year, 9% coupon bond, at a yield of9%, is 4.24 years.
Suppose we are an insurance company with a fixedcommitment in 4.24 years, for which we receive 100today.
By investing the 100 in the coupon bond, we can
immunise our returns over the next 4.24 yearsagainst shifts in interest rates.
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Using Modified Duration for
Hedging We can use Modified Duration for minimising the price
risk of bonds.
This can be done by matching the duration of a bond
portfolio with the duration of the liabilities funding thatportfolio.
If duration of the assets and liabilities are matched theportfolio is immunised against small changes in the
yield because the change in value of assets is exactlyoffset by the change in the value of liabilities.
The process of matching the duration of a bond portfolio(asset) with the liabilites that fund it, is known as
Immunisation. 54
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Using Modified Duration for
Hedging If we were to invest the money in a bond with a
shorter duration than the liability, we would besubject to reinvestment risk.
If we invested in a bond with a longer duration, wewould be subject to price risk.
Investing in a duration matched asset balancesreinvestment risk and price risk.
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Drawbacks of Using Duration
for Hedging The duration of a bond keeps changing as
the interest rates change.
the time passes.
Hence, if a portfolio is duration matched orimmunised at particular time, there is noguarantee that it will remain immunised as timepasses.
Thus immunisation has to be done on acontinuous basis which involves large transactioncosts
56
Drawbacks of Using Duration
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Drawbacks of Using Durationfor Hedging
Duration is an accurate measure only for small yieldchanges.
Duration estimates the changes in price assuming alinear relationship between the price of the bond and
the yield. But the actual relationship is non-linear. Hence, for
large changes in yield the change in price calculatedusing duration is not correct.
This is illustrated in the diagram on next slide. For asmall change in yield to y1, the actual price is veryclose to the predicted price.
But for a large change in yield to y2, the actual price
is much higher than the predicted price. 57
b k f
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Drawbacks of Using Durationfor Hedging
Yield
Price
Y0 Y1 Y2
Actual Price
PricePredicted byDuration
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Drawbacks of Using Duration
for Hedging Duration calculates the change in the value of a
portfolio assuming a parallel shift in yield curve, i.e.,
all the yields shift up or down by the same amount.
This means that duration hedged portfolios will notbe immunised against non-parallel shifts in the yieldcurve and could still lose value due to non-parallel
shifts in the yield curve. Parallel and non-parallel shifts in the yield curve are
illustrated on the next slide.
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Illustration: Shifts in Yield Curve
Yield
Maturity
Original YieldCurve
Downward
parallel Shift
Non-Parallel Shift : Steepening
Non-Parallel Shift : Flattening
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Convexity
Duration is an accurate measure only for small yieldchanges.
Can we come up with a measure that (combined withduration) allows us to do better approximation of pricethan using duration alone?
The answer is YES. The measure is called convexity.
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Convexity Duration is a measure of how price changes with
interest rates.
It is the first derivative of price with respect toyield.
Convexity measures how duration changes withinterest rates.
It is the second derivative of price with respect to
yield.1 d2P
C = --- ---------
P dy262
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Calculating Convexity
Convexity of a bond paying cashflow Ct in period t (discountedcash flow) can be obtained by the following formula -
1 ct
C = ---------- t(t+1) ----------------
k^2*(1+y/k)^2 P/(1+y)^2
(k = compounding frequency)
Convexity of a 5 year bond with coupon 5% and yield 10% is
C = 2103 / 81.05 / 1.1^2 = 21.4465
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Calculating Convexity
Time Cash Flow Disc.Factor
DCF t(t+1)*DCF
1 5 0.9091 4.5455 9.09
2 5 0.8264 4.1322 24.793 5 0.7513 3.7566 45.084 5 0.6830 3.4151 68.305 105 0.6209 65.1967 1955.90
SUM 2103.17
Price 81.05Hence, C = 2103 / 81.05 / (1 + 10%) ^ 2
= 21.4465
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C f C
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Convexity of a Zero Coupon
Bond The convexity of a zero-coupon bond can be easily
calculated from the formula :
T(T + 1)
C(zero coupon) = -----------(1 + y)^2
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Ch i P i D
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Change in Price Due to
Convexity Consider a bond with price P and convexity C. If the
yield on the bond changes by dy, the change in the priceof the bond will be given by
dP (due to convexity) = (1/2)*C*P*(dy)^2
It can be seen that the change in price due to theproperty of convexity is always positive.
Hence, convexity is a desirable property in a bond.Because of convexity the bond price rises at a faster rateand falls at lower rate with changes in the yield.
66
T t l Ch i P i f
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Total Change in Price of aBond with Yield
The total change in the price of a bond due to a change in yieldis the sum of two components
Change in price due to duration
dP = (-MD)*P*(dy)
Change in price due to convexity
dP = (1/2)*C*(dy)^2
Thus total change in price is given by
dP = (-MD)*P*(dy) + (1/2)*C*(dy)^2
This is also called the Taylors Rule of Expansion
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E ti ti P i M t
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Estimating Price Movementswith Duration and Convexity
Suppose, the price of the bond was P0 before theyield change. The new price P1 will be
P1 = P0 + (-MD)*P*(y-y0) + (1/2)*C*(dy)^2
The first term is the change in price due to the
duration or first derivative of price with respect toyield.
The second term is the change in price due to theconvexity or second derivative of price with respect
to yield. 68
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Estimating Price Movementswith Duration and Convexity
Using duration alone allows us to estimate pricemovements when yield changes are small.
Using convexity as well as duration allows us toimprove our estimates when yield movements arelarger.
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Example
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Example Consider a 5 year, 5% (annual) coupon bond, with
price 81.046 and yield 10 %. Now the yielddecreases to 8%. Calculate the new price.
Price change due to duration:
The bond's modified duration is 4.08,
dP = - 4.08 x (-.02) x 81.046 = 6.613
Price change due to convexity:
The bond's convexity is 21.447,
dP = 1/2 x 21.447 x (-.02)^2 = 0.348
Total Change = 6.961
The bond's value increases from 81.046 to 88.007
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Factors Affecting Convexity
Time to maturity
as time to maturity increases the convexity increases
Coupon rate
convexity decreases with increase in the coupon rate
Current level of interest rates (yield)
convexity decreases with increase in yield
For a given duration, the more spread out the cashflows, the higher the convexity
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Duration Vs. Convexity
Increasing the duration of a position increases itsexposure to the direction of interest rates. This isbecause the change in value of position due to duration
depends on the direction of interest rate change.
Increasing convexity increases a position's exposure tolarge movements (i.e. volatility). The direction is
unimportant. This is because the change in value ofposition due to convexity depends on the square ofinterest rate change.
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Yield Curve
Represents the plot of yield to maturity against varyingterms to maturity of bonds.
YTMs of traded bonds of varying maturities is computed
and plotted as a scatter plot. The yield curve is drawn through these points,
representing the average YTMs across terms in themarket