LECTURE 1 : THE BASICS

LECTURE 1 :LECTURE 1 :

THE BASICSTHE BASICS

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

Prices, returns, HPRPrices, returns, HPR Nominal and real variablesNominal and real variables Basic concepts : compounding, Basic concepts : compounding,

discounting, NPV, IRRdiscounting, NPV, IRR Key questions in financeKey questions in finance Investment appraisal Investment appraisal Valuating a firmValuating a firm

Calculating Rates of Calculating Rates of ReturnReturn Financial data is usually provided in Financial data is usually provided in

forms of prices (i.e. bond price, forms of prices (i.e. bond price, share price, FX, stock price index, share price, FX, stock price index, etc.)etc.)

Financial analysis is usually Financial analysis is usually conducted on rate of return conducted on rate of return – Statistical issues (spurious regression Statistical issues (spurious regression

results can occur)results can occur)– Easier to compare (more transparent)Easier to compare (more transparent)

Prices Prices Rate of Rate of ReturnReturn

Arithmetic rate of returnArithmetic rate of returnRRtt = (P = (Ptt - P - Pt-1t-1)/P)/Pt-1t-1

Continuous compounded rate of returnContinuous compounded rate of returnRRtt = ln(P = ln(Ptt/P/Pt-1t-1))

– get similar results, especially for small price get similar results, especially for small price changeschanges

– However, geometric rate of return preferredHowever, geometric rate of return preferred more economic meaningful (no negative prices)more economic meaningful (no negative prices) symmetric (important for FX)symmetric (important for FX)

Exercise : Prices Exercise : Prices Rate of ReturnRate of Return Assume 3 period horizon. Let Assume 3 period horizon. Let

PP00 = 100 = 100

PP11 = 110 = 110

PP22 = 100 = 100

Then : Then : – Geometric : Geometric :

RR11 = ln(110/100) = ??? and R = ln(110/100) = ??? and R22 = ln(100/110) = ??? = ln(100/110) = ???

– Arithmetic : Arithmetic : RR11 = (110-100)/100 = ??? and R = (110-100)/100 = ??? and R22 = (100-110)/110 = = (100-110)/110 =

??????

Nominal and Real Nominal and Real Returns Returns WW11

rr W W11/P/P11gg = [(W = [(W00

rrPP00gg) (1+R)] / P) (1+R)] / P11

gg

(1+R(1+Rrr) ) W W11rr/W/W00

rr = (1 + R)/(1+ = (1 + R)/(1+))

RRrr WW11rr/W/W00

rr = (R – = (R – )/(1+)/(1+) ) R – R –

Continuously compounded returns Continuously compounded returns ln(Wln(W11

rr/W/W00rr) ) R Rcc

rr = ln(1+R) – ln(P = ln(1+R) – ln(P11gg/P/P00

gg) ) = R= Rcc - - cc

Foreign InvestmentForeign Investment

WW11 = W = W00(1 + R(1 + RUSUS) S) S11 / S / S00 R (UK R (UK US) US) W W11/W/W00 – 1 = R – 1 = RUSUS + +

SS11/S/S00 + R + RUSUS((SS11/S/S00) ) R RUSUS + R + RFXFX

Nominal returns (UK residents) = Nominal returns (UK residents) = local currency (US) returns + local currency (US) returns + appreciation of USDappreciation of USD

Continuously compounded returns Continuously compounded returns RRcc (UK (UK US) = ln(W US) = ln(W11/W/W00) = R) = Rcc

USUS + + s s

Summary : Risk Free Summary : Risk Free Rate, Nominal vs Real Rate, Nominal vs Real ReturnsReturns

Risk Free Asset Risk Free Asset – Risk free asset = T-bill or bank depositRisk free asset = T-bill or bank deposit– Fisher equation : Fisher equation :

Nominal risk free return = real return + expected inflation Nominal risk free return = real return + expected inflation

Real return : rewards for ‘waiting’ (e.g 3% - fairly constant)Real return : rewards for ‘waiting’ (e.g 3% - fairly constant)

Indexed bonds earn a known real return (approx. equal to Indexed bonds earn a known real return (approx. equal to the long run growth rate of real GDP). the long run growth rate of real GDP).

Nominal Risky Return (e.g. equity)Nominal Risky Return (e.g. equity)Nominal “risky” return = risk free rate + risk premiumNominal “risky” return = risk free rate + risk premium

risk premium = “market risk” + liquidity risk + default riskrisk premium = “market risk” + liquidity risk + default risk

FTSE All Share Index : FTSE All Share Index : (Nominal) Stock Price (Nominal) Stock Price

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Holding Period Return Holding Period Return (Yield) : Stocks(Yield) : Stocks HHt+1t+1 = (P = (Pt+1t+1–P–Ptt)/P)/Ptt + D + Dt+1t+1/P/Ptt

1+H1+Ht+1t+1 = (P = (Pt+1t+1 + D + Dt+1t+1)/P)/Ptt

Y = A(1+HY = A(1+Ht+1t+1(1)(1))(1+H)(1+Ht+2t+2

(1)(1)) … (1+H) … (1+Ht+nt+n(1)(1)) )

Continuously compounded returns Continuously compounded returns – One period hOne period ht+1t+1 = ln(P = ln(Pt+1t+1/P/Ptt) = p) = pt+1t+1 – p – ptt

– N periods hN periods ht+nt+n = p = pt+n t+n - p- ptt = h = htt + h + ht+1t+1 + … + h + … + ht+nt+n

– where pwhere ptt = ln(P = ln(Ptt) )

Finance : What are the Finance : What are the key Questions ? key Questions ?

‘‘Big Questions’ : Big Questions’ : Valuation Valuation How do we decide on whether … How do we decide on whether …

– … … to undertake a new (physical) investment project ?to undertake a new (physical) investment project ?– ... to buy a potential ’takeover target’ ?... to buy a potential ’takeover target’ ?– … … to buy stocks, bonds and other financial to buy stocks, bonds and other financial

instruments (including foreign assets) ? instruments (including foreign assets) ? To determine the above we need to calculate To determine the above we need to calculate

the ‘correct’ or ‘fair’ value V of the future cash the ‘correct’ or ‘fair’ value V of the future cash flows from these ‘assets’.flows from these ‘assets’.

If V > P (price of stock) or V > capital cost If V > P (price of stock) or V > capital cost of project then purchase ‘asset’. of project then purchase ‘asset’.

‘‘Big Questions’ : RiskBig Questions’ : Risk

How do we take account of the ‘riskiness of How do we take account of the ‘riskiness of the future cash flows when determining the the future cash flows when determining the fair value of these assets (e.g. stocks, fair value of these assets (e.g. stocks, investment project) ? investment project) ?

A. : Use Discounted Present Value Model A. : Use Discounted Present Value Model (DPV) where the discount rate should reflect (DPV) where the discount rate should reflect the riskiness of the future cash flows from the riskiness of the future cash flows from the asset the asset CAPM CAPM

‘‘Big Questions’Big Questions’

Portfolio Theory : Portfolio Theory : – Can we combine several assets in order to reduce Can we combine several assets in order to reduce

risk while still maintaining some ‘return’ ? risk while still maintaining some ‘return’ ? Portfolio theory, international diversification Portfolio theory, international diversification

Hedging : Hedging : – Can we combine several assets in order to reduce Can we combine several assets in order to reduce

risk to (near) zero ? risk to (near) zero ? hedging with derivatives hedging with derivatives

Speculation : Speculation : – Can ‘stock pickers’ ‘beat the market’ return (i.e. Can ‘stock pickers’ ‘beat the market’ return (i.e.

index tracker on S&P500), over a run of bets, after index tracker on S&P500), over a run of bets, after correcting for risk and transaction costs ? correcting for risk and transaction costs ?

Compounding, Compounding, Discounting, NPV, IRRDiscounting, NPV, IRR

Time Value of Money : Time Value of Money : Cash Flows Cash Flows

Project 1

Time

Project 2

Project 3

Example : PV, FV, NPV, Example : PV, FV, NPV, IRRIRR

Question :Question : How much money must I invest How much money must I invest in a comparable investment of similar risk in a comparable investment of similar risk to duplicate exactly the cash flows of this to duplicate exactly the cash flows of this investments ? investments ?

Case :Case : You can invest in a company and your You can invest in a company and your investment (today) of £ 100,000 will be investment (today) of £ 100,000 will be worth (with certainty) £ 160,000 one year worth (with certainty) £ 160,000 one year from today. from today.

Similar investments earn 20% p.a. !Similar investments earn 20% p.a. !

Example : PV, FV, NPV, Example : PV, FV, NPV, IRR (Cont.)IRR (Cont.)

-100,000

+ 160,000r = 20% (or 0.2)

Time 0

Time 1

CompoundingCompounding

Example : Example : AA00 is the value today (say $1,000) is the value today (say $1,000)r is the interest rate (say 10% or 0.1)r is the interest rate (say 10% or 0.1)Value of $1,000 today (t = 0) in 1 year : Value of $1,000 today (t = 0) in 1 year :

TV1 = (1.10) $1,000 = $1,100TV1 = (1.10) $1,000 = $1,100

Value of $1,000 today (t = 0) in 2 years : Value of $1,000 today (t = 0) in 2 years : TV2 = (1.10) $1,100 = (1.10)TV2 = (1.10) $1,100 = (1.10)22 $1,000 = $ 1,210. $1,000 = $ 1,210.

Breakdown of Future Value Breakdown of Future Value $ 100 = 1$ 100 = 1stst years (interest) payments years (interest) payments$ 100 = 2$ 100 = 2ndnd year (interest) payments year (interest) payments$ 10 = 2$ 10 = 2ndnd year interest payments on $100 1 year interest payments on $100 1stst year year (interest) payments(interest) payments

DiscountingDiscounting

How much is $1,210 payable in 2 How much is $1,210 payable in 2 years worth years worth todaytoday ? ? – Suppose discount rate is 10% for the Suppose discount rate is 10% for the

next 2 years. next 2 years.

– DPV = VDPV = V22 / (1+r) / (1+r)22 = $1,210/(1.10) = $1,210/(1.10)22

– Hence DPV of $1,210 is $1,000Hence DPV of $1,210 is $1,000

– Discount factor dDiscount factor d22 = 1/(1+r) = 1/(1+r)22

Compounding Compounding FrequenciesFrequencies

Interest payment on a £10,000 loan (r = 6% p.a.)Interest payment on a £10,000 loan (r = 6% p.a.)

– Simple interest Simple interest £ 10,000 (1 + 0.06) £ 10,000 (1 + 0.06) = £ 10,600 = £ 10,600– Half yearly compounding Half yearly compounding

£ 10,000 (1 + 0.06/2)£ 10,000 (1 + 0.06/2)22 = £ 10,609 = £ 10,609– Quarterly compoundingQuarterly compounding

£ 10,000 (1 + 0.06/4)£ 10,000 (1 + 0.06/4)44 = £ 10,614 = £ 10,614– Monthly compounding Monthly compounding

£ 10,000 (1 + 0.06/12)£ 10,000 (1 + 0.06/12)12 12 = £ 10,617= £ 10,617– Daily compounding Daily compounding

£ 10,000 (1 + 0.06/365)£ 10,000 (1 + 0.06/365)365365 = £ 10,618.31 = £ 10,618.31– Continuous compounding Continuous compounding

£ 10,000 e£ 10,000 e0.060.06 = £ 10,618.37 = £ 10,618.37

Effective Annual Rate Effective Annual Rate

(1 + R(1 + Ree) = (1 + R/m)) = (1 + R/m)mm

Simple Rates – Simple Rates – Continuous Compounded Continuous Compounded RatesRates

AeAeRc(n)Rc(n) = A(1 + R/m) = A(1 + R/m)mnmn

RRcc = m ln(1 + R/m) = m ln(1 + R/m)

R = m(eR = m(eRc/mRc/m – 1) – 1)

FV, Compounding : FV, Compounding : SummarySummary Single payment Single payment

FVFVnn = $A(1 + R) = $A(1 + R)nn

FVFVnnmm = $A(1 + R/m) = $A(1 + R/m)mnmn

FVFVnncc = $A e = $A eRc(n)Rc(n)

Discounted Present Discounted Present Value (DPV) Value (DPV)

What is the value today of a stream of What is the value today of a stream of payments (assuming constant discount factor payments (assuming constant discount factor and non-risky receipts) ? and non-risky receipts) ?

DPV DPV = V= V11/(1+r) + V/(1+r) + V22/(1+r)/(1+r)22 + … + …

= d= d11 V V11 + d + d22 V V22 + … + …

d = ‘discount factor’ < 1d = ‘discount factor’ < 1

Discounting converts all future cash flows on Discounting converts all future cash flows on to a common basis (so they can then be to a common basis (so they can then be ‘added up’ and compared). ‘added up’ and compared).

AnnuityAnnuity

Future payments are constant in each Future payments are constant in each year : FVyear : FVii = $C = $C

First payment is at the end of the first yearFirst payment is at the end of the first year Ordinary annuity Ordinary annuity

DPV = C DPV = C 1/(1+r) 1/(1+r)ii Formula for sum of geometric progression Formula for sum of geometric progression

DPV = CADPV = CAn,rn,r where Awhere An,rn,r = (1/r) [1- 1/(1+r) = (1/r) [1- 1/(1+r)nn]]

DPV = C/rDPV = C/r for n for n ∞ ∞

Investment Appraisal Investment Appraisal (NPV and DPV)(NPV and DPV)

Consider the following investment Consider the following investment – Capital Cost : Cost = $2,000 (at time t= 0)Capital Cost : Cost = $2,000 (at time t= 0)– Cashflows : Cashflows :

Year 1 : VYear 1 : V11 = $1,100 = $1,100

Year 2 : VYear 2 : V22 = $1,210 = $1,210

Net Present Value (NPV) is defined as the discounted Net Present Value (NPV) is defined as the discounted present value less the capital costs. present value less the capital costs.

NPV = DPV - CostNPV = DPV - Cost

Investment Rule : If NPV > 0 then invest in the project. Investment Rule : If NPV > 0 then invest in the project.

Internal Rate of Return Internal Rate of Return (IRR)(IRR) Alternative way (to DPV) of evaluating Alternative way (to DPV) of evaluating

investment projectsinvestment projects Compares expected cash flows (CF) and Compares expected cash flows (CF) and

capital costs (KC)capital costs (KC)– Example : Example :

KC = - $ 2,000KC = - $ 2,000 (t = 0)(t = 0)CF1 = $ 1,100CF1 = $ 1,100 (t = 1)(t = 1)CF2 = $ 1,210CF2 = $ 1,210 (t = 2)(t = 2)

NPV (or DPV) = -$2,000 + ($ 1,100)/(1 + r)NPV (or DPV) = -$2,000 + ($ 1,100)/(1 + r)11 + ($ + ($ 1,210)/(1 + r)1,210)/(1 + r)22

IRR : $ 2,000 = ($ 1,100)/(1 + y)IRR : $ 2,000 = ($ 1,100)/(1 + y)11 + ($ 1,210)/(1 + y) + ($ 1,210)/(1 + y)22

Graphical Presentation : NPV and the Discount rate

Discount (loan) rate

NPV

08%10% 12%

Internal rate of return

Investment DecisionInvestment Decision

Invest in the project if : Invest in the project if : DPV > KCDPV > KC or NPV > 0 or NPV > 0

IRR > rIRR > r

if DPV = KC or if IRR is just equal the if DPV = KC or if IRR is just equal the opportunity cost of the fund, then opportunity cost of the fund, then investment project will just pay back investment project will just pay back the principal and interest on loan. the principal and interest on loan.

If DPV = KC If DPV = KC IRR = rIRR = r

Summary of NPV and Summary of NPV and IRRIRR NPV and IRR give identical decisions NPV and IRR give identical decisions

for independent projects with ‘normal for independent projects with ‘normal cash flows’ cash flows’

For cash flows which change sign more For cash flows which change sign more than once, the IRR gives multiple than once, the IRR gives multiple solutions and cannot be used solutions and cannot be used use use NPVNPV

For mutually exclusive projects use the For mutually exclusive projects use the NPV criterionNPV criterion

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapter 1 Economics’, Chapter 1

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 3 Derivatives Markets’, Chapter 3 and 11 and 11

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