LECTURE 1 : THE BASICS (Asset Pricing and Portfolio Theory)

Contents Prices, returns, HPR Prices, returns, HPR Nominal and real variables Nominal and real variables Basic concepts : compounding, discounting, NPV, IRR Basic concepts : compounding, discounting, NPV, IRR Key questions in finance Key questions in finance Investment appraisal Investment appraisal Valuating a firm Valuating a firm

Calculating Rates of Return Financial data is usually provided in forms of prices (i.e. bond price, share price, FX, stock price index, etc.) Financial data is usually provided in forms of prices (i.e. bond price, share price, FX, stock price index, etc.) Financial analysis is usually conducted on rate of return Financial analysis is usually conducted on rate of return –Statistical issues (spurious regression results can occur) –Easier to compare (more transparent)

Prices  Rate of Return Arithmetic rate of return Arithmetic rate of return R t = (P t - P t-1 )/P t-1 Continuous compounded rate of return Continuous compounded rate of return R t = ln(P t /P t-1 ) –get similar results, especially for small price changes –However, 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  Rate of Return Assume 3 period horizon. Let Assume 3 period horizon. Let P 0 = 100 P 0 = 100 P 1 = 110 P 1 = 110 P 2 = 100 P 2 = 100 Then : Then : –Geometric : R 1 = ln(110/100) = ??? and R 2 = ln(100/110) = ??? –Arithmetic : R 1 = ( )/100 = ??? and R 2 = ( )/110 = ???

Nominal and Real Returns W 1 r  W 1 /P 1 g = [(W 0 r P 0 g ) (1+R)] / P 1 g W 1 r  W 1 /P 1 g = [(W 0 r P 0 g ) (1+R)] / P 1 g (1+R r )  W 1 r /W 0 r = (1 + R)/(1+  ) (1+R r )  W 1 r /W 0 r = (1 + R)/(1+  ) R r   W 1 r /W 0 r = (R –  )/(1+  )  R –  R r   W 1 r /W 0 r = (R –  )/(1+  )  R –  Continuously compounded returns Continuously compounded returns ln(W 1 r /W 0 r )  R c r = ln(1+R) – ln(P 1 g /P 0 g ) = R c -  c

Foreign Investment W 1 = W 0 (1 + R US ) S 1 / S 0 W 1 = W 0 (1 + R US ) S 1 / S 0 R (UK  US)  W 1 /W 0 – 1 = R US +  S 1 /S 0 + R US (  S 1 /S 0 )  R US + R FX R (UK  US)  W 1 /W 0 – 1 = R US +  S 1 /S 0 + R US (  S 1 /S 0 )  R US + R FX Nominal returns (UK residents) = local currency (US) returns + appreciation of USD Nominal returns (UK residents) = local currency (US) returns + appreciation of USD Continuously compounded returns Continuously compounded returns R c (UK  US) = ln(W 1 /W 0 ) = R c US +  s

Summary : Risk Free Rate, Nominal vs Real Returns Risk Free Asset Risk Free Asset –Risk free asset = T-bill or bank deposit –Fisher equation : Nominal risk free return = real return + expected inflation Real return : rewards for ‘waiting’ (e.g 3% - fairly constant) Indexed bonds earn a known real return (approx. equal to 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 premium risk premium = “market risk” + liquidity risk + default risk

FTSE All Share Index : (Nominal) Stock Price

FTSE All Share Index : (Nominal) Returns

FTSE All Share Index : (Real) Stock Price

FTSE All Share Index : (Real) Returns

Holding Period Return (Yield) : Stocks H t+1 = (P t+1 –P t )/P t + D t+1 /P t H t+1 = (P t+1 –P t )/P t + D t+1 /P t 1+H t+1 = (P t+1 + D t+1 )/P t 1+H t+1 = (P t+1 + D t+1 )/P t Y = A(1+H t+1 (1) )(1+H t+2 (1) ) … (1+H t+n (1) ) Y = A(1+H t+1 (1) )(1+H t+2 (1) ) … (1+H t+n (1) ) Continuously compounded returns Continuously compounded returns –One period h t+1 = ln(P t+1 /P t ) = p t+1 – p t –N periods h t+n = p t+n - p t = h t + h t+1 + … + h t+n –where p t = ln(P t )

Finance : What are the key Questions ?

‘Big Questions’ : Valuation How do we decide on whether … How do we decide on whether … –… to undertake a new (physical) investment project ? –... to buy a potential ’takeover target’ ? –… to buy stocks, bonds and other financial instruments (including foreign assets) ? To determine the above we need to calculate the ‘correct’ or ‘fair’ value V of the future cash flows from these ‘assets’. To determine the above we need to calculate the ‘correct’ or ‘fair’ value V of the future cash flows from these ‘assets’. If V > P (price of stock) or V > capital cost of project then purchase ‘asset’.

‘Big Questions’ : Risk How do we take account of the ‘riskiness of the future cash flows when determining the fair value of these assets (e.g. stocks, investment project) ? How do we take account of the ‘riskiness of the future cash flows when determining the fair value of these assets (e.g. stocks, investment project) ? A. : Use Discounted Present Value Model (DPV) where the discount rate should reflect the riskiness of the future cash flows from the asset  CAPM A. : Use Discounted Present Value Model (DPV) where the discount rate should reflect the riskiness of the future cash flows from the asset  CAPM

‘Big Questions’ Portfolio Theory : Portfolio Theory : –Can we combine several assets in order to reduce risk while still maintaining some ‘return’ ?  Portfolio theory, international diversification Hedging : Hedging : –Can we combine several assets in order to reduce risk to (near) zero ?  hedging with derivatives Speculation : Speculation : –Can ‘stock pickers’ ‘beat the market’ return (i.e. index tracker on S&P500), over a run of bets, after correcting for risk and transaction costs ?

Compounding, Discounting, NPV, IRR

Time Value of Money : Cash Flows Project 1 Time Project 2 Project 3

Example : PV, FV, NPV, IRR Question : How much money must I invest in a comparable investment of similar risk to duplicate exactly the cash flows of this investments ? Case : You can invest in a company and your investment (today) of £ 100,000 will be worth (with certainty) £ 160,000 one year from today. Similar investments earn 20% p.a. !

Example : PV, FV, NPV, IRR (Cont.) -100, ,000 r = 20% (or 0.2) Time 0 Time 1

Compounding Example : Example : A 0 is the value today (say $1,000) r is the interest rate (say 10% or 0.1) Value of $1,000 today (t = 0) in 1 year : TV1 = (1.10) $1,000 = $1,100 TV1 = (1.10) $1,000 = $1,100 Value of $1,000 today (t = 0) in 2 years : TV2 = (1.10) $1,100 = (1.10) 2 $1,000 = $ 1,210. TV2 = (1.10) $1,100 = (1.10) 2 $1,000 = $ 1,210. Breakdown of Future Value $ 100 = 1 st years (interest) payments $ 100 = 2 nd year (interest) payments $ 10 = 2 nd year interest payments on $100 1 st year (interest) payments

Discounting How much is $1,210 payable in 2 years worth today ? How much is $1,210 payable in 2 years worth today ? –Suppose discount rate is 10% for the next 2 years. –DPV = V 2 / (1+r) 2 = $1,210/(1.10) 2 –Hence DPV of $1,210 is $1,000 –Discount factor d 2 = 1/(1+r) 2

Compounding Frequencies Interest payment on a £10,000 loan (r = 6% p.a.) –Simple interest £ 10,000 ( ) = £ 10,600 –Half yearly compounding £ 10,000 ( /2) 2 = £ 10,609 –Quarterly compounding £ 10,000 ( /4) 4 = £ 10,614 –Monthly compounding £ 10,000 ( /12) 12 = £ 10,617 –Daily compounding £ 10,000 ( /365) 365 = £ 10, –Continuous compounding £ 10,000 e 0.06 = £ 10,618.37

Effective Annual Rate (1 + R e ) = (1 + R/m) m

Simple Rates – Continuous Compounded Rates Ae Rc(n) = A(1 + R/m) mn R c = m ln(1 + R/m) R = m(e Rc/m – 1)

FV, Compounding : Summary Single payment Single payment FV n = $A(1 + R) n FV n m = $A(1 + R/m) mn FV n c = $A e Rc(n)

Discounted Present Value (DPV) What is the value today of a stream of payments (assuming constant discount factor and non-risky receipts) ? What is the value today of a stream of payments (assuming constant discount factor and non-risky receipts) ? DPV = V 1 /(1+r) + V 2 /(1+r) 2 + … = d 1 V 1 + d 2 V 2 + … d = ‘discount factor’ < 1 Discounting converts all future cash flows on to a common basis (so they can then be ‘added up’ and compared).

Annuity Future payments are constant in each year : FV i = $C Future payments are constant in each year : FV i = $C First payment is at the end of the first year First payment is at the end of the first year Ordinary annuity Ordinary annuity DPV = C  1/(1+r) i Formula for sum of geometric progression Formula for sum of geometric progression DPV = CA n,r where A n,r = (1/r) [1- 1/(1+r) n ] DPV = C/rfor n  ∞

Investment Appraisal (NPV and DPV) Consider the following investment Consider the following investment –Capital Cost : Cost = $2,000 (at time t= 0) –Cashflows : Year 1 : V 1 = $1,100 Year 2 : V 2 = $1,210 Net Present Value (NPV) is defined as the discounted present value less the capital costs. Net Present Value (NPV) is defined as the discounted present value less the capital costs. NPV = 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 (IRR) Alternative way (to DPV) of evaluating investment projects Alternative way (to DPV) of evaluating investment projects Compares expected cash flows (CF) and capital costs (KC) Compares expected cash flows (CF) and capital costs (KC) –Example : KC = - $ 2,000(t = 0) CF1 = $ 1,100(t = 1) CF2 = $ 1,210(t = 2) NPV (or DPV) = -$2,000 + ($ 1,100)/(1 + r) 1 + ($ 1,210)/(1 + r) 2 IRR : $ 2,000 = ($ 1,100)/(1 + y) 1 + ($ 1,210)/(1 + y) 2

Graphical Presentation : NPV and the Discount rate Discount (loan) rate NPV 0 8%10%12% Internal rate of return

Investment Decision Invest in the project if : Invest in the project if : DPV > KCor NPV > 0 IRR > r if DPV = KC or if IRR is just equal the opportunity cost of the fund, then investment project will just pay back the principal and interest on loan. If DPV = KC  IRR = r If DPV = KC  IRR = r

Summary of NPV and IRR NPV and IRR give identical decisions for independent projects with ‘normal cash flows’ NPV and IRR give identical decisions for independent projects with ‘normal cash flows’ For cash flows which change sign more than once, the IRR gives multiple solutions and cannot be used  use NPV For cash flows which change sign more than once, the IRR gives multiple solutions and cannot be used  use NPV For mutually exclusive projects use the NPV criterion For mutually exclusive projects use the NPV criterion

References Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapter 1 Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapter 1 Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 3 and 11 Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 3 and 11

END OF LECTURE