LECTURE 1 THE BASICS Asset Pricing and

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

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

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

Prices Rate of Return n Arithmetic rate of return Rt = (Pt - Pt-1)/Pt-1 n Continuous compounded rate of return Rt = ln(Pt/Pt-1) – get similar results, especially for small price changes – However, geometric rate of return preferred n n more economic meaningful (no negative prices) symmetric (important for FX)

Exercise : Prices Rate of Return n Assume 3 period horizon. Let n n P 0 = 100 P 1 = 110 P 2 = 100 Then : – Geometric : R 1 = ln(110/100) = ? ? ? and R 2 = ln(100/110) = ? ? ? – Arithmetic : R 1 = (110 -100)/100 = ? ? ? and R 2 = (100 -110)/110 = ? ? ?

Nominal and Real Returns W 1 r W 1/P 1 g = [(W 0 r. P 0 g) (1+R)] / P 1 g n (1+Rr) W 1 r/W 0 r = (1 + R)/(1+p) n Rr DW 1 r/W 0 r = (R – p)/(1+p) R – p n n Continuously compounded returns ln(W 1 r/W 0 r) Rcr = ln(1+R) – ln(P 1 g/P 0 g) = Rc - p c

Foreign Investment W 1 = W 0(1 + RUS) S 1 / S 0 n R (UK US) W 1/W 0 – 1 = RUS + DS 1/S 0 + RUS(DS 1/S 0) RUS + RFX n Nominal returns (UK residents) = local currency (US) returns + appreciation of USD n Continuously compounded returns n Rc (UK US) = ln(W 1/W 0) = Rc. US + Ds

Summary : Risk Free Rate, Nominal vs Real Returns n 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). n Nominal Risky Return (e. g. equity) Nominal “risky” return = risk free rate + 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 n Ht+1 = (Pt+1–Pt)/Pt + Dt+1/Pt 1+Ht+1 = (Pt+1 + Dt+1)/Pt Y = A(1+Ht+1(1))(1+Ht+2(1)) … (1+Ht+n(1)) n Continuously compounded returns n n – One period ht+1 = ln(Pt+1/Pt) = pt+1 – pt – N periods ht+n = pt+n - pt = ht + ht+1 + … + ht+n – where pt = ln(Pt)

Finance : What are the key Questions ?

‘Big Questions’ : Valuation n 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) ? n 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 n 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) ? n 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’ n Portfolio Theory : – Can we combine several assets in order to reduce risk while still maintaining some ‘return’ ? Portfolio theory, international diversification n Hedging : – Can we combine several assets in order to reduce risk to (near) zero ? hedging with derivatives n Speculation : – Can ‘stock pickers’ ‘beat the market’ return (i. e. index tracker on S&P 500), over a run of bets, after correcting for risk and transaction costs ?

Compounding, Discounting, NPV, IRR

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

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. ) r = 20% (or 0. 2) + 160, 000 Time 1 -100, 000

Compounding n 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 : n TV 1 = (1. 10) $1, 000 = $1, 100 Value of $1, 000 today (t = 0) in 2 years : n TV 2 = (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 n 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 (1 + 0. 06) = £ 10, 600 – Half yearly compounding £ 10, 000 (1 + 0. 06/2)2 = £ 10, 609 – Quarterly compounding £ 10, 000 (1 + 0. 06/4)4 = £ 10, 614 – Monthly compounding £ 10, 000 (1 + 0. 06/12)12 = £ 10, 617 – Daily compounding £ 10, 000 (1 + 0. 06/365)365 = £ 10, 618. 31 – Continuous compounding £ 10, 000 e 0. 06 = £ 10, 618. 37

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

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

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

Discounted Present Value (DPV) n 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 n n n Future payments are constant in each year : FVi = $C First payment is at the end of the first year Ordinary annuity DPV = C S 1/(1+r)i n Formula for sum of geometric progression DPV = CAn, r DPV = C/r where An, r = (1/r) [1 - 1/(1+r)n] for n ∞

Investment Appraisal (NPV and DPV) n 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 n Net Present Value (NPV) is defined as the discounted present value less the capital costs. NPV = DPV - Cost n Investment Rule : If NPV > 0 then invest in the project.

Internal Rate of Return (IRR) n n Alternative way (to DPV) of evaluating investment projects Compares expected cash flows (CF) and capital costs (KC) – Example : KC = - $ 2, 000 CF 1 = $ 1, 100 CF 2 = $ 1, 210 (t = 0) (t = 1) (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 NPV Internal rate of return 0 8% 10% 12% Discount (loan) rate

Investment Decision n Invest in the project if : DPV > KC or 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. n If DPV = KC IRR = r

Summary of NPV and IRR n n n 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 mutually exclusive projects use the NPV criterion

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

END OF LECTURE