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LECTURE 1 :
THE BASICS
(Asset Pricing and Portfolio Theory)
Contents
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Prices, returns, HPR
Nominal and real variables
Basic concepts : compounding,
discounting, NPV, IRR
Key questions in finance
Investment appraisal
Valuating a firm
Calculating Rates of
Return
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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
– Statistical issues (spurious regression
results can occur)
– Easier to compare (more transparent)
Prices  Rate of Return
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Arithmetic rate of return
Rt = (Pt - Pt-1)/Pt-1
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Continuous compounded rate of return
Rt = ln(Pt/Pt-1)
– get similar results, especially for small price
changes
– However, geometric rate of return preferred
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more economic meaningful (no negative prices)
symmetric (important for FX)
Exercise : Prices  Rate
of Return
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Assume 3 period horizon. Let
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P0 = 100
P1 = 110
P2 = 100
Then :
– Geometric :
R1 = ln(110/100) = ??? and R2 = ln(100/110) = ???
– Arithmetic :
R1 = (110-100)/100 = ??? and R2 = (100-110)/110 = ???
Nominal and Real Returns
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W1r  W1/P1g = [(W0rP0g) (1+R)] / P1g
(1+Rr)  W1r/W0r = (1 + R)/(1+p)
Rr  DW1r/W0r = (R – p)/(1+p)  R – p
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Continuously compounded returns
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ln(W1r/W0r)  Rcr = ln(1+R) – ln(P1g/P0g)
= Rc - pc
Foreign Investment
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W1 = W0(1 + RUS) S1 / S0
R (UK  US)  W1/W0 – 1 = RUS +
DS1/S0 + RUS(DS1/S0)  RUS + RFX
Nominal returns (UK residents) = local
currency (US) returns + appreciation of
USD
Continuously compounded returns
Rc (UK  US) = ln(W1/W0) = RcUS + Ds
Summary : Risk Free Rate,
Nominal vs Real Returns
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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).
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Nominal Risky Return (e.g. equity)
Nominal “risky” return = risk free rate + risk premium
risk premium = “market risk” + liquidity risk + default risk
Apr-04
Apr-01
Apr-98
Apr-95
Apr-92
Apr-89
Apr-86
Apr-83
Apr-80
Apr-77
Apr-74
Apr-71
Apr-68
Apr-65
Apr-62
FTSE All Share Index :
(Nominal) Stock Price
3500
3000
2500
2000
1500
1000
500
0
-0.1
-0.2
-0.3
-0.4
May-04
May-01
May-98
May-95
May-92
May-89
May-86
May-83
May-80
May-77
May-74
May-71
May-68
May-65
May-62
FTSE All Share Index :
(Nominal) Returns
0.5
0.4
0.3
0.2
0.1
0
Apr-04
Apr-01
Apr-98
Apr-95
Apr-92
Apr-89
Apr-86
Apr-83
Apr-80
Apr-77
Apr-74
Apr-71
Apr-68
Apr-65
Apr-62
FTSE All Share Index :
(Real) Stock Price
25
20
15
10
5
0
-0.1
-0.2
-0.3
-0.4
May-04
May-01
May-98
May-95
May-92
May-89
May-86
May-83
May-80
May-77
May-74
May-71
May-68
May-65
May-62
FTSE All Share Index :
(Real) Returns
0.5
0.4
0.3
0.2
0.1
0
Holding Period Return
(Yield) : Stocks
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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))
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Continuously compounded returns
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– 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
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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) ?
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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
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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
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CAPM
‘Big Questions’
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Portfolio Theory :
– Can we combine several assets in order to reduce risk
while still maintaining some ‘return’ ?
 Portfolio theory, international diversification
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Hedging :
– Can we combine several assets in order to reduce risk to
(near) zero ?
 hedging with derivatives
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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
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 0
Time 1
-100,000
Compounding
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Example :
A0 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 :
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TV1 = (1.10) $1,000 = $1,100
Value of $1,000 today (t = 0) in 2 years :
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TV2 = (1.10) $1,100 = (1.10)2 $1,000 = $ 1,210.
Breakdown of Future Value
$ 100 = 1st years (interest) payments
$ 100 = 2nd year (interest) payments
$ 10 = 2nd year interest payments on $100 1st year
(interest) payments
Discounting
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How much is $1,210 payable in 2
years worth today ?
– Suppose discount rate is 10% for the
next 2 years.
– DPV = V2 / (1+r)2 = $1,210/(1.10)2
– Hence DPV of $1,210 is $1,000
– Discount factor d2 = 1/(1+r)2
Compounding Frequencies
Interest payment on a £10,000 loan (r = 6% p.a.)
– Simple interest £ 10,000 (1 +
– Half yearly compounding
£ 10,000 (1 +
– Quarterly compounding
£ 10,000 (1 +
– Monthly compounding
£ 10,000 (1 +
– Daily compounding
£ 10,000 (1 +
– Continuous compounding
£ 10,000 e0.06
0.06)
= £ 10,600
0.06/2)2
= £ 10,609
0.06/4)4
= £ 10,614
0.06/12)12
= £ 10,617
0.06/365)365 = £ 10,618.31
= £ 10,618.37
Effective Annual Rate
(1 + Re) = (1 + R/m)m
Simple Rates – Continuous
Compounded Rates
AeRc(n) = A(1 + R/m)mn
Rc = m ln(1 + R/m)
R = m(eRc/m – 1)
FV, Compounding :
Summary
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Single payment
FVn = $A(1 + R)n
FVnm = $A(1 + R/m)mn
FVnc = $A eRc(n)
Discounted Present Value
(DPV)
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What is the value today of a stream of payments
(assuming constant discount factor and non-risky
receipts) ?
DPV = V1/(1+r) + V2/(1+r)2 + …
= d1 V1 + d2 V2 + …
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
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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
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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)
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Consider the following investment
– Capital Cost : Cost = $2,000 (at time t= 0)
– Cashflows :
Year 1 : V1 = $1,100
Year 2 : V2 = $1,210
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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.
Internal Rate of Return
(IRR)
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Alternative way (to DPV) of evaluating investment
projects
Compares expected cash flows (CF) and capital
costs (KC)
– Example :
KC = - $ 2,000
CF1 = $ 1,100
CF2 = $ 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
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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.
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If DPV = KC
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IRR = r
Summary of NPV and IRR
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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
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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
END OF LECTURE