Topic 1. Project Appraisal Techniques
Safe dollar is worth more than a risky one.
Capital budgeting — the choice about the real assets in which firms should
invest.
Real assets — investment projects that generate cash in the future, but which
are not traded on financial markets like financial assets.
Fundamental objective of firms: to maximise the value of cash invested.
Investments must have 'adequate' returns = at least equal to the returns
available to investors outside the firm in the financial markets.
Capital budgeting techniques = investment decision rules used by firms to
evaluate real assets:
➔ Net Present Value (NPV);
➔ Internal Rate of Return (IRR);
➔ Payback period
Net Present Value (NPV)
The Concept of Present Value
One dollar today is worth more than a dollar tomorrow, because the dollar today
can be invested, and start to earn interest immediately -> time value of money.
➔ Present value — the value today of a cash flow received in t years time:
◆ Obeys the principle of additivity: the present value of cash flows
combined is the sum of their individual present values
𝑃𝑉 =
𝐶
(1+𝑟)
𝑡
𝑃𝑉 = ∑
𝐶
(1+𝑟)
𝑁
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟)
Perpetuity — a stream of fixed cash flows, starting one period from today and
lasting forever:
𝑃𝑉 =
𝐶
𝑟
Growing perpetuity — a perpetual cash flow stream that grows at a constant
rate g over time:
𝑃𝑉 =
𝐶
𝑟−𝑔
Annuity — stream of constant cash flows, starting one period from today and
lasting for T periods:
𝑃𝑉 =
𝐶
𝑟
× (1 −
1
(1+𝑟)
𝑇
)
𝑡
The NPV technique uses the discounting principle based on the present value to
evaluate real assets:
𝑁
𝑁𝑃𝑉 = ∑
𝐶
𝑡=1 (1+𝑟)
𝑡
−1
➔ NPV > 0 ⇒ accept // NPV < 0 ⇒ reject // NPV = 0 ⇒ free to choose
➔ Cash flows are incremental, i.e. additional ⇒ no sunk costs, because
they are incurred whether or not the project is accepted;
➔ NPV method is the optimal technique for the valuation of real assets
because it enables the management to maximise the expected wealth of
shareholders.
The rate of return used to discount the expected cash inflows has to be
the rate of return offered by equivalent investment alternatives in the capital
market ⇒ the discount rate — opportunity cost of capital, where 'opportunity'
derives from the fact that it represents the return forgone by investing in the
project rather than in financial assets.
The channelling of funds from saver-lenders to spender-borrowers makes
both parties better off if the rules are followed:
1. NPV rule: firms have to invest in real assets with a positive NPV;
2. Rate of return rule: stock- and bond- holders have to invest in securities
with a return higher than the rate of return on equivalent investments in
the capital market.
NPV and Mutually Exclusive Projects
Mutually Exclusive Projects — set of projects of which only one can be chosen at
a given time.
The additivity property implies that the value of the firm = sum of the values of
separate projects ⇒ when there are mutually exclusive projects, the NPV method
indicates that the project with the largest positive NPV should be adopted,
because it generates the largest NPV of the firm's aggregated cash flows.
shareholders can reinvest their money at the project's own internal rate ⇒
it is implied that different rates can exist for projects with the same risk;
➔ In the evaluation of mutually exclusive projects, the IRR can lead to
choices that do not maximise shareholders' wealth
➔ The additivity property does not apply to IRR
➔ Can give either more than one or no solution because of possible
changes in the structure of cash flows over time (e.g. coal-mining
investment)
𝑁𝑃𝑉 (𝑋 + 𝑌) = 𝑁𝑃𝑉 (𝑋) + 𝑁𝑃𝑉 (𝑌)
NPV and Limited Funds
When there are limited funds, the choice of the projects can be made as follows:
1. Calculate the NPV of all feasible combinations of possible projects;
2. Choose the combination with the greatest NPV.
Internal Rate of Return (IRR)
The IRR — rate at which the present values of the cash inflows associated with a
project equal the cash investment:
𝑁
𝑁𝑃𝑉 = ∑
𝑡=1
𝐶
(1+𝐼𝑅𝑅)
−𝐼=0
➔ The calculation of the IRR involves an iterative process, which can be
viewed as a trial and error procedure
➔ The absence of IRR does not indicate whether a project creates or
destroys value
➔ The choice of projects is made by comparing the IRR to a required rate of
return — hurdle rate (R*) = risk-free interest rate where cash flows are
riskless:
◆ IRR > R* ⇒ accept // IRR < R* ⇒ reject
Linear interpolation method to estimate approximate value of IRR
➔ !!! Choose close discount rates
➔ The two points IRR will be interpolated between must involve a positive
NPV and Negative NPV so that the line between the points captures the
IRR:
𝐼𝑅𝑅 = 𝑟1 +
𝑁𝑃𝑉1
𝑁𝑃𝑉1+|𝑁𝑃𝑉2|
× (𝑟2 − 𝑟1)
where r1 — lower rate; r2 — higher rate
Limitations of the IRR method
➔ The discount rate is not the market-determined opportunity cost of capital
as in the NPV: the discount rate is the IRR ⇒ IRR assumption:
Payback Period
PP method evaluates real assets based on the number of years needed to
recover the initial capital investment for the project
➔ The firm decides a fixed payback period ⇒ accept if the project generates
enough cash flows in the pre-specified period and vise versa
Limitations of the PP method
➔ Ignores the cash flows after the cut-off date ⇒ the choice of the cut-off
date is extremely critical, and should be tailored to the project's life;
otherwise, poor, shorter-lived projects could be accepted at the expense
of good, longer-lived projects
➔ Ignores the time value of money ⇒ drawback could be avoided by
discounting the cash flows generated during the payback period —
discounted payback period
Profitability index
Index which allows to compare benefits from different projects relative to their
costs
➔ Shows how much value investor receives per dollar invested (same as
NPV turned into a percentage)
𝑃𝐼 =
𝑃𝑉
𝐼
=1+
𝑁𝑃𝑉
𝐼
➔ PI > 1 ⇒ accept
➔ Used in cases when funds are limited and projects are non-mutually
exclusive (i.e. range the projects by their PI and choose project with the
highest PI)
Topic 2. Valuation of Fixed Income Securities
Financial assets (securities) issued by spender-borrowers on the financial
markets to collect funds from the saver-lenders, in order to buy real assets.
Fixed Income security — security that promises cash flows of fixed amounts at
fixed dates.
➔ Involve default risk — the risk that the issuer will not meet his cash flow
obligations
◆ US Treasuries can be considered an approximation of default-free fixed
income securities
Types of risk-free bonds
Treasury bills
(T-bills)
Maturities: < 1 year
- no coupon
- only face value
Treasury notes (T-notes)
Maturities: 1-10 years
- semi-annual coupon
- coupon + face value
Treasury bonds (T-bonds)
Maturities: > 10 years
- semi-annual coupon
- coupon + face value
Financial assets:
➔ Government bonds;
◆ Investors know the future benefits with a high degree of certainty
➔ Corporate bonds;
◆ Issued by corporations;
◆ Involve default risk, i.e. the risk that the corporation fails to pay the
promised cash flows;
◆ The price of a corporate bond depends on promised cash flows, spot
rates, likelihood of default, risk premium associated to default;
◆ The valuation formulae are the same, the main differences:
● Expected cash flows — need to account for the potential default;
● Risk-adjusted discount rates.
➔ Preferred and Common stocks:
◆ Since estimation of future benefits is challenging, as a consequence of
the higher degree of uncertainty about the future benefits, a higher
discount rate has to be used in discounting those benefits to present
value.
Bonds
Coupon Bonds (very liquid instrument)
The value of coupon bonds equals the present value of future interest and
principal payments to be paid to the lender (bondholder) by the borrower (issuer
of the bond).
➔ Promises a periodic cash flow (coupon) and the face value at maturity;
◆ The time to maturity — the length of time until maturity
➔ The coupon rate is the ratio of the coupon to the face value;
➔ Coupon payments are typically semi-annual for US bonds and annual for
European bonds.
➔ A coupon bond with annual coupon payments:
◆ P = principal payment / face value;
◆ c = annual coupon rate, i.e. the interest rate, — a specified
percentage (= annual coupon rate %) of the principal;
◆ 𝑟𝑎𝑛𝑛 = annual compounded discount rate — the required annual
compounded rate of return on similar bonds ⇒ this is the return
investors give up when they buy a specific bond
𝑁−1
𝑃𝑐𝑏, 𝑎𝑛𝑛 = ∑
𝑐×𝑃
+
𝑡
𝑡=1 (1+𝑟𝑎𝑛𝑛)
𝑃×(1+𝑐)
𝑁
(1+𝑟𝑎𝑛𝑛)
➔ A coupon bond with semi-annual coupon payments:
◆
𝑐
2
= semi-annual coupon rate , or annual coupon rate /2;
◆ 𝑟𝑠𝑒𝑚 = semi-annual compounded discount rate
2𝑁−1
𝑃𝑐𝑏, 𝑠𝑒𝑚 = ∑
𝑐
2
𝑐
×𝑃
+
𝑡
𝑡=1 (1+𝑟𝑠𝑒𝑚)
𝑃×(1+ 2 )
2𝑁
(1+𝑟𝑧𝑒𝑚)
Zero Coupon Bonds, or Treasury Strips
Zero Coupon Bond promises a single cash flow, face value at some future date
(maturity).
➔ The price of a zero coupon bond:
𝑃𝑧𝑐𝑏 =
𝑃
𝑁
(1+𝑟)
Spot Rates and Yield Curve
Spot rates can be calculated from the prices of financial assets traded in the
market, namely from prices on zero coupon bonds:
𝑃=
𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒
𝑡
(1+𝑟𝑡)
Spot rate — rate prevailing in the market today for different future maturities
➔ To value fixed-income securities, investors need to know the spot rate for
each maturity ⇒ the term structure of spot rates, or yield curve,
represents the spot rates as a function of maturity
Yield Curve, or term structure of interest rates, plots the yields of bonds with
different maturities but same risk
➔ Usually constructed from gov-t (treasury) securities and is referred to as
the benchmark yield curve
➔ Slopes up: rates at higher maturities are higher than for short maturities;
➔ Inversion of the yield curve: some time before the economic recessions
short term maturities rates become higher than long term maturities;
➔ Spot rates at both long and short maturities tend to move together over
time.
Term Structure Theories
Expectations theory predicts that yield curves are as likely to be upward sloping
as the are to be downward sloping or flat
Investing in long-maturity bonds should return the same, on average, as investing
in short-maturities bonds and rolling over (reinvest)
➔ Investors don't prefer on particular maturity over another ⇒ they compare
returns ⇒ short term and long term bonds are perfect substitutes ⇒ two
equivalent investment strategies: buy $1 of 1-year bond and when it
expires — buy another one OR buy $1 of 2-year bond and hold it
◆ Forward rate — rate that we guarantee today for investing in the
𝑓
𝑡 𝑡+1
⇒ (1 + 𝑟1) (1 +
2
𝑓 ) = (1 + 𝑟1)
1 2
Interest rate on the long bond is the average of the interest on short bonds
expected over life of the long bond ⇒ for n-period bonds:
𝑟𝑛𝑡 =
➔
➔
➔
➔
𝑟1+ 1𝑓2 +
𝑓 + ... +
2 3
Yields for different bonds are determined by the demand and supply with no effect
from expected returns of other bonds ⇒ bonds are not perfect substitutes
➔ Markets for different maturity bonds are completely segmented
➔ Can explain all three forms of the yield curve (esp. for hump shaped
curves)
➔ The increase demand for long-maturity bonds ⇒ long rates decreased
Liquidity premium
Investors prefer short maturity bonds ⇒ term structure slopes up on average
➔ Predicts that yield curves will normally slope upwards, only rarely they will
be flat / downward sloping;
➔ Even default-free bonds are risky due to inflation and future interest rates
◆ Inflation risk: bondholders care about the real return (the
purchasing power of the return), not just the nominal $ value of
the coupons' payments;
◆ Interest rate risk: in case investor's payment horizon is NOT
equal to the maturity of the bond (capital gain/loss)
◆ Both risks require a compensation for holding a long-term bond (
α𝑡)
➔ Bonds are substitutes, but not perfect
Expectations Theory
future —
Market Segmentation
𝑓
𝑡+(𝑛−1) 𝑡
𝑛
Ignores inflation and interest rate risk;
Long rates reflect expectations of future short rates;
Slope up: market expect rates to rise;
Slope down: market expect rates to fall;
Valuation of Bonds via Zero Coupon Bonds
The bond price is determined using the PV rule ⇒ the PV rule gives the market
price based on the no arbitrage condition (i.e. arbitrage — making money out of
nothing // costless = self-financing strategy, riskless, profitable)
Example:
- Using PV rule the coupon bond price is determined by discounting the cash
flows and face value at the spot rates
- Using no arbitrage — the prices of instruments with the same risk and the same
cash flows should be the same
Synthetic Replication ⇒ construct a portfolio of three zero-coupon bonds that has
the same cash flow as the coupon bond ⇒ since cash flows are the same, the
price should be the same! ⇒ otherwise, arbitrage is possible
➔ The prices of all bonds can be obtained from the prices of zero-coupon
bonds;
➔ Any bond can be synthetically replicated by a portfolio of zero-coupon
bonds;
➔ Absence of arbitrage implies that a bond must have the same price as its
replicating portfolio ⇒ the same price given by the PV rule.
Valuation of Bonds via Coupon Bonds
➔ Main use of YTM:
◆ An alternative way to quote the price of a bond;
◆ The YTM of a zero-coupon bond is the spot rate corresponding to
the bond's time to maturity;
◆ The YTM is the return of investing in the bond and holding it until
maturity
➔ For a bond with annual coupon rate c and T years to maturity, the YTM is:
𝑃𝑚𝑎𝑟𝑘𝑒𝑡 =
𝐶𝑜𝑢𝑝𝑜𝑛
1
(1+𝑌𝑇𝑀)
+
𝐶𝑜𝑢𝑝𝑜𝑛
2
(1+𝑌𝑇𝑀)
+... +
𝐶𝑜𝑢𝑝𝑜𝑛 + 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒
𝑇
(1+𝑌𝑇𝑀)
𝑇
= ∑
𝐶𝑜𝑢𝑝𝑜𝑛
𝑡
𝑡=1 (1+𝑌𝑇𝑀)
+
𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢
(1+𝑌𝑇𝑀)
Relation between coupon, YTM, and the price
The relation between the coupon rate and the bond's YTM tells how the bond
price compares to the face value:
➔ If YTM > c, then the bond sells at a discount (price < face value);
➔ If YTM = c, then the bond sells at par (price = face value);
➔ If YTM < c, then the bond sells at premium (price > face value)
Relation between YTM and the Spot Rates
➔ YTM is a complicated average of the spot rates corresponding to the
different maturities;
➔ YTM is simple only if the bond is a zero coupon bond ⇒ equal to the
T-year spot rate.
YTM of Coupon Bond
The YTM of a coupon bond — return of investing in the bond, holding it until
maturity, and reinvesting the coupon at a rate equal to the YTM
➔ Problem 1: the YTM is not the return for any investment horizon other
than maturity
➔ Problem 2: the YTM is not the return even for investment horizon equal to
maturity because the future spot rates, at which the coupon will be
reinvested, may be different from the YTM
The YTM as a Tool to Compare Bonds
Using the YTM to compare bonds is correct only when they have the same
coupon and time to maturity ⇒ correct way to compare bonds is to use the PV
rule.
Yield To Maturity
The YTM of a bond is the discount rate that equates the PV of the bond's cash
flows to the bond's price, i.e. viewing the bond as an investment project, the YTM
= bond's IRR
Topic 3. Stock Valuation
Equity represents claims to shares of the net income and assets of a firm.
Each year some cash of the firm is reinvested to finance growth, some are
distributed to stockholders via dividends.
➔ Shareholders’ share increases with the company’s increase
Common stocks, i.e. ordinary shares
Discounted Cash Flow models assume that the value of a stock is equal to the
present value of the cash flows stockholders expect to receive from the firm.
Common stocks have unspecified cash flows
⇒ common stockholders have to consider (1) how much cash the firm will
generate now and in the future, (2) how much of that cash the firm will reinvest to
finance growth, and (3) how much the firm will distribute to stockholders via
dividends
⇒ common stocks simply deliver a stream of uncertain future dividends to
stockholders
However, there are two forms of payoffs:
➔ cash dividends;
➔ capital gains (losses).
Expected return on a stock over the next year (E(R)) — function of the
expected dividend and expected capital gain:
𝐸(𝑅) =
𝐷𝐼𝑉1+ 𝑃1−𝑃0
𝑃0
𝐷𝐼𝑉1 = expected dividend to be paid at time 1;
𝑃0 = current price of the stock;
𝑃1 − 𝑃0 = capital gain on the stock
The current price of a stock in terms of forecasted dividends and expected
price next year (equation represents market equilibrium condition: if it does not
hold, stocks will be undervalued or over-priced):
𝑃0 =
𝐷𝐼𝑉1+𝑃1
1+𝑟𝑒
𝑟𝑒 = required annual rate on similar equity stocks ⇒ this is the return that can be
earned in the financial market on securities of comparable risk ⇒ this discount
rate reflects not only the time value of money, but also the riskiness of the stock;
The current price of a stock in terms of dividends over N years and the price
at the end of year N:
𝑃0 =
𝐷𝐼𝑉1+𝑃1
1+𝑟𝑒
+
𝐷𝐼𝑉2
+
2
(1+𝑟𝑒)
𝐷𝐼𝑉3
3
(1+𝑟𝑒)
+... +
𝐷𝐼𝑉𝑁+𝑃𝑁
𝑁
(1+𝑟𝑒)
Dividend Discount Model
Dividend Discount Model — the value of common stocks equals the present
value of future periodic dividends that stockholders expect the firm to distribute
forever.
The price today of a common stock:
∞
𝑃0 = ∑
𝐷𝐼𝑉𝑡
𝑡
𝑡=1 (1+𝑟𝑒)
𝐷𝐼𝑉𝑡 = dividends to be paid annually at time t ⇒ since a common stock does not
have a fixed timeline, an infinite stream of dividends must be forecast.
➔ To implement this equation, investors need to know all the future
dividends received until infinity and the appropriate discount rate ⇒ it is
impractical to use the model in the purest form, hence, two simplifying
assumptions:
➔ Zero Growth Model — assumes a constant dividend stream
(𝐷𝐼𝑉1 = 𝐷𝐼𝑉2 =...)
𝑃0 =
𝐷𝐼𝑉1
𝑟𝑒
➔ Gordon Growth Model — dividends grow at a constant rate g
per annum
𝑃0 =
𝐷1
𝑟𝑒−𝑔
;
𝐷1 = dividend expected at time 1 ⇒ is the next or expected dividend payment in
the next period
➔ Gordon Growth Model with current dividend — sometimes
company announces its next dividend, but more often this will not
be known and investors have to estimate it from the current
dividend 𝐷0: 𝐷
= 𝐷0(1 + 𝑔)
1
◆ Problem 1: firm's growth may not be constant (e.g. small
firms with a new product may experience very rapid
growth in their early years and then growth may slow
down to more normal levels before possibly declining);
◆ Problem 2: some firms may not pay dividends ⇒
alternative valuation methods are required.
Two main determinants of a firm's growth rate:
➔ Size of the new investments made by a firm;
➔ Rate of return of the new investments
In practice, financial analysts may forecast dividends for a given period and add a
"terminal value":
➔ Forecast dividends for some foreseeable future using company, industry specific knowledge and specific market analytics;
𝑇
𝑃0 = ∑
𝐷𝑖𝑣𝑡
𝑡
𝑡=1 (1+𝑟𝑒)
+
𝑇𝑉𝑇
𝑇
(1+𝑟𝑒)
Where T — max number of years where accurate forecasting is possible
Growth Rate
➔ Historical growth rate in year t:
𝑔=
𝐷𝑡 +𝐷𝑡−1
𝐷𝑡−1
➔ Forecasted growth — same as historical but with forecasted growth rates
➔ Growth rate from ROE and Payout ratio:
𝑔 = 𝑅𝑂𝐸 × (1 − 𝑃𝑎𝑦𝑜𝑢𝑡)
◆ Book value of equity — the value of firm's equity in the firm's
books;
◆ Book value of assets — the value of firm's assets in the firm's
books;
◆ Net income — firm's earnings for the current year;
◆ Return on equity:
𝑅𝑂𝐸 =
𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦
◆ Payout ratio:
𝑃𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜 =
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠
𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
Expected Return
➔ An estimate of the expected return can be obtained from the CAPM:
𝑟𝑖 = 𝑟𝑓 + β𝑖(𝐸(𝑟𝑚) − 𝑟𝑓)
◆ Beta — sensitivity to market changes: from regression or
financial sources;
◆ Risk-free rate: in principle, take the same maturity as the cash
flows; however, in practice, one may use rates of longer-dated
bonds
◆ Estimate the risk premium using the historical average excess
return of the market over the risk-free interest rate
➔ Estimates can be also obtained from multi-factor models.
Terminal Value
➔ Perpetual growth method:
◆ Assume that from year T+1 the company is mature enough to
offer investors constant infinite dividend growth g
➔ Exit multiples method:
◆ Use comparable companies from the same industry;
◆ Calculate at how much you will be able to sell the company at the
end of year T
Topic 4. Portfolio Theory
Risk and Return Relationship
Risk and Return
Risk and return are the two main attributes of a financial security.
Return
➔ The actual return of a security is the amount received divided by the
amount invested, however when securities are originally acquired, their
returns are usually uncertain because their prices are subject to changes
⇒ a variety of return outcomes are likely to exist, each occurring with a
specific probability ⇒ the return measure of a risky asset — random
variable ⇒ the uncertainty of the return is measured by the expected
(average) return
𝐸(𝑅) = 𝑝1𝑅1 + 𝑝2𝑅2 +... + 𝑝𝑛𝑅𝑛
➔ Returns and their probabilities are not usually known ⇒ historical average
returns and number of past observations are often used as proxies;
➔ Return > -1 (which represents the complete loss of the original
investment in stock)!
Holding Period Return
𝑅 =
𝐷+𝑃𝑡−𝑃𝑡−1
𝑃𝑡−1
= 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛
σ=
A portfolio — combination of different assets held by an investor.
Construction of a stock portfolio includes:
➔ Computing portfolio returns;
➔ Benefits of diversification;
➔ Mean-variance optimization and portfolio frontier
◆ With risky stocks only;
◆ With risky stocks and a risk-free asset.
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 =
The variation around the expected return is the measure of the risk of a security
⇒ the measure of how far the actual return differs from the expected return
2
Portfolio Theory
Portfolio returns
Risk
2
➔ Relationship is positive: the performance of each security class
increases the risk ranking
◆ Occurs because investors require compensation for bearing risk
⇒ the higher the risk associated with a financial instrument, the
higher the return investors require to hold a financial asset
➔ Relationship changes over different holding periods: the level of risk
tends to reduce over longer holding periods
2
𝑝1(𝑅1 − 𝐸(𝑅)) + 𝑝2(𝑅2 − 𝐸(𝑅)) +... + 𝑝𝑛(𝑅𝑛 − 𝐸(𝑅))
𝑠ℎ𝑎𝑟𝑒 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡
𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
;
∑ 𝑤𝑖 = 1
The expected return of a portfolio — 𝐸(𝑅𝑝) — the weighted average of the
expected returns of the individual stocks of which the portfolio is composed,
where the weights are the portfolio weights:
𝐸(𝑅𝑃) = 𝑤1𝐸(𝑅1) + 𝑤2𝐸(𝑅2) +.. + 𝑤𝑛𝐸(𝑅𝑛)
Correlation measures the degree to which two returns tend to move together:
ρ1,2 =
➔
σ1,2
σ1σ2
, where σ1,2= E(XY)+E(X)E(Y)
ρ1,2 > 0 ⇒ two returns tend to move together;
➔ ρ1,2 < 0 ⇒ two returns tend to move in opposite directions;
➔ ρ1,2 = 0 ⇒ two returns are wholly unrelated.
The correlation determines the degree of curvature:
➔ The lower correlation, the more distended the curvature;
➔ When the investor can include in the portfolio a risk-free asset with a
return perfectly negatively correlated, the return-risk combinations are
represented by a pair of lines
The portfolio variance (the portfolio standard deviation is the square root of the
variance):
2
2 2
2 2
σ𝑝 = 𝑤1σ1 + 𝑤2σ2 + 2𝑤1 𝑤2 σ1 σ ρ1,2
2
Benefits of diversification
➔ The variance of the portfolio return < any individual asset variance
◆ By forming portfolios (or by including additional assets in the
portfolio), risk-averse investors are able to reduce risk —
diversification
◆ Diversification does not work in the extremely rare cases where
returns move perfectly together
➔ The portfolio variance falls as the number of assets held increases ⇒ can
be achieved even with a relatively small number of stocks
➔ For portfolios with a large number of assets, covariance among the
assets accounts for almost all of the portfolio's risk.
➔ Diversifiable risk = idiosyncratic = firm-specific, undiversifiable =
systematic
◆ The compensation (return) covers only systematic risk
➔ Agents have preferences that only involve expected portfolio returns and
return standard deviations;
➔ The utility function is increasing in expected portfolio returns, and
decreasing in return standard deviation.
Mean-standard deviation frontier shows expected return and risk that could be
achieved by different combinations (portfolios) of the two risky assets, and
enables understanding of how investors view the trade-offs between means and
standard deviations when they choose the portfolio weights of the assets to
include in their portfolios.
➔ Only the upper part of the mean-standard deviation frontier will
be of interest to risk-averse investors — efficient frontier — the
group of portfolios that both minimise risk for a given level of
expected return, and maximise expected return for a given level
of risk.
The set of all points representing portfolios made from N assets is called the
feasible set or feasible region
➔ For two assets feasible region = mean-standard deviation frontier
How to find the minimum-variance portfolio?
2
𝑤1 =
2
2
2
σ1+σ2−2σ1 σ ρ1,2
2
Short Sales
➔ Selling a stock we do not own
➔ To make profit on short sales, the price must go down between 0 and 1
Mean-variance optimization and portfolio frontier
The choice of stock portfolio includes 2 steps:
1. Determine the efficient frontier
a. Maximise return, keeping risk constant or
b. Minimise risk, keeping expected return constant
2. Identify the optimal portfolio (of two risky assets) for a given investor
a. To identify the optimal portfolio — the portfolio where the
investor's indifference curve is tangent to the mean-standard
deviation frontier: the set of preferences towards risk and return
is required — indifference curve
➔ Through the optimisation, the efficient frontier including the risk-free asset
is obtained — capital market line
Portfolio selection by Harry Markowitz (1952)
The mean-standard deviation portfolio theory:
σ2−σ1σ ρ1,2
Investor's Preferences
The concept of risk aversion is related to the behaviour of individuals under
uncertainty:
➔ Mean-variance framework:
𝑈 = 𝐸(𝑅) −
1
2
◆ Assets may be tradable but not investable because of restrictions
placed on certain kinds of investors (e.g. tradable stocks are
investable only for domestic investors)
2
𝐴σ — the utility of an investment
Two fund separation problem
The optimal portfolio of the mean-variance investor is a combination of a position
in the risk-free asset and a position in the tangent portfolio
Capital Market Line and Security Market Line
A capital allocation line includes all possible combinations of the risk-free asset
and an investor's optimal risky portfolio ⇒ the capital market line — special
case of CAL, where the risky portfolio is the market portfolio
𝐶𝐴𝐿: 𝐸(𝑅𝑝) = 𝑟𝑓 +
𝐶𝑀𝐿: 𝐸(𝑅𝑝) = 𝑟𝑓 +
𝐸(𝑅𝐴) −𝑟𝑓
σ𝐴
𝐸(𝑅𝑀) −𝑟𝑓
σ𝑀
· σ𝑃
· σ𝑃
➔ The risk-free asset is a debt security with no default / inflation / liquidity /
interest rate risk (e.g. US Treasury bills)
A security market line — graphical representation of the CAPM with beta,
reflecting systematic risk, on the x-axis and expected return on the y-axis
𝑆𝑀𝐿: 𝐸(𝑅𝑖) = 𝑟𝑓 + β𝑖 · [𝐸(𝑅𝑀) − 𝑟𝑓]
Capital Asset Pricing Model (CAPM)
Assumptions from portfolio theory:
➔ Investors maximise their utility only on the basis of expected portfolio
returns and return standard deviations;
➔ Unlimited amounts can be borrowed or loaned at the risk-free rate;
➔ Markets are perfect and frictionless (i.e. no taxes on sales and
purchases, no transaction costs and no short sales restrictions)
➔ Investors have homogeneous beliefs regarding future returns, which
means that all investors have the same information and assessment
about expected returns, standard deviations and correlations of all
feasible portfolios
Market Equilibrium
In the market equilibrium:
➔ Tangent portfolio = market portfolio, since total demand = tangent
portfolio and total supply = market portfolio
◆ Market portfolio includes all risky assets or anything that has
value (stocks, bonds, real estate, human capital, etc);
◆ Not all assets are tradable, not all tradable assets are investable
(e.g. human capital is un-investable asset);
Where β
=
σ𝑖,𝑀
2
σ𝑀
— beta of the risky asset, which measures the sensitivity of an
individual security to the market movements (systematic factor risk)
Beta of a portfolio: β
𝑝
= ∑ 𝑤𝑖β𝑖
Multifactor Models and APT
All common variations in stock returns are generated by movements in a set of
factors:
𝑅𝐼 = 𝑎𝑖 + 𝑏𝑖,1𝐹1 +... + 𝑏𝑖,𝑛𝐹𝑛 + ε𝑖
𝑎𝑖 — expected return on the stock if all F = 0;
𝑏𝑖,1 — sensitivity of stock return to changes in respective factor
𝐹1 — factors representing the source of systematic risk
ε𝑖 — random error
APT outcome
Assets with the same factor sensitivities must offer the same expected returns in
financial market equilibrium:
Where λ — risk premium