Portfolio leveraging refers to the strategy where an investor borrows funds at the risk-free rate of return
and invests all the available funds in a risky security (or portfolio)
Example: Consider the following information on a risk-free security (F) and risky security (A)
Calculate the expected return and standard deviation of return of a portfolio if an investor with $10,000
borrows $5,000 at the risk-free rate and invests $15,000 in the risky security
Proportion of funds invested in risky security: 𝑤𝐴 = $15,000/$10,000 = 1.5 or 150%
Proportion of funds borrowed at risk-free rate: 𝑤𝐹 = −$5,000/$10,000 = −0.5 or −50%
Expected return of the portfolio: 𝐸 (𝑟𝑝) = −0.5 (0.06) + 1.5 (0.12) = 15%
Standard deviation of returns of the portfolio:
Short selling refers to borrowing (typically via a broker) shares, selling them now with a contractual
obligation to buy them back later at (an expected) lower price
– Note that being long a security means you have purchased it and being short a security means you have
sold it
– Short selling is like risky borrowing to leverage a portfolio
– This leveraging increases portfolio risk!
Borrowing and short selling a security A and investing proceeds in security B implies wA < 0% and wB >
100% such that wA + wB = 1
You are given the following information on two stocks which have a return correlation of +0.5
Calculate the expected return and standard deviation of returns for the following portfolios
– An investor invests $5,000 each in stocks A and B (base case)
– An investor with $10,000 borrows $5,000 worth of stock A and invests the total available funds in stock B
– An investor with $10,000 borrows $10,000 worth of stock A and invests the total available funds in stock
B
Base case: An investor with $10,000 invests $5,000 each in stocks A and B implying that wA = wB = 0.5
There is approximately a 95% probability that the realised return will lie in the range (E(rp) ± 2σ) = (−15.4%,
45.4%)
Note: Even though stocks A and B are positively correlated there is still some reduction in risk [as σ <
17.5%]
Short-selling case 1: An investor with $10,000, borrows stock A and sells short $5,000 worth of A, investing
$15,000 in stock B implying wA = −0.50 and wB = 1.50
There is a 95% probability that the realised return will lie in the range (E(rp) ± 2σ) = (−29%, 79%)\
Short-selling case 2: An investor with $10,000 borrows stock A and sells short $10,000 worth of A, investing
$20,000 in stock B implying wA = −1 and wB = 2
There is a 95% probability that the realised return will lie in the range (E(rp) ± 2σ) = (−40%, 100%)
TAKEAWAYS:
A leveraged position involves borrowing funds and investing both the borrowed funds and your own funds
into a risky asset
– Results in:
• E(rp) > E(r risky asset) provided cost of borrowing is lower then E(r risky asset)
• σp> σrisky asset
Short selling involves, borrowing shares, selling them on market and going back into the market to
repurchase to deliver the shares back to lender
– Similarly to a borrow-and-invest strategy, it can increase an investor’s E(rp) and σp
Illustration:
– Consider portfolios of 2, 5, 10, 20, 50, 100 and 500 stocks where you invest 1/N of your wealth in each
stock, where each stock has a standard deviation of return of 10%, and where each pair of stocks has a
return correlation of 0.6.
– Calculate the standard deviations for these portfolios.
– What general relation between portfolio standard deviation and the number of stocks is being illustrated
here?
– Assumptions:
Illustration:
2-asset portfolio
4-asset portfolio
Under these assumptions
We can show that:
As N becomes very large
– The first term approaches zero
– The second term approaches σj,k as (N-1)/N approaches 1
In large portfolios, return covariances determine portfolio risk
– As a portfolio becomes large in size its total risk (standard deviation) falls, but at a declining rate
TAKEAWAYS:
- As a portfolio becomes more and more diversified, it is the covariance between the assets that
becomes more important than the stand-alone risk of the assets themselves
- Incremental diversification benefits reduce as the number of assets in the portfolio increase
- Diversification benefits reflect the reduction in unsystematic risk (a.k.a diversifiable or firm-specific
or idiosyncratic risk)
- Even with a well-diversified portfolio risk remains – this is systematic risk (a.k.a. nondiversifiable or
covariance or market risk)
The Capital Asset Pricing Model (CAPM)
● The CAPM is a theoretical model that can be used to “price” individual securities
– “Pricing” a security here means estimating its required rate of return (using the CAPM) and then obtaining
a price estimate based on the security’s future expected cash flows (that is, dividends and future price)
● The CAPM relates a security’s required rate of return to its non-diversifiable or systematic risk
– The higher the systematic risk the higher the required rate of return
– Note: Total risk (σ) is no longer relevant to pricing securities, or portfolios of securities
Investors are risk-averse individuals who maximize the expected utility of their end-of-period wealth
• Investors make portfolio decisions on the basis only of expected return and the variance (or standard
deviation) of returns
– Investors have homogeneous expectations about the volatilities, correlations and expected returns of
securities
– The returns on these securities are jointly normally distributed
– Capital markets are perfect as there are no taxes, transactions costs or government interference
– Unlimited borrowing and lending at a risk-free rate is possible
– Investors only hold efficient portfolios of securities that are all traded in financial markets
An investor’s preferred investment set is now a combination of the riskfree security and the risky portfolio M
(tangent to the line through rf)
• The capital market line, rfM, now dominates FF’
– In equilibrium, portfolio M must be the (value-weighted) market portfolio of all risky securities
– Why is M the market portfolio?
• Suppose security A comprises 5% of all risky securities by its market cap
• Suppose that only 2% of portfolio M is composed of security A
• That is, 3/5th of security A is not held by anyone
– Security A is now in excess supply
– The price of security A will fall, and its expected return will rise so investors will be willing to hold the
remaining 3/5th of A as part of the risky portfolio M
– In equilibrium, M must be the market portfolio
– The CAPM can be written as...
• Expected return on A = Risk-free return + Risk premium
– Risk premium = Amount of risk × Market price of risk
• The amount of risk is measured by the scaled covariance of the security returns with the returns on
the market portfolio (the beta of the security, β)
• The market price of risk (or the market risk premium) is the return above the risk-free rate that
investors earn for holding the (risky) market portfolio
• Higher the market price of risk and/or higher the amount of risk, greater the risk premium
– The security market line (SML) equation is
Illustration
– Assume that it is equally likely that the market portfolio’s return will increase by 5% and decline by 3%
– Consider three types of firms: S, T and I.
• Type S firms’ returns move by +6% on average when the economy is strong and by –4% when the
economy is weak.
• Type T firms’ returns move by +8% on average when the economy is strong and +4% when the
economy is weak.
• Type I firms’ returns do not move with the market. What are their betas?
– Uncertainty related to economy’s strength produces a 5% – (–3%) = +8% change in the return of the
market portfolio
– Type S firms’ returns change on average by: 6% – (–4%) = +10%
– Type T firms’ returns change on average by 8%– (4%)= +4%
– Type S firms’ beta, βS = ΔRj/ΔRm = 0.10/0.08 = 1.25
– Type T firms’ beta, βT = 0.04/0.08 = 0.5
Interpretation: A 1% change in the return of the market portfolio will likely lead to a 1.25% (0.5%) change in
the type S (T) firms’ returns, on average
What about type I firms?
– The market price of risk is measured as 𝐸(𝑟𝑀) −𝑟𝑓
– The systematic risk of asset j is measured by beta; 𝛽𝑗
– Since 𝜎𝑗,𝑚 = 𝜌𝑗,𝑚 𝜎𝑗 𝜎𝑚 we also have:
βi = 1: Security has the same systematic risk as the market portfolio
βi = 0: Security has zero systematic risk
βi < 1: Security has lower systematic risk than the market portfolio
βi > 1: Security has higher systematic risk than the market portfolio
The diagrammatic representation of the CAPM is the SML
Example:
– Assume that the risk-free rate is 7% and the expected market return is 12%.
– What is the market risk premium?
– Locate the expected returns for securities with the following betas on the SML
• Security A: βA = 1.5
• Security B: βB = 0.5
• Security C: βC = –0.5
– Will an investor ever invest in a security like security C? Explain.
– Given: rf = 7% , E(rm) = 12%
– Market risk premium = E(rm) – rf = 0.12 – 0.07 = 5.0%
– The SML is...𝐸 𝑟𝑗 = 𝑟𝑓 + 𝐸(𝑟𝑀) −𝑟𝑓 𝛽𝑗
– E(rA) = 0.07 + (0.12 – 0.07)1.5 = 14.5%
– E(rB) = 0.07 + (0.12 – 0.07)0.5 = 9.5%
– E(rC) = 0.07 + (0.12 – 0.07)(–0.5) = 4.5% << 7% = rf!
Recall that in Week 4 we used a required rate of return of 9% p.a. to value the shares of Computershare
(ASX: CPU). An online broker’s estimate of CPU’s beta is 0.7. Using a risk-free return of 4% and an
expected market risk premium of 7% p.a. what is CPU’s required rate of return?
Given:
What would you expect to happen to the security market line, required returns and security prices if the
following events occurred one after the other...
– Event 1: There is an unexpected increase in the market risk premium as a result of scepticism related to
the Commonwealth government’s fiscal stimuli
– Event 2: The Reserve Bank lowers the cash rate, a move also not expected by the market
– Event 3: A few months later, as the economy and financial markets recover, the Reserve Bank raises the
cash rate by 0.5%, a move widely expected by the market
TAKEAWAYS
- Efficient frontier gives the risk/return combinations of all efficient portfolios – representing optimum
rates of E(r) at each risk level
- The CML provides the risk-return tradeoff for efficient portfolios once allowing for borrowing/lending
at rf
- Under a set of very restrictive assumptions the CAPM provides a link between the risk and E(r) for
all assets
-
The risk that is priced in the CAPM is the risk that can’t be eliminated via diversification –
systematic/covariance/market/beta risk and is measured as;
-
The diagrammatic representation of the CAPM is the SML
An individual asset’s beta can be estimated using the market model regression which is:
Where: 𝛼𝑗 is the intercept, 𝛽𝑗 is the regression’s slope and 𝑒𝑗,𝑡 is the error term
Note that the market model is an empirical model while the CAPM is a theoretical model
1. Using the more recent data from Jan 2018 – Dec 2020, what would you expect to happen to the return
on BHP if the market’s return suddenly falls by 1%.
• A sudden drop in the market portfolio’s return of 1% is an unexpected move and you’d expect BHP’s
return to drop by around 1% (= 1 x 0.98)
2. What would the effect of the above changes be on an equally-weighted portfolio of BHP and ANZ Bank?
Recall what happens to total risk (σP) when we combine assets into a portfolio
But systematic risk – as measured by beta – is undiversifiable risk.
The beta of a portfolio of assets is simply the weighted average of the betas of the assets in that portfolio
What would be the expected effect of a 1% fall in market return be on an equally-weighted portfolio of BHP
and ANZ Bank?
So if there were a sudden drop in the market portfolio’s return of 1% you would expect a fall in your
portfolio’s return of 1.105%
In addition to the beta, the other parameters that need to be estimated are...
– The risk-free rate, 𝑟𝑓
– The market risk premium, 𝐸(𝑟𝑀) −𝑟𝑓
The risk-free rate, 𝑟𝑓
– Generally estimated as the yield to maturity on long-term government bonds
– The 10-year bond rate in Australia, 10- or 30-year rates in the US
The market risk premium, 𝐸(𝑟𝑀) −𝑟𝑓
– Generally estimated as an historical average premium
The next few slides show information related to observed/historical risk premiums
CAPM and security selection
Example
– Around November 2018 Asjeet was considering whether to purchase more shares of Sonic Healthcare
(ASX: SHL).
– Sonic is a health care company that offers laboratory medicine/pathology and radiology/diagnostic
imaging services to clinicians, hospitals, community health services and their patients. It operates 233
primary care clinics in Australia, the US, Germany, and internationally.
– Asjeet used Sonic’s estimated beta of 0.7, a risk-free rate of 4% and an expected market risk premium of
7% to estimate the required return on its shares.
– At that time, the consensus estimate of Sonic’s dividend next year was $0.85 which was expected to grow
at 5.5% p.a. for the foreseeable future
Example, continued
– Using the SML, the required return is...
𝐸 𝑟𝑗 = 𝑟𝑓 + 𝐸(𝑟𝑀) −𝑟𝑓 𝛽𝑗 = 0.04 + 0.07x0.7 = 8.9%
– Recall (from Week 4) that the estimated price using constant dividend growth model is...
– So, the expected return on equity is...
– The expected return implied by a current price of $22.50 is...
– From the CAPM, the equilibrium required return is 8.9%
– So, the expected return of 9.3% will fall to the equilibrium required return of 8.9%
– Assuming that the expected dividend and its growth rate are unchanged you’d expected Sonic’s price to
rise so at that time (Nov 2018) Sonic shares were underpriced or undervalued
– How do things change if the growth in dividends (g) is estimated to be 4.5% p.a. instead of 5.5% p.a.?
– You’d expect the return of 8.3% to rise to the equilibrium level of 8.9%
– You’d expect Sonic’s price to fall (why?) implying that the shares were overpriced or overvalued at that
time