Module 4: Constructing an Investment Portfolio—Part 1 Flashcards
Define “risk averse” investors and describe their behavior as it relates to investment choices, risks, and returns.
Historical data shows that risky assets command a risk premium, implying that most investors are “risk averse.”
A risk-averse investor is one who will only accept a risky investment if it provides a premium over the return available from a risk-free investment (risk premium). The greater the risk, the greater the premium that is required.
Use the utility value formula to determine how an investor with a risk aversion index of 4 will assess the following two investment options:
- Investment A: Portfolio with an expected rate of return of 20% and a standard deviation of 20%
- Investment B: T-bills with a guaranteed rate of return of 7% and a standard deviation of 0.
Using the utility value formula:
U = E(r) − ½ Aσ^2
Where:
U = Utility value
E(r) = Expected return
A = Investor’s risk aversion index
σ = Standard deviation
Investment A utility value:
U = .20 − (½ × 4 × .202)
= .20 − (2 × .04)
= .20 − .08
= .12
Investment B utility value:
U = .07 − (½ × 4 × 0)
= .07 − (2 × 0)
= .07 − 0
= .07
An investor with a risk aversion index of 4 will choose Investment A because it has a higher utility value.
Describe a tool called the mean variance (M-V) criterion that can be used to compare investment portfolios.
The mean-variance (M-V) criterion can be used to compare a portfolio of investments.
The mean-variance criterion compares portfolios by favouring those with the characteristics of higher expected returns (E(r)) and lower volatilities, as measured by σ (standard deviation). A risk-averse investor seeks the highest expected return, given the portfolio’s standard deviation, or conversely, the lowest standard deviation for a given expected return.
The M-V criterion states that an investor prefers Portfolio A over Portfolio B if:
E(rA) ≥ E(rB) and σA ≤ σB
(It can also be said that one investment portfolio “dominates” another portfolio, when using the M-V criterion.)
Where:
E(rA) = Expected return of A
E(rB) = Expected return of B
≥ = Equal to or greater than
σA = Standard deviation of A
σB = Standard deviation of B
≤ = Equal to or smaller than
Using this criterion, if two investments have the same expected return but one has a lower standard deviation, the one with the lower standard deviation is the better choice.
Use the mean-variance (M-V) criterion to identify the dominant investment portfolio among a group of three alternatives described by their expected returns and standard deviations shown below:
Portfolio 1: E(r) = 10% and σ = 15%
Portfolio 2: E(r) = 15% and σ = 10%
Portfolio 3: E(r) = 15% and σ = 15%.
Portfolio 1 can be ruled out because its expected return is lower than both 2 and 3 and its standard deviation is equal to or greater than both 2 and 3.
Portfolios 2 and 3 can then be compared using the following M-V criterion:
E(rA) ≥ E(rB) and σA ≤ σB
Where:
E(rA) = Expected return of A
E(rB) = Expected return of B
≥ = Equal to or greater than
σA = Standard deviation of A
σB = Standard deviation of B
≤ = Equal to or smaller than
Portfolio 2 dominates Portfolio 3 if E(r2) ≥ E(r3) and σ2 ≤ σ3
15% ≥ 15% and 10% ≤ 15%, and as a result Portfolio 2 will be preferred by a risk-averse investor and is seen as the dominant Portfolio within the group.
Outline the three steps involved in portfolio construction and comment on how the risk of an investment portfolio might be controlled.
Constructing an investment portfolio can be viewed as a three-step, top-down process:
(a) Allocation of available funds between risk-free assets and risky assets (for example, between safe assets such as Treasury bills and risky assets such as stocks.) This process is referred to as the “capital allocation” decision.
(b) Allocation of funds within the risky portfolio across broad asset classes (e.g., Canadian stocks, international stocks and long-term bonds). Also referred to as “asset allocation.”
(c) Selection of individual securities within each asset class.
“Capital allocation” involves the decision-making around the proportion of an overall portfolio to devote to risky versus risk-free assets. “Asset allocation” chooses the asset classes to hold within the risky portion of the portfolio. “Security selection” picks the particular securities to hold within each chosen asset class.
The most straightforward way to control the risk of a portfolio is through the fraction of the overall portfolio invested in safe securities versus the fraction invested in risky assets. Most investment professionals consider these decisions the most important part of portfolio construction.
Describe which money market instruments are viewed as risk-free assets and the rationale for this industry practice
It is common practice to view Treasury bills as “the” risk-free asset. Their short-term nature makes their prices insensitive to interest rate fluctuations. An investor can lock in a short-term nominal return by buying a T-bill and holding it to maturity. Moreover, inflation uncertainty over the course of a few weeks, or even months, is negligible compared with the uncertainty of stock market returns.
Many investors use a broad range of money market instruments as risk-free assets. All the money market instruments are virtually free of interest rate risk because of their short maturities and are fairly safe in terms of default or credit risk.
Money market funds hold a variety of low-risk instruments so investors may also treat these funds as their effective risk-free asset.
Identify the expected returns and standard deviations of two potential assets for your investment portfolio: (1) a risk-free Treasury bill (T-Bill) with an expected return of 3%, and (2) a stock with an expected return of 10% and a standard deviation of 20%. You have decided to invest entirely in one asset or the other.
If you invest 100% in the risk-free asset, your expected return on your portfolio would be 3% and your risk would be 0%. The calculation of risk for this portfolio is simple because the standard deviation of a risk-free asset is 0%.
Likewise, investing 100% in the stock would give you an expected return on your portfolio of 10% and standard deviation of 20%.
Calculate the expected return and standard deviation for a portfolio comprising 25% allocated to a T-bill with an expected return of 3% and 75% allocated to a stock with an expected return of 10% and a standard deviation of 20%. Explain the result in the context of the two portfolio components.
Using the expected return of complete portfolio formula:
E(rc) = yrp +(1 − y) rf
Where:
E(rc) = Expected return of complete portfolio
y = Proportion of complete portfolio allocated to risky portfolio
rp = Rate of return of risk-free portfolio
rf = Rate of return of risk-free portfolio
E(rc) = (.75 × .10) + (1 − .75) × .03
= .075 + (.25 × .03)
= .0825 or 8.25%
The expected return is 8.25%.
The standard deviation (σ) on the complete portfolio is the proportion of the risky asset in that portfolio multiplied by the standard deviation of the risky asset:
σc = yσp
Where:
y = Proportion of complete portfolio invested in risky asset
σ = standard deviation of risky asset
= 75% × 20% = 15%.
The standard deviation of the risk-free portfolio is 0%.
A portfolio comprised of 25% in T-bills and 75% in stock has an expected return of 8.25%, and its standard deviation is 15%. As expected, both the expected return and standard deviation are higher than those for T-bills alone and lower than those for stock alone.
Assuming a risk-free rate of 6%, calculate the expected return on two alternative scenarios each comprising a risky portfolio with an expected return of 10% and a standard deviation of 20%. Scenarios (a) and (b) allocate a different proportion to the risky portfolio, as follows:
(a) 80% is allocated to the risky portfolio.
(b) 60% is allocated to the risky portfolio.
Explain the difference in expected returns between the two scenarios.
Using the utility value formula:
E(rc) = yrp + (1 − y)rf
Where:
E(rc)= Expected return of complete portfolio
y = Proportion of complete portfolio allocated to risky portfolio
rp = Rate of return of risky portfolio
rf = Rate of return of risk-free portfolio
The expected return of scenario (a) with an 80% allocation to the risky portfolio is:
E(rc) = (.80 × .10) + (1 − .80) × .06
= (.08) + (.2 × .06)
= (.08) + (.012)
= .092 or 9.2%
The expected return of scenario (b) with a 60% allocation to the risky portfolio is:
E(rc) = (.60 × .010) + (1 − .60) × .06
= (.06) + (.4 ×.06)
= (.06) + (.024)
= .084 or 8.4%
Scenario (b) holds more assets in the lower return, risk-free assets than scenario (a). This leads to a reduction in the expected return on the complete portfolio for scenario (b) from that of scenario (a).
Calculate the standard deviation and explain the risk-return trade-off between two alternative portfolios that comprise a risk-free T-bill with an expected return of 3% and a stock with an expected return of 10% and a standard deviation of 20%. The alternative portfolios differ in the allocation to the risky portfolio as follows:
Portfolio (1) allocates 80% to the risky portfolio.
Portfolio (2) allocates 60% to the risky portfolio.
The standard deviation of the complete portfolio is the standard deviation of the risky asset multiplied by the weight of the risky asset in that portfolio (the standard deviation of the risk-free portfolio is 0%). The expected return of the complete portfolio is the sum of the expected return of the risky asset multiplied by the weight of the risky asset in that portfolio and the expected return of the risk-free asset multiplied by its weight in the portfolio.
Portfolio (1), with 80% allocated to the risky portfolio, has a standard deviation of 16% (20% × 80%). Its expected return is (.80 × .10) + (.20 × .03) = .086, or 8.6%.
Portfolio (2) with 60% allocated to the risky portfolio has a standard deviation of 12% (20% × 60%). Its expected return is (.60 × .10) + (.40 × .03) = .072, or 7.2%.
Comparing the two portfolios shows that holding a greater proportion of the total portfolio in lower return risk-free asset reduces the risk (i.e., reduces the portfolio’s standard deviation); however, it also reduces the portfolio’s expected return. The standard deviation of the two portfolios are different because the stock allocations in each are different. Note that there is no change in the standard deviation of the stock.
Define “capital allocation line” (CAL) and explain its purpose.
The capital allocation line (CAL) shows all the risk-return combinations available to investors given three variables (1) the rate of return on a risk-free asset, (2) the expected return of a risky asset, and (3) the standard deviation of a risky asset.
The CAL is created by plotting expected returns on the y-axis and standard deviations on the x-axis. The capital allocation line then shows the expected return and standard deviation of all possible portfolios using the risk-free and risky assets whose characteristics have been used to construct the graph.
The slope of the capital allocation line is also known as the Sharpe ratio. Recall from Module 3 that the Sharpe ratio measures the trade-off between risk and return. It identifies the increase in the expected return of the chosen portfolio for each additional unit of standard deviation, that is, the measure of incremental return per incremental risk. A CAL with a steeper slope means that investors receive higher expected return in exchange for taking on more risk.
Explain why the capital allocation choices of individuals with different risk aversion indices will be different despite those same investors making choices from an identical investment opportunity set as depicted by the capital asset line (CAL).
The CAL provides a graph of all feasible risk-return combinations available for allocation of capital between risk-free and risky investments. Investors can use the CAL to choose one optimal combination, trading off risk and return.
Each investor’s risk aversion index will affect how they allocate their investment between the risk-free asset and the risky portfolio. More risk-averse investors (with higher risk aversion indices) will choose to hold less of the risky asset and more of the risk-free asset than will in investors whose risk aversion indices are lower. Investors will choose the allocation to the risky asset that maximizes their utility function.
Determine your optimal portfolio of risky and risk-free assets, given that your risk aversion index is 3, the rate of return on your risk-free portfolio is 7%, the expected return on your risky portfolio is 15% and its standard deviation is 22%.
Your optimal portfolio will be based on the allocation to the risky asset that maximizes your utility function. To do this, use the formula for the optimal portfolio of risky and risk-free assets:
y* = E(rp)−rfAσ2p
Where:
y* = Optimal allocation to risky assets
E(rp) = Expected return of risky portfolio
rf = Rate of return of risk-free portfolio
A = Investor’s risk aversion index
σp = Standard deviation of portfolio return
y* =(.15−.07)/(3×.222)
y* =(.08)/(.1452)
= .5510 or 55%
Given your risk aversion index and the risk-return characteristics of your options, your optimal portfolio is 55% risky assets and 45% risk-free assets.
Determine the expected return and standard deviation of your optimal portfolio comprising risky assets of 55%, assuming the rate of return on your risky portfolio is 15% and its standard deviation is 22%, and the rate of return on your risk-free portfolio is 7%.
The expected return of the optimal portfolio is determined using the same formula as the one used to determine the expected return on the complete portfolio, using the proportion of optimal portfolio invested in risky assets:
E(ro) = yorp + (1 − yo) rf
Where:
E(ro) = Expected return of optimal portfolio
yo = Proportion of optimal portfolio invested in risky portfolio
rp = Rate of return of risky portfolio
rf = Rate of return of risk-free portfolio
= (.55 × .15) + (1 − .55) × .07
= .0825 + .0315
= 11.4%
The expected return on your optimal portfolio is 11.4%
The standard deviation on your optimal portfolio is determined the same way as the standard deviation of the complete portfolio – the proportion (p) of the risky asset in that portfolio multiplied by the standard deviation of the risky asset (σ). The standard deviation of the risk-free portfolio is 0%. As a result, the standard deviation of your optimal portfolio is 12.1% = (55% × 22%).
Determine your optimal portfolio based on your risk aversion index of 6, assuming that the rate of return on your risk-free portfolio is 7%, the rate of return on your risky portfolio is 15% and its standard deviation is 22%.
Using the optimal portfolio of risky and risk-free assets formula:
y* = E(rp)−rfAσ2p
Where:
y* = Optimal allocation to risky assets
E(rp) = Expected return of risky portfolio
rf = Rate of return of risk-free portfolio
A = Investor’s risk aversion index
σp = Standard deviation of portfolio return
For your portfolio:
y* =(.15−.07)/(6×.222)
y* =(.08)/(.2904)
= .275 or 27.5%
Based on your risk aversion index and the risk-return characteristics of your options, your optimal portfolio is a combination of 27.5% risky assets and 72.5% risk-free assets (100% - 27.5%).
