FIM Exam: Risk & Return Practice

A focused collection of past exam questions and solutions on CAPM and Beta.

Overall Assessment and Preparation Guide

Difficulty Level

The past questions on Return, Risk, and the Security Market Line range from Basic to Intermediate. They primarily test your ability to apply core formulas directly. There are no highly theoretical or abstract questions, so the focus should be on practical application.

High-Priority Areas for Preparation

Based on the questions, these are the key areas you should master to confidently solve similar problems:

  • CAPM Formula: The most frequently tested topic is the Capital Asset Pricing Model (CAPM). You must be able to use the formula to find the required return, the beta, or the market risk premium when other variables are given.
  • Portfolio Beta: Another high-priority area is calculating a portfolio's beta as a weighted average of its constituent assets' betas. This concept is fundamental to understanding how diversification affects systematic risk.
  • Portfolio Expected Return: Similar to portfolio beta, you should be able to calculate a portfolio's expected return as a weighted average of its assets' expected returns.
  • Interpretation of Betas and Weights: Beyond just calculation, you need to understand the meaning of the results. Be prepared to explain what a beta greater than 1 means, or what a negative portfolio weight implies (e.g., a leveraged position).
  • Relationship between Risk and Return: Understand the core principle that expected return is a reward for bearing systematic risk, not total risk. Questions often test this conceptual understanding.

Question 4(b) - May 2024: CAPM and Portfolio Beta

Problem Statement: A stock has a beta of 1.08 and an expected return of 11.6 percent. A risk-free asset currently earns 3.6 percent.

Required:

(i) What is the expected return on a portfolio that is equally invested in the two assets?

(ii) If a portfolio of the two assets has a beta of .50, what are the portfolio weights?

(iii) If a portfolio of the two assets has an expected return of 10.5 percent, what is its beta?

(iv) If a portfolio of the two assets has a beta of 2.16, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain.

Theoretical Foundation: This problem tests your ability to apply the Capital Asset Pricing Model (CAPM) and the principles of portfolio construction. A portfolio's expected return and beta are weighted averages of the individual assets' returns and betas. The CAPM formula is given by: $E(R_i) = R_f + \beta_i \times (E(R_M) - R_f)$.

Step-by-step Solution:

First, we need to find the market risk premium, $(E(R_M) - R_f)$. We can use the CAPM formula with the given stock's data:

$11.6\% = 3.6\% + 1.08 \times (E(R_M) - R_f)$ $8.0\% = 1.08 \times (E(R_M) - R_f)$ $E(R_M) - R_f = \frac{8.0\%}{1.08} \approx 7.41\%$

i. Expected return for an equally invested portfolio:

The portfolio weights are 0.5 for the stock and 0.5 for the risk-free asset.

$E(R_P) = w_{stock} \times E(R_{stock}) + w_{rf} \times R_f$ $E(R_P) = (0.5 \times 11.6\%) + (0.5 \times 3.6\%)$ $E(R_P) = 5.8\% + 1.8\% = \textbf{7.6\%}$

ii. Portfolio weights for a beta of 0.50:

The beta of a risk-free asset is 0. We set up the portfolio beta equation and solve for the weight of the stock, $w_{stock}$.

$\beta_P = w_{stock} \times \beta_{stock} + (1 - w_{stock}) \times \beta_{rf}$ $0.50 = (w_{stock} \times 1.08) + ((1 - w_{stock}) \times 0)$ $w_{stock} = \frac{0.50}{1.08} \approx 0.463$ or \textbf{46.3%}. $w_{rf} = 1 - 0.463 = 0.537$ or \textbf{53.7%}.

iii. Portfolio beta for an expected return of 10.5%:

First, we need to find the portfolio weights by setting up the expected return equation and solving for the stock's weight.

$E(R_P) = w_{stock} \times E(R_{stock}) + (1 - w_{stock}) \times R_f$ $10.5\% = w_{stock}(11.6\%) + (1 - w_{stock})(3.6\%)$ $10.5 = 11.6w_{stock} + 3.6 - 3.6w_{stock}$ $6.9 = 8w_{stock}$ $w_{stock} = \frac{6.9}{8} = 0.8625$

Now, we can calculate the portfolio beta using this weight.

$\beta_P = (0.8625 \times 1.08) + ((1 - 0.8625) \times 0) = \textbf{0.9315}$

iv. Portfolio weights for a beta of 2.16:

We set up the portfolio beta equation again and solve for the stock's weight.

$\beta_P = w_{stock} \times \beta_{stock} + (1 - w_{stock}) \times \beta_{rf}$ $2.16 = (w_{stock} \times 1.08) + ((1 - w_{stock}) \times 0)$ $w_{stock} = \frac{2.16}{1.08} = 2$ or \textbf{200%}. $w_{rf} = 1 - 2 = -1$ or \textbf{-100%}.

Interpretation of Weights: A portfolio weight of 200% for the stock and -100% for the risk-free asset means the investor is using financial leverage. They have borrowed an amount equal to the total portfolio value at the risk-free rate and have invested this borrowed amount, along with their own capital, in the risky stock. This amplifies both the potential returns and the potential losses.

Question 1(ix) - Jan 2024: Required Return using CAPM

Problem Statement: The risk-free rate of interest, $k_{RF}$, is 6 percent. The overall stock market has an expected return of 12 percent. Hazlett, Inc. has a beta of 1.2. What is the required return of Hazlett, Inc. stock?

a) 12.0%

b) 12.2%

c) 12.8%

d) 13.2%

e) 13.5%

Theoretical Foundation: This is a direct application of the Capital Asset Pricing Model (CAPM). The required return on an asset is equal to the risk-free rate plus a risk premium that is proportional to the asset's systematic risk (beta).

Step-by-step Solution:

The formula for CAPM is: $R_{i} = R_{f} + \beta \times (R_{M} - R_{f})$

  • $R_{f} = 6\%$ (risk-free rate)
  • $R_{M} = 12\%$ (market expected return)
  • $\beta = 1.2$ (beta of Hazlett, Inc.)
  • $R_{M} - R_{f} = 12\% - 6\% = 6\%$ (market risk premium)

Now, substitute the values into the formula:

$R_{i} = 6\% + 1.2 \times (12\% - 6\%)$ $R_{i} = 6\% + 1.2 \times 6\%$ $R_{i} = 6\% + 7.2\% = \textbf{13.2\%}$

The correct answer is (d) 13.2%.

Question 4(c) - Sep 2023: Holding Company Beta & Required Return

Problem Statement: ECRI Corporation is a holding company with four main subsidiaries. The percentage of its capital invested in each of the subsidiaries, and their respective betas, are as follows: Subsidiary, Percentage of Capital, Beta. Electric utility: 60%, 0.70. Cable company: 25%, 0.90. Real estate development: 10%, 1.30. International/special projects: 5%, 1.50. Required: (i) What is the holding company’s beta? (ii) If the risk-free rate is 6 percent and the market risk premium is 5 percent, what is the holding company’s required rate of return? (iii) ECRI is considering a change in its strategic focus; it will reduce its reliance on the electric utility subsidiary, so the percentage of its capital in this subsidiary will be reduced to 50 percent. At the same time, it will increase its reliance on the international/special projects division, so the percentage of its capital in that subsidiary will rise to 15 percent. What will the company’s required rate of return be after these changes?

Theoretical Foundation: This problem requires you to apply the principle that a portfolio's beta is a weighted average of the individual asset betas. The weights are the percentages of capital invested in each subsidiary. For part (ii) and (iii), you will use the CAPM to find the required return.

Step-by-step Solution:

i. Holding company's beta:

The beta of a portfolio is the weighted average of the betas of its individual components. The formula is: $\beta_{portfolio} = \sum_{i=1}^n w_i \beta_i$.

$\beta_{ECRI} = (0.60 \times 0.70) + (0.25 \times 0.90) + (0.10 \times 1.30) + (0.05 \times 1.50)$ $= 0.42 + 0.225 + 0.13 + 0.075 = \textbf{0.85}$

ii. Holding company's required rate of return:

Using the CAPM formula, $R_{i} = R_f + \beta_i \times (\text{Market Risk Premium})$:

$R_i = 6\% + 0.85 \times 5\%$ $R_i = 6\% + 4.25\% = \textbf{10.25\%}$

iii. Required rate of return after strategic change:

We first need to determine the new portfolio weights. The total change in weights is a reduction of 10% from the electric utility and an increase of 10% for international projects. The weights for the cable and real estate subsidiaries remain the same.

  • New weight for Electric utility: 60% - 10% = 50%
  • New weight for International/special projects: 5% + 10% = 15%
  • Weights for the other two remain constant: 25% + 10% = 35%

Recalculate the portfolio beta with the new weights:

$\beta_{new} = (0.50 \times 0.70) + (0.25 \times 0.90) + (0.10 \times 1.30) + (0.15 \times 1.50)$ $= 0.35 + 0.225 + 0.13 + 0.225 = \textbf{0.93}$

Now, calculate the new required rate of return using the new beta:

$R_{i\_new} = 6\% + 0.93 \times 5\%$ $R_{i\_new} = 6\% + 4.65\% = \textbf{10.65\%}$

Question 5(a) - May 2023: Beta and Expected Return

Problem Statement: If a company's beta were to double, would its expected return double?

Theoretical Foundation: This question tests your understanding of the linear relationship described by the CAPM. The expected return is a function of both the risk-free rate and the risk premium. While the risk premium is directly proportional to beta, the expected return is not.

Step-by-step Solution:

No, a company's expected return would not double if its beta were to double. The relationship is defined by the CAPM formula: $E(R_i) = R_f + \beta_i \times (E(R_M) - R_f)$.

Tricky Area:

The key here is that the expected return has two components: the risk-free rate and the risk premium. If beta doubles, only the risk premium part of the equation doubles. The risk-free rate remains constant. Because the risk-free rate is typically a positive value, the overall expected return will increase, but it will be less than double the original expected return.

Question 4(c) - May 2023: Stock Valuation with CAPM

Problem Statement: Sorbond Industries has a beta of 1.45. The risk-free rate is 8 percent and the expected return on the market portfolio is 13 percent. The company currently pays a dividend of Tk. 2 a share, and investors expect it to experience a growth in dividends of 10 percent per annum for many years to come. Required: (i) What is the stock's required rate of return according to the CAPM? (ii) What is the stock's present market price per share, assuming this required return? (iii) What would happen to the required return and to market price per share if the beta were 0.80? (Assume that all else stays the same.)

Theoretical Foundation: This problem combines two key financial models: the CAPM to determine the required rate of return (or discount rate) and the Gordon Growth Model (also known as the dividend growth model) to value the stock based on that rate.

Step-by-step Solution:

i. Required rate of return (CAPM):

We use the CAPM formula: $k_e = R_f + \beta \times (E(R_M) - R_f)$.

$k_e = 8\% + 1.45 \times (13\% - 8\%)$ $k_e = 8\% + 1.45 \times 5\% = 8\% + 7.25\% = \textbf{15.25\%}$

ii. Present market price per share:

We use the Gordon Growth Model: $P_0 = \frac{D_1}{k_e - g}$. First, we need to calculate the next period's dividend, $D_1$.

$D_1 = D_0 \times (1 + g) = 2 \times (1 + 0.10) = \text{Tk. } 2.20$ $P_0 = \frac{2.20}{0.1525 - 0.10} = \frac{2.20}{0.0525} \approx \textbf{Tk. 41.90}$

iii. Impact of beta change:

If beta were 0.80, the new required return would be:

$k_{e\_new} = 8\% + 0.80 \times 5\% = 8\% + 4\% = \textbf{12\%}$

The new market price would be:

$P_{0\_new} = \frac{2.20}{0.12 - 0.10} = \frac{2.20}{0.02} = \textbf{Tk. 110.00}$

When the beta decreases, the required rate of return decreases (from 15.25% to 12%), and the market price per share increases (from Tk. 41.90 to Tk. 110.00).

Question 1(ii) - Sep 2022: Diversifiable Risk

Problem Statement: Which type of risk is avoidable through proper diversification?

a) Systematic risk

b) Unsystematic risk

c) Total risk

d) Market risk

e) Portfolio risk

Theoretical Foundation: The principle of diversification states that unique, company-specific risks can be eliminated by combining assets into a portfolio. The risk that remains is systematic risk.

Step-by-step Solution:

The correct answer is (b) Unsystematic risk. This type of risk, also known as diversifiable or firm-specific risk, is unique to a specific company or industry. By holding a well-diversified portfolio of assets, the unsystematic risks of each asset tend to cancel each other out, thereby reducing the overall risk of the portfolio. Systematic risk (or market risk) affects the entire market and cannot be diversified away.

Question 1(viii) - May 2022: Common Stock vs. Debt Return

Problem Statement: The common stock of a company must provide a higher expected return than the debt of the same company because:

a) There is less demand for stock than for bonds

b) There is greater demand for stock than for bonds

c) There is more systematic risk involved for the common stock

d) There is a market premium required for bonds

e) There is higher floatation cost for stocks

Theoretical Foundation: This question relates to the core risk-return trade-off. In finance, assets with higher risk are expected to provide higher returns. The level of risk is defined by the priority of claims on a company's assets and earnings.

Step-by-step Solution:

The correct answer is (c) There is more systematic risk involved for the common stock. Equity holders (stockholders) have a lower priority claim on a firm's assets and earnings than bondholders (debt holders). This makes common stock inherently riskier than debt. To compensate for this higher risk, investors demand a higher expected return on equity. The relevant risk is systematic risk, as unsystematic risk can be diversified away.