Past Exam Questions Masterclass

Solutions to past exam questions from professional financial examinations.

Exam Questions & Solutions

Question 4(c) - Jan 2025: Dividend Growth Model

Problem Statement: Newline Plc is expected to pay equal dividends at the end of each of the next two years. Thereafter, the dividend will grow at a constant annual rate of 3.5 percent, forever. The current stock price is \$59. What is next year's dividend payment if the required rate of return is 11 percent?

Theoretical Foundation: This is a classic multi-stage dividend discount model problem. The value of the stock today is the sum of the present value of the dividends paid during the non-constant growth period and the present value of the stock price at the end of that period, which is calculated using the constant growth (Gordon Growth) model.

Solution:

Given: $P_0 = \$59$, $R = 11\%$, $g = 3.5\%$. $D_1 = D_2$. We need to solve for $D_1$.

Step 1: Set up the valuation formula.

$$P_0 = \frac{D_1}{(1+R)} + \frac{D_2}{(1+R)^2} + \frac{P_2}{(1+R)^2}$$

Step 2: Define the future stock price ($P_2$) in terms of $D_1$.

The constant growth model applies starting at year 2, so the price at year 2 depends on the dividend in year 3 ($D_3$). Since the dividend in year 2 and year 1 are the same ($D_1=D_2$), $D_3$ is the first dividend to grow at a constant rate.

$$D_3 = D_2(1+g) = D_1(1.035)$$ $$P_2 = \frac{D_3}{R - g} = \frac{D_1(1.035)}{0.11 - 0.035}$$

Tricky Area:

It's easy to confuse which dividend to use for the Gordon Growth Model. Remember that the formula $P_t = \frac{D_{t+1}}{R-g}$ always uses the dividend one period after the price you are trying to find. Here, we are finding $P_2$, so we need $D_3$.

Step 3: Substitute and solve for $D_1$.

Substitute the expression for $P_2$ back into the main formula and solve for the unknown $D_1$.

$$59 = \frac{D_1}{1.11} + \frac{D_1}{1.11^2} + \frac{\frac{D_1(1.035)}{0.11-0.035}}{1.11^2}$$

Calculating the coefficients for $D_1$:

$$59 = D_1(0.9009 + 0.8116 + \frac{1.035}{0.075 \times 1.2321}) = D_1(0.9009 + 0.8116 + 11.2007)$$

$$59 = D_1(12.9132)$$

$$D_1 = \frac{59}{12.9132} \approx \textbf{\$4.57}$$

Question 1(v) - Jan 2024: Constant Growth Model Assumptions

Problem Statement: The constant growth model of equity valuation assumes that:

  1. the dividends paid by the company remain constant
  2. the dividends paid by the company grow at a constant rate
  3. the cost of equity may be less than or equal to the growth rate
  4. the cost of equity remains constant
  5. the dividend payout ratio remains constant

Theoretical Foundation: This question tests your fundamental understanding of the core assumptions of the Gordon Growth Model.

Solution:

The correct answer is b) the dividends paid by the company grow at a constant rate. This is the defining assumption of the model, which allows us to simplify the infinite stream of future dividends into a single, computable value.

The other options are incorrect for the following reasons:

  • a) This is the assumption for a zero-growth stock, which is a special case of the dividend discount model but not the defining assumption of the constant growth model.
  • c) The model requires that the cost of equity ($R$) is strictly greater than the growth rate ($g$) to produce a finite, positive stock price. If $R \leq g$, the formula breaks down and yields a non-sensical or infinite value.
  • d) While the model assumes a constant required return, this isn't the primary defining assumption of the model itself. The constant growth rate of dividends is the most direct and crucial assumption.
  • e) While a constant payout ratio often leads to a constant growth rate, it's not a direct assumption of the model itself. A company could maintain a constant dividend growth rate even if the payout ratio changes over time, as long as the earnings growth rate adjusts accordingly.

Question 1(ii) - Sep 2023: Intrinsic Value of a Perpetual Bond

Problem Statement: What is the intrinsic value of a Tk. 1,000 face value, 8% coupon paying perpetual bond if an investor's required rate of return is 6%? (Assume annual interest payments and discounting)

  1. Tk. 1,333.33
  2. Tk. 1,000
  3. Tk. 750
  4. Tk. 75.47
  5. cannot be determined

Theoretical Foundation: The intrinsic value of a perpetual bond is calculated using the perpetuity formula, which is a core concept in the time value of money.

Solution:

Tricky Area:

Don't get confused by the Tk. 1,000 face value. This is only used to calculate the annual coupon payment. The value of a perpetuity is the cash flow divided by the discount rate, not the face value.

Step 1: Calculate the annual coupon payment.

$$\text{Annual Coupon Payment} = \text{Face Value} \times \text{Coupon Rate} = \text{Tk. } 1,000 \times 8\% = \text{Tk. } 80$$

Step 2: Use the perpetuity formula to find the intrinsic value.

$$\text{Intrinsic Value} = \frac{\text{Annual Coupon Payment}}{\text{Required Rate of Return}} = \frac{80}{0.06} = \textbf{Tk. 1,333.33}$$

The correct answer is a) Tk. 1,333.33.

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

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 integrates two key models: the Capital Asset Pricing Model (CAPM) to find the required return, and the Dividend Discount Model (DDM), specifically the Gordon Growth Model, to find the stock price.

Solution:

Given: $\beta = 1.45$, $R_f = 8\%$, $R_M = 13\%$, $D_0 = \text{Tk. } 2$, $g = 10\%$.

i. Required rate of return (CAPM):

We use the Capital Asset Pricing Model formula to find the required rate of return ($R$).

$$R = R_f + \beta \times (R_M - R_f)$$ $$R = 0.08 + 1.45 \times (0.13 - 0.08) = 0.08 + 1.45 \times 0.05 = 0.08 + 0.0725 = 0.1525 \text{ or } \textbf{15.25\%}$$

Tricky Area:

It's crucial to calculate the required return first, as it is a required input for the next part of the problem. Don't mix up the given returns with the risk-free rate and market return.

ii. Present market price per share:

We use the Gordon Growth Model with the required return from part (i).

Step 1: Calculate the next dividend ($D_1$).

$$D_1 = D_0 \times (1+g) = 2 \times (1+0.10) = \text{Tk. } 2.20$$

Step 2: Calculate the stock price ($P_0$).

$$P_0 = \frac{D_1}{R - g} = \frac{2.20}{0.1525 - 0.10} = \frac{2.20}{0.0525} \approx \textbf{Tk. 41.90}$$

iii. Impact of beta change:

If beta changes to 0.80, the new required return is:

$$R_{new} = 0.08 + 0.80 \times (0.13 - 0.08) = 0.08 + 0.80 \times 0.05 = 0.08 + 0.04 = 0.12 \text{ or } \textbf{12\%}$$

The new market price is:

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

Key Takeaway:

A lower beta indicates lower risk. This results in a lower required rate of return ($R_{new} < R$). When the required return in the denominator of the DDM decreases, the stock price increases. This demonstrates the inverse relationship between risk (beta) and stock valuation.

Question 1(iii) - Sep 2022: Constant Growth Dividend Model

Problem Statement: United Airlines will pay a Tk. 4 dividend next year on its common stock, which is currently selling at Tk. 100 per share. What is the market's required return on this investment if the dividend is expected to grow at 5% forever?

  1. 12 percent
  2. 9 percent
  3. 7 percent
  4. 5 percent
  5. 4 percent

Theoretical Foundation: This problem applies the rearranged Gordon Growth Model to find the total required return, which is composed of the dividend yield and the capital gains yield.

Solution:

Tricky Area:

Pay close attention to whether the dividend given is $D_0$ (just paid) or $D_1$ (next year's dividend). In this case, the problem states the dividend will be paid "next year," so it is $D_1$. You do not need to calculate $D_1$ from $D_0$.

Given: $P_0 = \text{Tk. } 100$, $D_1 = \text{Tk. } 4$, $g = 5\% = 0.05$.

We rearrange the formula for the Gordon Growth Model to solve for the required return ($R$):

$$R = \frac{D_1}{P_0} + g$$ $$R = \frac{4}{100} + 0.05 = 0.04 + 0.05 = 0.09 \text{ or } \textbf{9\%}$$

The market's required return is b) 9 percent.

Question 4(b) - May 2022: Stock Yield Analysis

Problem Statement: Just today, Acme Rocket, Inc.'s common stock paid a Tk.1 annual dividend per share and had a closing price of Tk. 20. Assume that the market expects this company's annual dividend to grow at a constant 6 percent rate forever. Required: (i) Determine the implied yield on this common stock. (ii) What is the expected dividend yield? (iii) What is the expected capital gains yield?

Theoretical Foundation: This problem requires an understanding of the relationship between total return, dividend yield, and capital gains yield, as defined by the Gordon Growth Model.

Solution:

Tricky Area:

The dividend given is the one that was "just today" paid, which means it is $D_0$. You must calculate $D_1$ before finding any yields or the total return. This is a common point of error.

Given: $D_0 = \text{Tk. } 1$, $P_0 = \text{Tk. } 20$, $g = 6\% = 0.06$.

i. Implied yield (Total Required Return):

The implied yield is the required rate of return ($R$), which we can find using the Gordon Growth Model formula: $R = \frac{D_1}{P_0} + g$.

First, we need to calculate $D_1$: $$D_1 = D_0 \times (1+g) = 1.00 \times (1.06) = \text{Tk. } 1.06$$

$$R = \frac{1.06}{20} + 0.06 = 0.053 + 0.06 = 0.113 \text{ or } \textbf{11.3\%}$$

The implied yield is 11.3%.

ii. Expected dividend yield:

The dividend yield is the next period's dividend divided by the current price.

$$\text{Dividend Yield} = \frac{D_1}{P_0} = \frac{1.06}{20} = 0.053 \text{ or } \textbf{5.3\%}$$

The expected dividend yield is 5.3%.

iii. Expected capital gains yield:

In the Gordon Growth Model, the capital gains yield is equal to the constant growth rate.

$$\text{Capital Gains Yield} = g = \textbf{6\%}$$

The expected capital gains yield is 6%.