Math Problem Statement

A Stock is currently selling for $39.51 in the market.   It has a beta of 1.5. The market risk premium is 7.1 percent, and T-bills are currently yielding 4.1 percent. The most recent dividend paid was $4 per share and dividends are expected to grow indefinitely at 4.2%

What is your estimate of your expected return on this investment if you hold the stock for exactly one year ?

Solution

To estimate the expected return on the stock if you hold it for exactly one year, we can use the Capital Asset Pricing Model (CAPM) to find the required return and then factor in the dividend growth model for the stock price appreciation. Here are the steps:


Step 1: Calculate the Required Return Using CAPM

The formula for CAPM is:

re=rf+β(rmrf)r_e = r_f + \beta (r_m - r_f)

Where:

  • rer_e = Required return (expected return under CAPM)
  • rfr_f = Risk-free rate (T-bill yield = 4.1% or 0.041)
  • β\beta = Beta of the stock (1.5)
  • rmrfr_m - r_f = Market risk premium (7.1% or 0.071)

Substituting the values:

re=0.041+1.5×0.071r_e = 0.041 + 1.5 \times 0.071

re=0.041+0.1065=0.1475(14.75%)r_e = 0.041 + 0.1065 = 0.1475 \, (14.75\%)


Step 2: Estimate the Stock Price in One Year

Using the dividend growth model, the price of the stock at any time tt is given by:

Pt=Dt+1regP_t = \frac{D_{t+1}}{r_e - g}

Where:

  • PtP_t = Price at time tt
  • Dt+1D_{t+1} = Dividend expected next year
  • rer_e = Required return (from CAPM = 0.1475)
  • gg = Dividend growth rate (4.2% or 0.042)

The dividend next year, Dt+1D_{t+1}, is calculated as:

Dt+1=D0×(1+g)D_{t+1} = D_0 \times (1 + g)

Substitute D0=4D_0 = 4 and g=0.042g = 0.042:

Dt+1=4×(1+0.042)=4×1.042=4.168D_{t+1} = 4 \times (1 + 0.042) = 4 \times 1.042 = 4.168

Now calculate the expected price in one year (P1P_1):

P1=4.1680.14750.042P_1 = \frac{4.168}{0.1475 - 0.042}

P1=4.1680.105539.49P_1 = \frac{4.168}{0.1055} \approx 39.49


Step 3: Estimate the Total Expected Return for One Year

The total return consists of:

  1. Dividend Yield: Dt+1P0\frac{D_{t+1}}{P_0}
  2. Capital Gain: P1P0P0\frac{P_1 - P_0}{P_0}

Here:

  • P0=39.51P_0 = 39.51
  • Dt+1=4.168D_{t+1} = 4.168
  • P139.49P_1 \approx 39.49

Dividend Yield:

Dividend Yield=Dt+1P0=4.16839.510.1055(10.55%)\text{Dividend Yield} = \frac{D_{t+1}}{P_0} = \frac{4.168}{39.51} \approx 0.1055 \, (10.55\%)

Capital Gain:

Capital Gain=P1P0P0=39.4939.5139.510.00051(0.051%)\text{Capital Gain} = \frac{P_1 - P_0}{P_0} = \frac{39.49 - 39.51}{39.51} \approx -0.00051 \, (-0.051\%)

Total Return:

Total Return=Dividend Yield+Capital Gain\text{Total Return} = \text{Dividend Yield} + \text{Capital Gain}

Total Return=0.10550.000510.1050(10.50%)\text{Total Return} = 0.1055 - 0.00051 \approx 0.1050 \, (10.50\%)


Final Answer:

The expected return on the investment if you hold the stock for one year is approximately:

10.50%\boxed{10.50\%}


Additional Questions:

  1. How does the dividend growth rate impact the expected return?
  2. Why is the capital gain negative in this scenario despite the dividend payment?
  3. Can the CAPM model be applied to predict returns over a longer time horizon?
  4. What assumptions are made in using the dividend growth model?
  5. How sensitive is the expected return calculation to changes in the beta or market risk premium?

Tip: When estimating stock returns, always combine both the capital appreciation and dividend components for a complete picture.

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Math Problem Analysis

Mathematical Concepts

Financial Mathematics
Capital Asset Pricing Model (CAPM)
Dividend Discount Model (DDM)

Formulas

CAPM: r_e = r_f + β(r_m - r_f)
Dividend Growth Model: P_t = D_{t+1} / (r_e - g)
Total Return: Dividend Yield + Capital Gain

Theorems

Capital Asset Pricing Model (CAPM)
Dividend Discount Model (DDM)

Suitable Grade Level

Undergraduate Finance or Business Courses