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Rob's s1452 Notes

✏️ CAPM, Dividends, and Holding Period Return

The key equation in the CAPM tells you the return that you expect to get on any security.

E(rs)=rf+β(E(rM)−rf)E(r_s) = r_f + β(E(r_M)-r_f)

The actual return you get is your Holding Period Return (HPR):

HPR=P1−P0+DP0=  P1−P0P0  +  DP0↑capital gainearnings yield↑\begin{aligned} HPR=\frac{P_1-P_0+D}{P_0} = &\; \textcolor{green}{\frac{P_1-P_0}{P_0}} \;+\; \textcolor{red}{\frac{D}{P_0}} \\ &\quad\textcolor{green}{\text{↑capital gain}} \\ &\quad\textcolor{red}{\text{earnings yield↑}} \end{aligned}

The CAPM says that the Holding Period return is determined by the CAPM formula:

P1−P0+DP0=rf+β(E(rM)−rf)\frac{P_1-P_0+D}{P_0} = r_f + β(E(r_M)-r_f)

✏️ Suppose rf=1%r_f=1\%, β=1.1β=1.1, and the market risk premium is 11%11\%.
If you expect a stock to be worth $100 in 1 year and you expect a $4 dividend, what is the fair price for the stock today?

✔ Click here to view answer

We start by calculating the Expected Return using CAPM. Because we are given the risk premium, E(rM)−rf=11%E(r_M)-r_f=11\%, rather than the expected return, we plug 11% directly into the formula without subtracting 1%.

E(rs)=rf+β(E(rM)−rf)=1%+1.1(11%)=13.1%E(r_s) = r_f + β(E(r_M)-r_f) = 1\% + 1.1(11\%) = 13.1\%

Next, we “Plug and Chug:“

Plug and chug: (help)
  1. Equation:

    HPR=P1−P0+DP0HPR=\frac{P_1-P_0+D}{P_0}

  2. Plug:🔌

    13.1%=100−P0+4P013.1\%=\frac{100-P_0+4}{P_0}

  3. Solve: 🚂

    13.1%×P0=100−P0+413.1\% × P_0 = 100 - P_0 + 4
    1.131×P0=1041.131 × P_0 = 104
    P0=1041.131=91.95P_0 = \frac{104}{1.131} = 91.95

  4. Reflect: 🧠

    The fair price for the stock is $91.95 today. Plugging this number back into the HPR formula, we confirm that this price will yield a return of 13.1% (within rounding error): $100−$91.95+$4$91.95=0.13105≈13.1%\frac{\$100-\$91.95+\$4}{\$91.95} = 0.13105 \approx 13.1\%

✏️ Continuing the last problem, suppose that the actual market price of the stock is $90. Is it overpriced or underpriced?

✔ Click here to view answer

This stock should have a return of 12%12\%:
  CAPM=r=12%CAPM = r = 12\%
However, we expect it to earn 15.56%15.56\%
  HPR=100−90+490=15.56%HPR = \frac{100-90+4}{90} = 15.56\%
A bond currently has a YTM of 15.56%. The CAPM says it should have a return of 12%. It’s return is higher than it should be. This must be because it’s currently underpriced. You should buy it, because you are buying it cheaply.
Because the stock is doing better than we would expect, it is outperforming and has a positive alpha:
  α=15.56%−12%=3.56%α= 15.56\% - 12\% = 3.56\%
The only way for a security to have a positive alpha is if it is currently underpriced. Therefore, this security is underpriced.

Another way to think about it is that if you know the future value of a stock, then:

  • Anything with a return higher than you would expect has a high return because it is cheap (undervalued).
  • Anything with a lower return than you would expect has a high return because it is expensive (overvalued).

Suppose a stock will be worth $10 in a year (you will receive a dividend of $1 and can sell it for $9). We can think about it as a “fixed income” instrument (ie a bond) in this case.

  • When it’s current price goes down (relative to its future value), it’s return goes up.
  • When it’s current price goes up, it’s return goes down. Just like a bond…

Based on its nondiversifiable risk it should have a return of 11.11%. If it had a price of $9 today, it would get that return.

With a price above $9, it is overpriced and will have a return less than 11.11%. (🧠: overpriced = low return).

With a price below $9, it is underpriced and will have a return higher than 11.11% (🧠: underpriced = high return).

Another way to approach the underpriced/overpriced question is to ask what the return is. If you know how much the stock is going to be worth next year and the current stock price leads to a return greater than CAPM would predict, it means that the current stock price is too low. Remember, lower prices always mean higher returns.

It’s just like bonds:

Price is too low ←→ Return is too high.

For example, suppose a security is worth $10 next year. You calculate out that based on its current price of $9, the stock will have a return of 11.11%. However, based on its nondiversifiable risk (β)(β), you feel that it should have a return of 15%. In this case***, IS OVERPRICED***. To figure out the price that would give it a return of 15%, just take the present value:

PS=$10.0001.000+15.00%=$8.7P_S = \frac{\$10.000}{1.000+15.00\%} = \$8.7

Sure enough, it’s actual price is $9, higher than the price that would give it it’s “fair” return of 15%.

✏️ Suppose rf=1%r_f=1\%, β=1.1β=1.1, and the market risk premium is 10%10\%.
If a stock is worth $90 today and you expect a $4 dividend, what does the CAPM predict it will be worth in 1 year?

✔ Click here to view answer

We start by calculating the Expected Return using CAPM:

E(rs)=rf+β(E(rM)−rf)=12%E(r_s) = r_f + β(E(r_M)-r_f) = 12\%

Next, we “Plug and Chug:“

Plug and chug: (help)
  1. Equation:

    HPR=P1−P0+DP0HPR = \frac{P_1-P_0+D}{P_0}

  2. Plug:🔌

    12%=P1−$90+4$9012\% = \frac{P_1-\$90+4}{\$90}

  3. Solve: 🚂

    12%×$90=P1−90+412\% × \$90 = P_1 - 90 + 4
    $10.8=P1−86\$10.8 = P_1 - 86
    P1=10.8+86=96.8P_1 = 10.8 + 86 = 96.8
    96.8−90+490=0.12\frac{96.8-90+4}{90}=0.12

  4. Reflect: 🧠

    The market expects a capital gain of $6.8. Together with a dividend of $4, the total return of 4+6.8=10.8 is roughly 12% of our $90 investment. That sounds reasonable.