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Rob's s1452 Notes

✏️ Examples (CAPM)

✏️ You are considering investing in two portfolios: A and B:
  rF=3%r_F = 3\%
  E(rA)=18%E(r_A) = 18\%
  σA=25%σ_A = 25\%
  E(rB)=13%E(r_B) = 13\%
  VarB=.0169Var_B = .0169
  ρAB=.15ρ_{AB} = .15
What is the standard deviation of portfolio B and the covariance between A and B?

✔ Click here to view answer σB=.0169.5=0.13σ_B = .0169^.5 = 0.13Cov(rA,rB)=σA×σB×ρA,B=25%×13%×.15=0.0049Cov_{(rA,rB)} = σ_A × σ_B × ρ_{A,B} = 25\% × 13\% × .15 = 0.0049

✏️ Suppose you invest wA=60%wA=60\% in portfolio A and wB=40%wB=40\% in portfolio B. What is the Expected return, variance, standard deviation, risk premium, and Sharpe ratio of the resulting portfolio?

✔ Click here to view answer E(rp)=(w1×Er1)+(w2×Er2)=60%×18%+40%×13%=16%\begin{aligned} E(r_p) &= (w_1×Er1) + (w_2×Er_2) \\ &= 60\% × 18\% + 40\% × 13\% \\ &= 16\% \end{aligned}Var(rp)=σ12×w12+σ22×w22+2w1×w2×σ1×σ2×Corr1,2=25%2×60%2+13%2×40%2+2×40%60%×25%×13%×.15=0.0275\begin{aligned} Var(r_p) &= σ_1^2 × w_1^2 + σ_2^2 × w2^2 + 2w_1 × w_2 × σ_1 × σ_2 × Corr_{1,2} \\ &= 25\%^2 × 60\%^2 + 13\%^2 × 40\%^2 + 2 × 40\% 60\% × 25\% × 13\% × .15 \\ &= 0.0275 \end{aligned}σ=Var.5=.0275.5=0.1658=16.58%\begin{aligned} σ &= Var ^.5 \\ &= .0275^.5 \\ &= 0.1658 \\ &= 16.58\% \end{aligned}Risk Premium =E(rc)−rF=16%−3%=13%\begin{aligned} \text{Risk Premium }&= E(r_c) - r_F \\ &= 16\% - 3\% \\ &= 13\% \end{aligned}Sharpe ratio = Reward to Risk Ratio=Er−rfσ=16%−3%16.58%=.7841\begin{aligned} \text{Sharpe ratio }&=\text{ Reward to Risk Ratio} \\ &= \frac{E_r-r_f}{σ} \\ &= \frac{16\%-3\%}{16.58\%} \\ &= .7841 \end{aligned}

✏️ You are considering investing in a stock. Its current price is $78 and you expect that next year it will pay a dividend of $3 and have a price of $85. It has a ββ of 1.11.1. rF=3%r_F = 3\% and E(rM)=11%E(r_M)=11\%. Is this stock overpriced or underpriced?

Note: for a discussion of questions like this, see: ✏️ CAPM, Dividends, and Holding Period Return

✔ Click here to view answer

We have the tools to calculate HPR and to calculate what the CAPM says a fair return would be. Let’s calculate both and then compare them to see whether the stock is overpriced or underpriced.

HPR=(NewPrice −OldPrice + Dividend)OldPrice =$85−$78+$3$78=13%\begin{aligned} HPR &= \frac{(\text{NewPrice }- \text{OldPrice }+\text{ Dividend})}{\text{OldPrice }} \\ &= \frac{\$85-\$78+\$3}{\$78} \\ &= 13\% \end{aligned}CAPME(rS)=3%+1.1×(11%−3%)=11.8%\begin{aligned} CAPM E(r_S) &= 3\% + 1.1 × (11\%-3\%) \\ &= 11.8\% \end{aligned}

The stock “should” have a return of 11.8%, but it actually will have a return of 13%. The only way for it to have a return this high is if it is currently underpriced and is currently a good deal. Underpriced. It has an alpha of 13%−11.8%=1.2%13\%-11.8\% = 1.2\%

✏️ Consider the above stock. If it is currently underpriced, what would a fair price for it be?

✔ Click here to view answer

We have equations that connect up all of the relevant variables, so let’s just “Plug and Chug.”

Plug and chug: (help)
  1. Equation:

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

  2. Plug:🔌

    Fair HPR=11.8%=$85−P0+$3P0\text{Fair } HPR = 11.8\% = \frac{\$85 - P_0 + \$3}{P_0}

  3. Solve: 🚂

    11.8%×P0=($85−P0+$3)11.8\% × P_0 = (\$85-P_0 +\$3)
    11.8%×P0=$88−P011.8\% × P_0 = \$ 88 - P_0
    .118×P0+1×P0=$88.118 × P_0 + 1 × P_0 = \$88
    1.118×P0=$881.118 × P_0 = \$88
    P0=$881.118=$78.71P_0 = \frac{\$88}{1.118} = \$78.71

  4. Reflect: 🧠

    We think that the stock should be _$78.71_ right now. Earlier, when P0P_0 was $78.00, the stock was underpriced and the HPR was 13%, which was higher than the CAPM suggested. Now, at the higher price of $78.71, it’s not quite as good of a deal and only has an HPR of 11.8%, in line with the CAPM’s prediction.

✏️ In the previous problem, we assumed that the market was wrong about the current price of the stock - ie that the stock was underpriced. Let’s instead consider that perhaps our projection of the future price ($85) is incorrect. Based on the actual current price fo the stock and on CAPM’s prediction of E(rS)E(rS), what future price would you predict for the stock? In other words, what would the future stock price have to be to give the stock an 11.8% return?

✔ Click here to view answer

Plug and chug: (help)

Plug and chug: (help)
  1. Equation:

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

  2. Plug:🔌

    Fair HPR=11.8%=P1−$78+$3$78\text{Fair }HPR = 11.8\% = \frac{P_1 - \$78 + \$3}{\$78}

  3. Solve: 🚂

    11.8%×$78=$9.2=P1−$78+$311.8\% × \$78 = \$9.2 = P_1 - \$78 + \$3
    $9.2=P1−$78+$3\$9.2 = P_1 - \$78 + \$3
    p1=$9.2+$78−$3=$84.2p_1 = \$9.2 + \$78 - \$3 = \$84.2

  4. Reflect: 🧠

    The market seems to think that the future price will be $84.2.

✏️ Given the numbers we’ve discussed before, would an Expected Return of 14% for this stock be consistent with the CAPM?

✔ Click here to view answer

We know that rF=3%r_F=3\%, β=1.1β=1.1 and E(rM)=11%E(r_M)=11\% from above.
Based on this, the CAPM implies that E(rS)=3%+1.1×(11%−3%)=11.8%E(r_S) = 3\% + 1.1 × (11\%-3\%) = 11.8\%
This is not consistent with having an expected return of 14%14\%.