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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(r_S), 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\%.