✏️ 2 Risky and 1 Riskless asset

This is definitely a stretch question, but people said it tied things together nicely. It shows a full optimization with two risky assets and a risk-free asset.

Big Picture Strategy

✏️ Consider the following assets:

	E(r)	σ	Corr w/T-Bill	Corr w/ Bond	Proportions
T-Bill	3%	0	1	0	30%
Bond fund	4%	8%	0	1	20.4%
Stock Fund	9%	15%	0	0.1	49.6%

How would you approach finding $E(r_c)$ and $σ_C$?

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Strategy:

1. Calculate $E(r_p)$ and $σ_P$ for the risky portfolio
   • From “optimal risky portfolio” lecture:
   $$σ_p = SQRT((σ_1^2 \times w_1^2) + (σ_2^2 \times w_2^2)+ (2 \times w_1 \times w_2 \times σ_1 \times σ_2 \times ρ_{1,2}))$$ $$E(r_p) = (Er_1 \times w_1) + (Er_2 \times w_2)\text{...}$$
   2. And then do a CAL calculation for $E(r_C)$ and $σ_C$ using these new numbers.
   $$Er_c = y \times Er_P + (1-y) \times r_F$$ $$σ_c = σ_p \times y$$

Step 1: the risky portfolio

✏️ Find $E(r_c)$ and $σ_C$ for the optimal risky portfolio.

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In the complete portfolio,
Bonds: 20.4% of COMPLETE portfolio
Stocks: 49.6% of COMPLETE portfolio
Bills: 30% of COMPLETE portfolio.

Therefore, Risk assets are 70% of complete portfolio.
$w_1 = \frac{20.4\%}{70\%} = 29.14\%$ of risky assets are bonds.
$w_2 = \frac{49.6\%}{70\%} = 70.86\%$ of risky assets are stocks.

$$\begin{aligned} E(r_p) &= (Er_1 \times w_1) + (Er_2 \times w_2)\text{...} \\ &=(4\% \times 29.14\%) + (9\% \times 70.86\%) \\ &= 0.07543 \end{aligned}$$$$\begin{aligned} σ_p &= SQRT((8\%^2 \times 29.14\%^2) + (15\%^2 \times 70.86\%^2) + (2 \times 29.14\% \times 70.86\%*8\%*15\%*.1)) \\ &=11.1\end{aligned}$$

Step 2: The Complete Portfolio

✏️ Find $E(r_c)$ and $σ_C$ for the optimal complete portfolio.

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30% risk free
70% risky

$$$$$$\begin{aligned} Er_c &= 70\% \times 0.07543 + 30\% \times 3\% \\ &= 6.18\% \end{aligned}$$

Let’s check our calculations by calculating it another way:

$$\begin{aligned} E(r_p) &= (Er_1 \times w_1) + (Er_2 \times w_2) + (Er_3 \times w_3) \\ &= 3\% \times 30\% + 4\% \times 20.4\% + 9\% \times 49.6\% \\ &= 6.18\% \end{aligned}$$

Great!

$$σ_c = σ_p \times y$$$$σ_c = 11.1\% \times 70\% = 7.77\%$$