✏️ CAL Examples

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Name

Equation

Example

Expected Return, $E(r_C)$

$E(r_C) = r_F + y (E(r_P)-r_F)$
or
$E(r_C) = y E(r_P) + (1-y) r_F$

= .04 + .50*(.12-.04) = .08 or 8%
or
= (.12 * .50) + (.04 * .50) = 8%

Standard Deviation of Complete Port, $σ_c$

$σ_c = y σ_p$

= .20 * .50 = .10 or 10%

Variance, $Var$

$Var = σ^2$ = σ^2

Var = 12%^2 = .0144

Standard Deviation, $σ$

$σ = \sqrt{Var} = Var^{\frac{1}{2}}$

σ = .0144^.5 = 12%

Risk Premium

Risk Premium $= E(r_C)-r_F$

= 12% - 2% = 10%

Sharpe Ratio, $S$

$S = \frac{E(r)-r_F}{σ}$

= (.12-.04)/.20 = (8/20) = .40

Equation for CAL

$E(r_C) = r_F + \frac{E(r_P) - r_F}{σ_P}σ_C$

Sharpe ratio = .8
E(r_C) = r_F + Sσ_C = 2% + .8 * 15% = 14%

Notation

$r_F$ = Return of Risk Free Assets

$r_P$ = Return of Portfolio of Risky Assets

$r_C$ = Return of Complete Portfolio

$E(r_P) / E(r_C)$ = Expected Return of Risky/Complete Portfolio

Occasionally I use $Er_P/Er_C$ as shorthand for $E(r_P)/E(r_C)$

$y$ = % of Portfolio in Risky Assets

$1-y$ = % in Risk-Free Assets

$σ/σ_P/σ_C$ = Standard Deviation

$S$ = Sharpe Ratio

Variance = Standard Deviation^2

Standard Deviation = SQRT of Variance

→ Complete formula page

✏️ You manage the following risky portfolio:

$σ_P$	$E(r_p)$
$0.11107$	$0.07543$
$r_f = 3\%$

A client walks in and says that they only want a $sd = 5\%$. What percentage of their assets should be in the risky portfolio?

✔ Click here to view answer

1. Equation → $σ_c = y × σ_p$ a. We want $y$, so we need a formula with $y$ in it. We can’t use $E_{rc} = rF + y × (ErP-rF)$ because we don’t know $y$ or $E_{rc}$. However, we can use $σ_c = σ_p × y$ because we know both risk levels.
2. Plug 🔌 → $5\% = 11.10\% × y$
3. Solve 🚂 → $y = \frac{5\%}{11.10\%} = 45.045\%$
4. 🧠 → This client will invest 45% of their net worth in risk assets. ✅

✏️ You manage the following risky portfolio:

$σ_P$	$E(r_p)$
0.11107	0.07543
$r_f = 3\%$

A client walks in and says that they only want a $Er_c = 12\%$. What percentage of their assets should be in the risky portfolio?

✔ Click here to view answer

We approach this by finding an equation and plugging numbers into it, then solving for y. This approach is a key tool for solving questions like this. It is often called “Plug and Chug,” because all you have to do is plug numbers into an equation and chug through a little algebra until you get the answer. Click the “help” link below for details.

Plug and chug: (help)

1. Equation:
   $Er_c = r_F + y (Er_P - r_F)$
2. Plug:🔌
   $12\% = 3\% + y (7.543\% - 3\%)$
3. Solve: 🚂
   $12\% = 3\% + y (7.543\% - 3\%)$
   $9\%=y×4.543$
   $y= \frac{9\%}{4.543\%}=198.107\%$
4. Reflect: 🧠
   This client will invest 198% of their net worth in risk assets.

To achieve these returns, the client would have to invest 198.107% of their assets in your portfolio. It is unlikely that they would be able to borrow this much money at anything near $r_F$, so this target is likely unrealistic. You may have to guide them toward more realistic goals.

… However, if they could borrow $y=198.107\%$ of their assets at the risk free rate, and did so, their final expected return would be

$$\begin{aligned} E(r_C) &= r_F + y (E(r_P)-r_F)\\ &= 3\% + 198.107\% (7.543\%-3\%) \\ &= 12.00\% \;\;\text{✅} \end{aligned}$$

✏️ Suppose that 50% of your complete portfolio is in risky assets and 50% of your risky assets are equities. What percent of your money is invested in equities?

✔ Click here to view answer

Half of your money is in risk assets and half of that is in equities, so you have $50\% × 50\%=25\%$ in equities.
In general:
% of complete portfolio in equities = % in risk asset $×$ weight of equities in risk assets. ✅

✏️ Suppose that 80% of your complete portfolio is in risky assets and that 60% of your risky assets are equities. What percent of your money is invested in equities?

✔ Click here to view answer

60% of 80% $= 60\% × 80\%=48\%$ ✅