🔎 Where did our two formulas come from?

Click/tap for Lecture 1 Formulas

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

The two most-important formulas from this week tell us how our expected return and standard deviation are based on the percentage of our wealth we allocate to the risky asset (we invest the rest in risk-free assets).

$$E(r_C) = r_{F} + y (E(r_P) - r_F)$$ $$σ_C = {y σ}_{P}$$

Where do they come from?

Expected Return

Well, a basic principle of finance is that if you split your money between two assets, your complete return is a weighted average of the returns of the two assets:

$$r_C = yr_P + (1-y) × r_F$$

An example might help.

✏️ Suppose you are investing in two stocks. $x=70\%$ of your money is in an asset earning $r_1 =3\%$ and the remainder, $(1-70\%)=30\%$ is in an asset earning $r_2 = 12\%$. What is your total return?

✔ Click here to view answer

$$r = xr_1 + (1-x) × r_2 = 70\% × 3\% + (1-70\%) × 12\% = 5.7\%$$

Based on the above formula, we can use “a powerful mathematical result” and some algebra to get:

$$E(r_C) = y × E(r_P) + (1-y) × r_F$$

This looks similar to the formula we started with, right? With a bit more algebra, we get

$$E(r_C) = r_F + y [E(r_P)-r_F]$$

Standard Deviation

For variance, we do the same thing. We go back to:

$$r_C = yr_P + (1-y) × r_F$$

… and we apply a formula we will see a couple of times:

$$Var(aX + bY) = a^2 × Var(X) + b^2 × Var(Y)$$

With a bit of algebra, we get our formula for variance:

$$Var(r_C) = y_2 × Var(r_P)$$

But we want standard deviation, so we take the square root of both sides:

$$\colorbox{#FFD300}{\textcolor{black}{$σ_C = y × σ_P$}}$$

What was the math this all was relying on?

The math used in the background is a type of math from probability called “random variables.” Whenever you have a quantity, such as a return or price that you can’t predict, you can think of it as a random variable. If you do,

$E(aX + bY) = aE(X) + bE(Y)$

$Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$

$Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2ab\left(σ_xσ_yρ_{x,y}\right)$

These are important equations, so of course we would apply the mathematics of random variables to analyze finance and gain insights into optimal portfolio composition and diversification.