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(rCβ)=rFβ+y(E(rPβ)βrFβ)ΟCβ=yΟPβ
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:
rCβ=yrPβ+(1βy)ΓrFβ
An example might help.
βοΈ Suppose you are investing in two stocks. x=70% of your money is in an asset earning r1β=3% and the remainder, (1β70%)=30% is in an asset earning r2β=12%. What is your total return?
β Click here to view answerr=xr1β+(1βx)Γr2β=70%Γ3%+(1β70%)Γ12%=5.7%
Based on the above formula, we can use βa powerful mathematical resultβ and some algebra to get:
E(rCβ)=yΓE(rPβ)+(1βy)ΓrFβ
This looks similar to the formula we started with, right? With a bit more algebra, we get
E(rCβ)=rFβ+y[E(rPβ)βrFβ]
Standard Deviation
For variance, we do the same thing. We go back to:
rCβ=yrPβ+(1βy)ΓrFβ
β¦ and we apply a formula we will see a couple of times:
Var(aX+bY)=a2ΓVar(X)+b2ΓVar(Y)
With a bit of algebra, we get our formula for variance:
Var(rCβ)=y2βΓVar(rPβ)
But we want standard deviation, so we take the square root of both sides:
ΟCβ=yΓΟPββ
What was the math this all was relying on?
The math used in the background is a type of math from probability called βrandom variables.β Whenever you have a quantity, such as a return or price that you canβt predict, you can think of it as a random variable. If you do,
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.