π Where did our two formulas come from?
or
or
= (.12 * .50) + (.04 * .50) = 8%
E(rC) = rF + SΟC = 2% + .8 * 15% = 14%
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).
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:
An example might help.
βοΈ Suppose you are investing in two stocks. of your money is in an asset earning and the remainder, is in an asset earning . What is your total return?
β Click here to view answer
Based on the above formula, we can use βa powerful mathematical resultβ and some algebra to get:
This looks similar to the formula we started with, right? With a bit more algebra, we get
Standard Deviation
For variance, we do the same thing. We go back to:
β¦ and we apply a formula we will see a couple of times:
Applying the second formula to the first, we get our formula for variance:
But we want standard deviation. Recalling that , we get:
Taking the square root of both sides, we get our main equation:
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.
Note that when, as Bruce mentions in the slides, when two variables are independent, the Covariance and correlation between them is zero. Therefore, the second and third formulas both become:
This is the equation we used, above.
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