Weighted Averages

Another recurring mathematical theme in the CFA is the weighted average (probably because it is so universally useful). It appears in portfolio management, expected returns, WACC, indexing etc. It essentially takes the components of a group, takes the proportional weight of each and determines the value of the aggregate. Example:

S = (wα x vα) + (wε x vε) + (wρ x vρ) + ...
Where w is the weight of of each component expressed as a percentage of the whole and v is the value of each component.

For portfolio management, each component is an individual security's expected rate of return.

[Example] A portfolio is make of three stocks A, B and C. A has an expected return of 8% and makes up 20% of the portfolio. B has an expected return of 10% and makes up half of the portfolio. Finally C has a return of 12%. What is the expected return of the portfolio?

[Solution] E (Rp) = wA x E (RA) + wB x E (RB) + wC x E (RC)
= 8% x 20% + 10% + 50% + 12% x 30%
= 1.6% + 5% + 3.6
= 10.2%

For weighted average cost of capital (WACC), each component (debt, mezz and equity financing) of cost is weighted by proportion again:
[Example] A company issues bonds with at a cost of 4% which accounts for half of their capital. The required rate of return for their projects is 9% and they have an issue of preferred shares out for of 6%. They have twice as many common shares issued as preferred. If their tax rate is 30%, what is their WACC? Assuming that preferred shares are treated as debt, what is their total financial leverage ratio?

[Solution] Note in this case: we = 2wp and wd = 50%
wd = 50% = 100% - wp - we
wp = 16.7%
we = 33.3%

WACC = wd x kd x (1 - tax rate) + wp x kp + we x ke
= 50% x 4% x (1 - 30%) + 16.7% x 6% + 33.3% x 9%
= 1.2 + 1% + 3%
= 5.2%

Financial Leverage of Assets (FLA) = A / E
On a percentage basis:
A = wd + wp + we = 100%
FLA = 100% / 33.3%
= 3

[Example] An market weighted index is composed of three stocks A, B and C. A is worth $50 and composes 50% of the index. B is worth $10 and is 30% of the index. If C increases in value by 15%, what is the increase in the index?

[Solution] Initial Index = 100%
Final Index = 50% + 30% + 20% x (1.15) = 103%

The index increases by 3%