Just about any combination of risk and return can be found by altering the percentage allocated to just two investments, provided one is risk free like a U.S. Government Security and the other is risky like a common stock. We will measure risk by using the standard deviation of returns.
Standard deviation, for our purposes, is a measure of the variability of returns. If stock A has a return one year of 20% and a loss of 10% the next, it has a greater standard deviation than stock B with a return of 5% one year and a loss of 2% the next. Stock A has more variable returns, so it is riskier and has a higher standard deviation.
In this table, you'll see the information on the two investments that we'll work with. One is a risk-free government security, and the other is the risky stock A:
| Security | Abbreviation | Return | Std. Deviation |
|---|---|---|---|
| US Govt T bill | Rf | 3% | 0 |
| Stock A | Rr | 20% | 10 |
We can calculate the expected return for any allocation of funds between these two securities by using this formula:
Er = W(Rf) * Er(Rf) + W(Rr) * Er(Rr)
This is where
- Er = expected return, and
- W = weight or percentage allocated to the investment
So if we allocate 50% of our money to both securities, the expected return is:
Er = 0.5 * 0.03 + 0.5 * 0.20
Er= 0.015 + 0.10
= 0.115 = 11.5%
Likewise, we can calculate the standard deviation of our 50/50 allocation with this formula:
E(Sd) = W(Rf) * Sd(Rf) + W(Rr) * Sd(Rr)
This is where:
- E(Sd) = expected standard deviation
The calculation is easy since the Sd of the risk-free security is zero:
E(Sd) = 0.5 * 0 + 0.5 *0.10 = 0.05 = 5
All of the combinations of risk and return for these two securities can be plotted on a capital allocation line (CAL) , where by changing the weight for each security you move along the line to find the level of risk and return that suits you. Check out this image; it's how it looks for these two securities:
 |
Conservative risk averse investors will choose an allocation on the lower left portion of the CAL, while those who can handle risk will move to the upper right.
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