Risk-Return Analysis: Definition & Methods

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  • 0:03 Risk and Return
  • 0:50 The Capital Allocation…
  • 3:22 The Efficient Frontier
  • 4:44 Diverse Portfolios
  • 5:19 Lesson Summary
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Lesson Transcript
Instructor: James Walsh

M.B.A. Veteran Business and Economics teacher at a number of community colleges and in the for profit sector.

To get a higher return in the investment world you have to take more risk. Two tools that help the investor balance the two are the capital allocation line and the efficient frontier. Let's see how these tools work with examples and illustrations.

Risk and Return

A central issue in investing is finding the right combination of risk and return. An investment like a U.S. Government Security has a small percentage return, but it's considered risk free, as the U.S. Government hasn't defaulted on an obligation in its 250 year history.

To get a better return than government securities, many invest in the stock market. But we all know that stock prices go up and down and if you buy in at the wrong time you can lose a good bit of money. Stocks can get you the better return, but they are more risky. To get more return in the investment world you just have to take more risk.

So what can help an investor balance the two? Let's look at some tools that will help us get at the answer.

The Capital Allocation Line (CAL)

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:

The risk-free asset is at the lower left end of the line and the risky asset is on the upper right.

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.

The Efficient Frontier

Now we will broaden our example to include all of the portfolios possible from combinations of different stocks. Each portfolio's expected return and standard deviation can be calculated like we just did, except that there will be correlations between the stock's price movements to adjust for. The returns and standard deviations can be plotted on a graph like this one:

The top half of the bullet shaped curve is the efficient frontier.
Efficient frontier

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