Expected Return and Variability Together
In Expected Value from a Table of Relative Frequencies, you used probabilities to calculate a random variable’s mean. In earlier tutorials on Variance of a Discrete Random Variable and Interpreting Standard Deviation of a Random Variable, you learned how to describe variability around that mean. Here, you will use both summaries to compare two possible investments.
For an investment, let \(X\) represent its return over a specified period. Its expected value, \(\mu_X\), is the probability-weighted average return across many repetitions of the chance process under the model. Its standard deviation, \(\sigma_X\), describes the typical distance of returns from that expected value. A larger standard deviation indicates more variability in returns.
In this comparison, variability is one way to describe investment risk. It is important not to confuse that description with a complete measure of risk: standard deviation counts returns above and below the mean as deviations. It does not tell you by itself how likely a loss is, how large a loss might be, or whether an investor can tolerate one.
The two summaries answer different questions. The mean describes the distribution’s probability-weighted center; the standard deviation describes its spread in the original units. If one investment has a higher expected return and a lower standard deviation than another, it is better on both of these particular measures. If one has a higher mean but also a higher standard deviation, the summaries show a tradeoff rather than a single obvious winner.
A Consistent Comparison Process
Before comparing values, make sure each random variable describes returns over the same time period. Comparing an expected one-year return with a standard deviation for a monthly return would mix different time scales. Also use consistent units: dollars with dollars, or percentage returns with percentage points.
State what each random variable measures and specify the time period and units.
Confirm that its probabilities are valid and add to 1, as in Checking Whether a Probability Distribution Is Valid.
Use the probability-weighted mean and the standard deviation procedure developed earlier in this unit. Keep the summaries paired with the correct investment.
Describe which investment has the higher expected return and which has more variability. Do not treat one summary as a replacement for the other.
State the tradeoff in context. Mean and standard deviation describe the distributions but do not determine an investor’s preferences.
For each distribution, the mean is \(\mu_X=\sum xP(X=x)\), and the variance is \(\sigma_X^2=\sum (x-\mu_X)^2P(X=x)\), with standard deviation \(\sigma_X=\sqrt{\sigma_X^2}\). These formulas were developed in the earlier tutorials on the mean, variance, and standard deviation of a discrete random variable. The examples show the calculations so you can compare the summaries carefully.
Worked Example: Two Investments in Dollars
Worked Example: Two Investments in Dollars
Suppose two fictional investments have the following possible net gains over one year. Let \(A\) and \(B\) be the net gain, in dollars, from one year in each investment. Compare their expected gains and standard deviations.
| Net gain \(x\) | \(P(A=x)\) | \(P(B=x)\) |
|---|---|---|
| \(-\$200\) | 0.20 | 0.10 |
| \(\$100\) | 0.50 | 0.50 |
| \(\$300\) | 0.30 | 0.40 |
State. We will compare the modeled average one-year net gain and the variability of the net gain for investment \(A\) and investment \(B\).
Plan. Each set of probabilities is nonnegative and sums to 1: for \(A\), \(0.20+0.50+0.30=1\); for \(B\), \(0.10+0.50+0.40=1\). The returns are both measured in dollars over one year, so their means and standard deviations can be compared directly. Calculate each mean, then calculate each variance and take its square root.
Do: calculate the means. For investment \(A\), the probability-weighted mean is:
For investment \(B\), the calculation is:
Do: calculate the standard deviations. For \(A\), the deviations from its mean of \(\$100\) are \(-\$300\), \(\$0\), and \(\$200\). Thus:
For \(B\), the deviations from its mean of \(\$150\) are \(-\$350\), \(-\$50\), and \(\$150\). Therefore:
Check the variances using \(\sigma_X^2=\sum x^2P(X=x)-\mu_X^2\). For \(A\), \((-200)^2(0.20)+(100)^2(0.50)+(300)^2(0.30)=8{,}000+5{,}000+27{,}000=40{,}000\), and \(40{,}000-100^2=30{,}000\). For \(B\), the corresponding sum is \(4{,}000+5{,}000+36{,}000=45{,}000\), and \(45{,}000-150^2=22{,}500\). Both checks agree.
Conclude. Under these models, investment \(B\) has a higher expected one-year net gain (\(\$150\) compared with \(\$100\)) and a lower standard deviation (\(\$150\) compared with about \(\$173.21\)). By these two summaries, \(B\) is better on both average gain and variability. This comparison does not guarantee that \(B\) will earn more in a particular year.
Worked Example: Higher Average Return, More Variability
Worked Example: Higher Average Return, More Variability
A fictional investor compares two funds using modeled annual returns. Let \(S\) and \(R\) be the return, in percent, for one year in a steadier fund and a riskier fund, respectively. The values in the table are percentage returns: for example, \(-5\) means a loss of 5% for the year.
| Annual return \(x\), in percent | \(P(S=x)\) | \(P(R=x)\) |
|---|---|---|
| \(-5\) | 0.10 | 0.20 |
| \(6\) | 0.60 | 0.50 |
| \(12\) | 0.30 | 0.30 |
| \(40\) | — | — |
The final row is not needed for these distributions; it is omitted from both probability models. Each distribution’s listed probabilities sum to 1.00. Calculate the mean and standard deviation for each fund.
Calculate the mean for \(S\). Its expected return is:
Calculate the standard deviation for \(S\). The deviations from \(6.7\) percentage points are \(-11.7\), \(-0.7\), and \(5.3\). The variance and standard deviation are:
Calculate the mean and standard deviation for \(R\). Its expected return is:
The deviations from \(5.6\) percentage points are \(-10.6\), \(0.4\), and \(6.4\). Thus:
Check the variance calculations using the alternative formula. For \(S\), the weighted sum of squared returns is \(25(0.10)+36(0.60)+144(0.30)=2.5+21.6+43.2=67.3\); \(67.3-(6.7)^2=67.3-44.89=22.41\). For \(R\), it is \(25(0.20)+36(0.50)+144(0.30)=5+18+43.2=66.2\); \(66.2-(5.6)^2=66.2-31.36=34.84\). These match the variance calculations from weighted squared deviations.
The steadier fund \(S\) has a higher expected annual return, \(6.7\%\) compared with \(5.6\%\), and a lower standard deviation, about 4.73 compared with 5.90 percentage points. The standard deviations are in percentage points, the same original units as the returns. This modeled comparison favors \(S\) on both average return and variability, but it does not predict the return in any one year.
Worked Example: Same Mean, Different Spread
Worked Example: Same Mean, Different Spread
Two other fictional investments have the following modeled one-year returns. Let \(C\) and \(D\) be returns, in percent. Compare their expected returns and variability.
| Return \(x\), in percent | \(P(C=x)\) | \(P(D=x)\) |
|---|---|---|
| \(-10\) | 0.25 | — |
| 0 | — | 0.50 |
| 10 | 0.50 | — |
| 20 | — | 0.50 |
| 30 | 0.25 | — |
Each distribution is valid: the probabilities for \(C\) total \(0.25+0.50+0.25=1\), and the probabilities for \(D\) total \(0.50+0.50=1\).
Calculate the means. For \(C\), \(\mu_C=(-10)(0.25)+(10)(0.50)+(30)(0.25)=-2.5+5+7.5=10\%\). For \(D\), \(\mu_D=(0)(0.50)+(20)(0.50)=10\%\). The two modeled expected returns are equal.
Calculate the standard deviations. For \(C\), the deviations from 10 are \(-20\), \(0\), and \(20\):
For \(D\), the deviations from 10 are \(-10\) and \(10\):
Check using weighted squared returns. For \(C\), \(100(0.25)+100(0.50)+900(0.25)=25+50+225=300\), and \(300-10^2=200\). For \(D\), \(0^2(0.50)+20^2(0.50)=200\), and \(200-10^2=100\). The results agree.
Both investments have the same expected return, 10%, but \(C\) has the larger standard deviation, about 14.14 percentage points compared with 10 percentage points for \(D\). According to these models, returns for \(C\) are more variable around the same expected return. The comparison does not say that every outcome for \(C\) is worse: greater variability includes values farther below and above the mean.
Common Mistakes and AP Exam Tips
- Comparing only expected returns. A higher mean does not describe how much returns vary. A full comparison reports both the expected return and standard deviation for each option.
- Calling standard deviation the chance of losing money. Standard deviation measures typical distance from the mean, not the probability of a loss. If asked about the probability of a loss, use the distribution to add the probabilities of the outcomes that are losses.
- Assuming more variability means every outcome is worse. Standard deviation reflects deviations on both sides of the mean. It does not distinguish favorable high returns from unfavorable low returns.
- Mixing time periods or units. Compare returns over the same specified period. A standard deviation measured in percentage points should be interpreted alongside percentage returns, not dollar gains.
- Choosing a universal “best” investment from two summaries. If one option offers a higher mean but also a higher standard deviation, say that it has a higher expected return and more variability. Which tradeoff is acceptable depends on the investor’s goals and tolerance for risk.
- Confusing variance and standard deviation. Variance is in squared units; standard deviation is in the original units. To describe typical variation in dollars or percentage points, interpret the standard deviation.
A full-credit comparison names both investments, states which has the higher expected return and which has the greater standard deviation, includes the units and time period, and explains the result in context. Avoid saying that an expected value is a guaranteed outcome or that a smaller standard deviation guarantees a smaller loss.
Key Takeaway
Expected value and standard deviation describe complementary features of an investment’s modeled return distribution. Compare means to discuss average return and standard deviations to discuss variability. A higher mean and lower standard deviation favor an option on both measures; a higher mean paired with a higher standard deviation presents a tradeoff.
Check Your Understanding
Use the fictional return distributions below. Returns are measured in percent over one year. Let \(M\) and \(N\) be the returns for two investment options.
| Return \(x\), in percent | \(P(M=x)\) | \(P(N=x)\) |
|---|---|---|
| \(-10\) | 0.20 | 0.10 |
| 5 | 0.50 | 0.60 |
| 20 | 0.30 | 0.30 |
- Check that each distribution’s probabilities are valid and sum to 1.
- Calculate \(\mu_M\) and \(\mu_N\), showing the probability-weighted products.
- Calculate \(\sigma_M\) and \(\sigma_N\), showing the weighted squared deviations and square roots.
- Which investment has the higher expected return, and which has more variability? State the comparison in context and include units.
- Explain why the standard deviation is not the probability of a negative return, and why the two summaries alone do not settle every investor’s choice.