Risk-Reward Trade-Offs

Why Risk and Reward Are Considered Together

An organization may accept uncertainty because an action offers the possibility of achieving a worthwhile objective. A hypothetical insurer, for example, might consider a new product line because it could generate additional earnings, while recognizing that actual results depend on uncertain claims and expenses. The possibility of reward can make accepting risk reasonable, but it does not make the reward certain.

A useful starting point is to describe reward as a measurable outcome, such as investment return or earnings, and risk as uncertainty about that outcome. In a simple model, variability in possible returns can summarize uncertainty. That is only one way to describe risk: variability alone does not show whether outcomes are favorable or unfavorable relative to an objective. Earlier material on defining risk relative to objectives is relevant here: the same uncertain outcome can matter differently depending on what the organization is trying to achieve.

The central idea is a possible trade-off, not a universal law. Accepting greater uncertainty can create a chance of greater reward, but it can also create a chance of a worse result. A higher expected return, when one is present, does not guarantee a higher return in any particular realization. This distinction is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: In a risk-reward discussion, reward is a favorable outcome relevant to an objective, often represented by a return or gain. Risk is uncertainty about outcomes relevant to that objective. A risk-reward trade-off describes a decision to accept uncertainty in pursuit of possible reward; it does not assert that greater risk must produce greater reward.

Expected Return Is Not a Promise

Let $R$ denote a one-period return, expressed as a decimal: a return of 8% is $R=0.08$. If the possible returns are $r_1,\ldots,r_n$, with probabilities $p_1,\ldots,p_n$, then the expected return is the probability-weighted average of those outcomes. The probabilities must be nonnegative and sum to 1. The expectation summarizes the model’s average outcome over repeated comparable instances; it is not a prediction that the next outcome will equal that average.

E[R]=\sum_{j=1}^{n}p_jr_j, \qquad \operatorname{Var}(R)=\sum_{j=1}^{n}p_j\bigl(r_j-E[R]\bigr)^2, \qquad \operatorname{SD}(R)=\sqrt{\operatorname{Var}(R)}.

The standard deviation measures spread around the expected return. It treats favorable and unfavorable deviations symmetrically, so it is not by itself a measure of loss or of the probability of failing an objective. A risk description should therefore identify the outcomes and objective being considered, rather than presenting a variability statistic as a complete account of risk.

Worked Example: A Higher Expected Return With a Chance of Loss

A hypothetical fund considers investing $100,000 for one year. Under a two-outcome model, the return is 8% with probability 0.5 and -2% with probability 0.5. A guaranteed alternative would return 2.5%. Calculate the fund’s expected return, its standard deviation, and the expected dollar gain. Explain what the figures do and do not establish.

Method. Use the probability-weighted mean for return, calculate deviations from that mean for the variance, and then scale the return by the $100,000 investment.

Expected return. The two probabilities sum to $0.5+0.5=1$. Thus,

$$ E[R]=0.5(0.08)+0.5(-0.02)=0.04-0.01=0.03. $$

The expected return is 3%. The guaranteed alternative returns 2.5%, so the modeled fund has a higher expected return by $3\%-2.5\%=0.5$ percentage points.

Standard deviation. The deviations from the 3% mean are $0.08-0.03=0.05$ and $-0.02-0.03=-0.05$. Therefore,

\operatorname{Var}(R)=0.5(0.05)^2+0.5(-0.05)^2 =0.5(0.0025)+0.5(0.0025)=0.0025, \qquad \operatorname{SD}(R)=\sqrt{0.0025}=0.05.

The standard deviation is 5 percentage points. The expected dollar gain is $100{,}000(0.03)=\$3{,}000$. The possible dollar gains are $100{,}000(0.08)=\$8{,}000$ and $100{,}000(-0.02)=-\$2{,}000$, each with probability 0.5. Their average is $0.5(\$8{,}000)+0.5(-\$2{,}000)=\$3{,}000$, which checks the expected-gain calculation.

The fund’s higher expected return comes with a chance of a negative result in this model. It does not establish that the fund will outperform the guaranteed alternative in the one-year outcome.

How the Two-Outcome Formula Works

For two outcomes, let $r_H$ occur with probability $p$ and $r_L$ occur with probability $1-p$. The mean is $m=pr_H+(1-p)r_L$. Write $\Delta=r_H-r_L$. Subtracting the mean from each outcome gives $r_H-m=(1-p)\Delta$ and $r_L-m=-p\Delta$. Substituting those deviations into the variance definition gives:

\begin{aligned} \operatorname{Var}(R) &=p\bigl((1-p)\Delta\bigr)^2+(1-p)(-p\Delta)^2\\ &=p(1-p)^2\Delta^2+(1-p)p^2\Delta^2\\ &=p(1-p)\bigl((1-p)+p\bigr)\Delta^2\\ &=p(1-p)(r_H-r_L)^2. \end{aligned}

This derivation shows why the spread depends on both the gap between outcomes and their probabilities. A large gap does not necessarily imply the same variability in every model: the probability assigned to each outcome matters too. The formula describes variability under the specified two-outcome assumptions; it does not establish that the assumptions are realistic or that the outcomes capture every relevant risk.

Worked Example: Converting Return Variability to Dollars

A hypothetical investment has two equally likely one-year returns: 31.21% and -11.21%. The investment amount is $2,000,000. Calculate the expected return, standard deviation of return, expected dollar gain, and standard deviation of the dollar gain.

Method. Treat the returns as a two-outcome distribution with probability 0.5 for each outcome. Compute the mean and deviations in decimal return units, then scale by the investment amount.

Expected return. In decimals, the outcomes are $0.3121$ and $-0.1121$. Therefore,

$$ E[R]=0.5(0.3121)+0.5(-0.1121) =0.15605-0.05605=0.1000. $$

The expected return is 10%. The deviations from the mean are $0.3121-0.1000=0.2121$ and $-0.1121-0.1000=-0.2121$. Thus,

\operatorname{Var}(R)=0.5(0.2121)^2+0.5(-0.2121)^2 =(0.2121)^2=0.04498641, \qquad \operatorname{SD}(R)=0.2121.

The expected dollar gain is $2{,}000{,}000(0.1000)=\$200{,}000$. Since the dollar gain is $G=2{,}000{,}000R$, its standard deviation is $2{,}000{,}000(0.2121)=\$424{,}200$. Equivalently, the two possible dollar gains are $2{,}000{,}000(0.3121)=\$624{,}200$ and $2{,}000{,}000(-0.1121)=-\$224{,}200$. Their deviations from the $200,000 mean are $+\$424{,}200$ and $-\$424{,}200$, confirming the standard deviation.

The result depends on the stated two-outcome model and the investment amount. A percentage standard deviation is not a dollar amount until it is scaled by the exposure. The possible negative gain also shows that a positive expected gain is not assured.

Why Greater Uncertainty Does Not Guarantee Greater Reward

The trade-off is best understood as a possibility that may influence a decision, rather than as a rule linking risk and realized return. A business action can have substantial uncertainty and still have a low or negative expected reward. Alternatively, a less uncertain action can produce a favorable result. Results depend on the probabilities and sizes of outcomes, the time horizon, and whether the model reflects the risks relevant to the objective.

A positive expected return can also coexist with frequent disappointing outcomes. The following hypothetical project illustrates why an average alone does not describe what an organization should expect to experience in a particular period.

Worked Example: A Positive Average With Frequent Shortfalls

A hypothetical project requires $500,000 for one year. Its return is 40% with probability 0.3 and 0% with probability 0.7. A separate objective is to earn at least 5% over the year. Calculate the expected return, the probability of falling short of the objective, the expected dollar gain, and the standard deviation of return.

Method. Use the two stated probabilities and returns to calculate the mean and variance. Separately identify which modeled outcomes fail the 5% objective.

Expected return.

$$ E[R]=0.3(0.40)+0.7(0)=0.12. $$

The expected return is 12%. The 0% outcome is below the 5% objective and has probability 0.7; the 40% outcome exceeds it. Therefore the probability of falling short is 0.7.

Expected dollar gain. The expected gain is $500{,}000(0.12)=\$60{,}000$. The modeled dollar gains are $500{,}000(0.40)=\$200{,}000$ with probability 0.3 and $0 with probability 0.7. Their weighted average is $0.3(\$200{,}000)+0.7(\$0)=\$60{,}000$, checking the result.

Standard deviation. The deviations from the 12% mean are $0.40-0.12=0.28$ and $0-0.12=-0.12$. Hence,

\operatorname{Var}(R)=0.3(0.28)^2+0.7(-0.12)^2 =0.3(0.0784)+0.7(0.0144) =0.02352+0.01008=0.0336,

so $\operatorname{SD}(R)=\sqrt{0.0336}\approx0.1833$, or about 18.33 percentage points. The expected return is positive and the model has a 70% probability of falling short of the objective. Neither figure alone determines whether accepting the project is appropriate; that judgment depends on the objective and the organization’s willingness and ability to bear the possible outcomes.

Common Mistakes and Practical Interpretation

Treating expected return as a guaranteed result. In the first example, the expected return is 3%, but the model’s actual outcomes are 8% and -2%. The mean is not one of those outcomes.
Using only the best-case outcome. In the $2,000,000 example, reporting 31.21% as the expected return ignores the equally likely -11.21% outcome. The correct calculation is $0.5(31.21\%)+0.5(-11.21\%)=10\%$.
Assuming variability is always harmful. Standard deviation counts deviations above and below the mean. A high standard deviation signals spread, not the direction or objective-based significance of each outcome.
Assuming greater risk must produce greater reward. Greater uncertainty can accompany a possibility of greater reward, but outcomes and probabilities determine the expected result. There is no guarantee that an uncertain action will pay off.
Ignoring the objective. A 0% return may be acceptable for one objective and a shortfall for another. State the relevant target before interpreting favorable or unfavorable results.

When considering a risk-reward trade-off, keep the model’s expected outcome separate from its possible outcomes. Ask what reward is being sought, what unfavorable outcomes could occur, and whether the stated probabilities and time period are appropriate for the decision. These questions help prevent a potential reward from being mistaken for a promised result.

Check Your Understanding

For each original problem, calculate the requested quantities from the stated model and interpret the result in context.

A hypothetical venture has a return of 12% with probability 0.4 and 0% with probability 0.6. Find the expected return and standard deviation. (Answer: Expected return 4.8%; standard deviation approximately 5.8788%.)
A hypothetical asset has equally likely returns of 18% and -6%. For a $250,000 investment, find the expected return, standard deviation of return, and expected dollar gain. (Answer: Expected return 6%; standard deviation 12%; expected dollar gain $15,000.)
A hypothetical project returns 50% with probability 0.2 and -5% with probability 0.8. For a $400,000 investment, find the expected return, probability of a negative return, and expected dollar gain. (Answer: Expected return 6%; probability of a negative return 0.8; expected dollar gain $24,000.)
A hypothetical portfolio has an expected return of 3% and a return standard deviation of 8%. For a $1,500,000 exposure, find the expected dollar gain and dollar standard deviation. (Answer: Expected dollar gain $45,000; dollar standard deviation $120,000.)
A hypothetical proposal has a 25% chance of a 32% return and a 75% chance of a 0% return. A stated objective is to earn at least 10%. Find the expected return and probability of falling short of the objective. Does the expected return guarantee meeting the objective? (Answer: Expected return 8%; probability of falling short 0.75; no, the expected return does not guarantee meeting the objective.)

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