5.1Lesson 5.1 · Module 5 — Risk and Return

What Risk and Return Actually Mean

Total shareholder return, realized versus expected return, risk as uncertainty, and why investors demand a risk premium.

Total return = (distribution + price change) / beginning price
Expected return = probability-weighted average
Risk premium = expected return − risk-free rate
Risk affects the discount rate used in valuation

Start

Learning objectives

By the end of this lesson, you should be able to:

1Compute total shareholder return from beginning price, distribution, and ending price.
2Distinguish realized return from expected return as a probability-weighted average.
3Explain risk as uncertainty about the return an investor will actually receive.
4Define the risk premium and the realized excess return relative to a risk-free rate.
5Connect higher risk to a higher discount rate and therefore a lower present value.
6Recognize the limitations of standard deviation as a measure of risk.

5.1.1Section 1

Total shareholder return

When you own a stock, your return over a period comes from two sources: any distribution the company pays (a dividend or buyback), and the change in price. Together these make up the total shareholder return.

Total return

R = \frac{D _{1} + P _{1} - P _{0}}{P _{0}}

The total return is the cash distribution plus the price change, divided by the price you paid at the start.

$P_{0}$: Beginning price (what you paid).
$P_{1}$: Ending price (what it is worth at period end).
$D_{1}$: Cash distribution received during the period.
$R$: Total holding-period return.

R = \frac{1 + 54 - 50}{50} = \frac{5}{50} = 0.10

Total return = 10%

The $2 dividend yield (2%) plus the $4 capital gain (8%) add up to the 10% total return. The two components are independent ways to be paid — they always sum to total return.

We can decompose the return into two yields. The dividend yield is D₁ / P₀, and the capital gain yield is (P₁ − P₀) / P₀. In the example: 2% + 8% = 10%.

Try itTotal return worksheet

Practice: a stock you bought for $80

You bought a share for $80. A year later it trades at $74 and paid a $2 dividend. Compute each component of the return, then the total.

Dividend yield (D₁ / P₀)

Attempt 1 of 3

Capital gain yield ((P₁ − P₀) / P₀)

Attempt 1 of 3

Total return R

Attempt 1 of 3

5.1.2Section 2

Realized versus expected return

Definition · Realized return

The return that actually occurred over a period. It is a historical fact — the prices and cash flows that happened.

Definition · Expected return

The return an investor anticipates looking forward, calculated as a probability-weighted average across possible future states.

Expected return

E [R] = s \sum p_{s} R_{s}

The expected return multiplies each possible return by the probability of that state, then sums across all states.

$p_{s}$: Probability of state s occurring.
$R_{s}$: Return if state s occurs.
$E [R]$: Expected (probability-weighted) return.

Consider a stock whose returns depend on the economy next year:

State	Probability	Return	Contribution
Strong	30%	+25%	0.30 × 25% = 7.5%
Normal	50%	+8%	0.50 × 8% = 4.0%
Recession	20%	−20%	0.20 × (−20%) = −4.0%
Sum	100%		E[R] = 7.5%

Adding the three contributions: 7.5% + 4.0% − 4.0% = 7.5%. The expected return is 7.5% — but no single state actually produces exactly 7.5%. The realized return will be one of +25%, +8%, or −20%.

Try itState-of-the-world exercise

Verify the expected return

The table lists three states. First confirm the probabilities sum to 100%, then work out each state's weighted contribution and the resulting expected return.

Do the probabilities sum to 100%? (enter the sum as a percent)

Attempt 1 of 3

Weighted contribution of the Strong state (p × R)

Attempt 1 of 3

Expected return E[R] (sum of all contributions)

Attempt 1 of 3

If the realized return turned out to be −20%, does this contradict E[R] = 7.5%?

5.1.3Section 3

Expected return is not a smooth path

A positive expected return does not mean the stock goes up steadily every year. The realized path can be jagged — strong years, painful years, and quiet years — while the long-run average still reflects the expectation.

Year	1	2	3	4
Return	+25%	−15%	+12%	+6%

One year loses 15%. Another gains 25%. There is no guarantee in any single year — only a distribution of possible outcomes centered on the expected return. This gap between the expectation and the actual outcome is exactly what we mean by risk.

5.1.4Section 4

Risk as uncertainty

Two investments can have the same expected return but very different risk. Consider:

Investment A

A guaranteed 8% return. No uncertainty — you always get 8%.

E[R] = 8%

Investment B

50% chance of +30%, 50% chance of −14%. The same expectation, but outcomes are spread far apart.

0.5 × 30% + 0.5 × (−14%) = 8%

Both have E[R] = 8%, but investment B is plainly riskier: its outcomes (−14% or +30%) are far more spread out than A's certain 8%. An investor who cannot tolerate losing 14% would prefer A even though the expectation is identical.

Definition · Risk

Risk is the uncertainty in the return the investor will actually receive. Two investments with the same expected return can have different risk because their possible outcomes are spread differently.

5.1.5Section 5

Risk premium and excess return

Expected risk premium

Expected risk premium = E [R_{i}] - r_{f}

The extra return investors expect to earn above the risk-free rate as compensation for bearing risk.

$E [R_{i}]$: Expected return on the risky asset.
$r_{f}$: Risk-free rate (e.g. short-term government bills).

10% - 4% = 6%

Expected premium = 6%

Realized excess return

Realized excess return = R_{i} - r_{f}

How much the actually-realized return beat (or missed) the risk-free rate.

$R_{i}$: Realized return on the risky asset.
$r_{f}$: Risk-free rate over the same period.

- 12% - 4% = - 16%

Realized excess = −16%

Suppose the risk-free rate is 4% and a stock has an expected return of 10%. The expected risk premium is 6%. But if the stock actually returns −12%, the realized excess return is −16% — the investor did far worse than holding the risk-free asset. A positive expected premium does not guarantee a positive realized excess return.

Try itReturn classification exercise

Distinguish the four return concepts

Each quantity below looks similar but means something different. Compute each one. The risk-free rate is 4%.

A stock you are considering has an 11% expected return. What is the expected risk premium? (r_f = 4%)

Attempt 1 of 3

The stock actually returned 3%. What is the realized excess return? (r_f = 4%)

Attempt 1 of 3

Expected return

The probability-weighted average of possible future returns. A forward-looking expectation — not a promise.

Realized return

The return that actually happened. A backward-looking fact about prices and cash flows.

Expected risk premium

Expected return minus the risk-free rate. The compensation expected for bearing risk — also forward-looking.

Realized excess return

Realized return minus the risk-free rate. How much you actually beat (or missed) the risk-free asset.

5.1.6Section 6

Connection to the cost of equity

In Module 4 you valued equities by discounting expected payoffs at a required return. That required return is the risk-free rate plus a risk premium:

Cost of equity (required return)

r_{e} = r_{f} + risk premium

The rate investors demand for holding a risky equity is the risk-free rate plus compensation for bearing that equity's risk.

$r_{e}$: Cost of equity / required return.
$r_{f}$: Risk-free rate.

Two companies with the same expected payoff can have different values today, because risk changes the discount rate.

Company	E[payoff]	Cost of equity	P₀ = payoff / (1 + r_e)
Safer	$110	6%	$110 / 1.06 = $103.77
Riskier	$110	12%	$110 / 1.12 = $98.21

Same expected payoff, but the riskier company is worth less today: $98.21 < $103.77. Higher risk → higher required return → heavier discounting → lower present value, all else equal.

5.1.7Section 7

Limitations of volatility as a risk measure

In the next lesson we will measure risk with standard deviation (volatility). It is a useful starting point, but it is not a complete picture of risk. Volatility treats large gains and large losses symmetrically as “fluctuations,” which understates the dangers investors actually fear.

Permanent capital loss — a position you cannot recover from.
Downside risk and drawdowns — the pain of large peak-to-trough drops.
Negative skewness — rare but severe losses hidden by an otherwise calm average.
Illiquidity — assets you cannot sell when you need to.
Leverage — losses amplified by borrowed money.
Tail events — extreme outcomes far outside the normal range.

Standard deviation counts a +40% gain and a −40% loss as equally “risky.” But investors experience losses far more painfully than equivalent gains. Volatility is the introduction — not the final word — on risk.

Try itConcept check

Pulling it together

Given r_f = 3%, E[R] = 9%, and a realized return of -5%, compute the two excess-return measures and interpret them.

Expected risk premium (E[R] − r_f)

Attempt 1 of 3

Realized excess return (R − r_f)

Attempt 1 of 3

Try itMastery check

Pass with 3 of 5 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
P₀ = $50, D₁ = $1, P₁ = $54. What is the total return?
02
E[R] = 10%, $r_{f}$ = 4%. What is the expected risk premium?
03
Which of the following is NOT guaranteed in advance: the expected risk premium, or the realized excess return?
04
A riskier company with the same expected payoff as a safer company should have:
05
Realized return = −12%, $r_{f}$ = 4%. What is the realized excess return?

Lesson summary

1Total return = (distribution + price change) / beginning price.
2Expected return is the probability-weighted average of possible future returns.
3Realized return is what actually occurred — it may differ from expected.
4Risk is the uncertainty in the return the investor will actually receive.
5Risk premium = expected return − risk-free rate.
6A positive expected premium does not guarantee a positive realized excess return.
7Higher risk raises the discount rate, lowering current value, all else equal.

Replay this lesson Continue to Measuring Historical Return and Volatility