Risk and return · RiskAtoms Academy

By the end you will be able to

1Explain why investors demand a premium for bearing risk, using expected return and volatility.
2Calculate the expected return and standard deviation of an investment from a set of scenarios.
3Calculate a risk premium and a Sharpe ratio, and use them to compare two investments.
4Show how mixing a risky asset with a risk-free one moves expected return and volatility together.

Why it matters

Would you rather have £100 for certain, or toss a coin for £200 or nothing? On average the coin pays the same. Most people still take the £100. That simple preference sets the price of every risky asset in the world.

Core explanation

Nobody bears risk for free

In the coin toss, both choices have the same average, £100. The difference is the spread: the gamble can leave you £100 above or below that average. People who prefer the sure thing are risk averse, and most people and most investors are, at least when the stakes are large.

Now apply that to markets. A short-term government bill pays a known amount: it is close to a risk-free investment. A share might rise 30% or fall 20%. If both offered the same return on average, risk-averse investors would all buy the bill. The share's price would fall until its expected return was high enough to tempt them. That extra is the risk premium.

The core idea

Prices adjust so that riskier assets offer higher expected returns. That does not mean they deliver higher returns every year: if they did, they would not be risky.

To put numbers on this you need two measures, both met in Level 1: Foundations of Finance:

Expected return: what you earn on average, weighting each possible outcome by its chance.
Volatility, measured by the standard deviation: how widely the outcomes swing around that average. L1.5.06: Equity returns and volatility estimated it from past returns; here we start from scenarios.

The ratio of the two tells you how well you are paid for the risk you take. The Sharpe ratio divides the risk premium by the volatility: extra return for each point of volatility. Higher is better.

Investment	What drives the premium	Typical risk
Treasury bill	None: close to risk-free	Very low
Government bond, 10 years	Rates may rise and prices fall	Low to moderate
Corporate bond	Rates, plus default and liquidity	Moderate
Shares	Profits and the economy	High

Think firstCommon confusions

The formal version

Each scenario $i$ has a probability $p_i$ and a return $R_i$ . The expected value is the subject of M-C02: Expectation; here it is built one piece at a time.

The volatility uses the variance from M-C03: Variance and standard deviation, weighted by the same probabilities.

Two more measures follow on a few lines. $r_f$ is the risk-free rate:

\begin{aligned} \text{Risk premium} &= \mathbb{E}[R] - r_f \\ \text{Sharpe ratio} &= \frac{\mathbb{E}[R] - r_f}{\sigma} \end{aligned}

Mix a share $w$ of your money into the risky asset and keep the rest in the risk-free bill. The bill adds nothing to the swings, so:

\begin{aligned} \mathbb{E}[R_{\text{mix}}] &= r_f + w\,(\mathbb{E}[R] - r_f) \\ \sigma_{\text{mix}} &= w\,\sigma \end{aligned}

From data instead of scenarios

When volatility is estimated from past returns, as in L1.5.06: Equity returns and volatility, use the sample standard deviation, which divides by

n - 1

. To turn a daily figure into an annual one, multiply by the square root of the number of trading days, here 252: a daily volatility of 1.1% is

1.1\% \times \sqrt{252} = 17.46\%

a year.

Try it

Shape a risky fund with three scenarios, then choose how much of your money to put in it. Start with the defaults and slide your share from 0% to 100%: your dot walks up the line. Then make the bad year worse and watch the Sharpe ratio fall. Illustrative numbers.

Try itRisk and return explorer

The risky fund, in three scenarios (chances fixed at 25%, 50%, 25%):

Good year30.0%

Normal year10.0%

Bad year−20.0%

Risk-free rate3.0%

Share of your money in the fund100%

The fund

Expected return7.5%

Volatility (σ)17.9%

Risk premium4.5%

Sharpe ratio0.25

Your mix

Expected return7.5%

Volatility (σ)17.9%

In a bad year−20.0%

Expected return against volatilityEvery mix of the bill and the fund lies on the line. Its slope is the Sharpe ratio: extra return for each point of volatility.

Worked example

A share fund, a bond fund and a Treasury bill

A Treasury bill pays 3% for sure. Two funds face the same three scenarios next year: a boom (chance 25%), a normal year (50%) and a bust (25%). Returns are illustrative, in % for one year.

Scenario	Chance	Share fund	Bond fund
Boom	25%	30%	9%
Normal	50%	10%	6%
Bust	25%	−20%	0%

Expected return, risk and reward per unit of risk

Step 1 of 7

1
Expected return of the share fund
$0.25 \times 30 + 0.5 \times 10 + 0.25 \times (-20) = 7.5\%$

Regulatory lens · looking ahead

Banks apply the same trade-off to their own business. Capital rules make them hold more capital against riskier assets (risk-weighted assets, L1.8.06: Capital, equity and leverage), so a riskier loan must earn more to give the same return on that capital. Many banks measure this with risk-adjusted return on capital. The rules themselves are introduced in M2.7: Introduction to regulation.

Sparks

SparkThe St Petersburg paradox

A coin is tossed until it lands heads; the prize doubles with every tail. The expected prize is infinite, yet few people would pay more than a few pounds to play. In 1738 Daniel Bernoulli explained why: each extra pound matters less to you the more you have. That idea, diminishing marginal utility, is the root of risk aversion.

SparkThe premium that is too big

In 1985 Rajnish Mehra and Edward Prescott found that US shares had beaten short-term bills by about 6 percentage points a year from 1889 to 1978, far more than standard models of risk aversion could explain. Economists still debate this “equity premium puzzle”.

Key takeaways

Most investors are risk averse, so riskier assets must offer higher expected returns: a risk premium.
Expected return is the probability-weighted average of the outcomes; volatility is their standard deviation.
Risk premium = expected return − risk-free rate. Sharpe ratio = premium ÷ volatility.
Mixing a risky asset with a risk-free one moves expected return and volatility together along a straight line.
From data, use the sample standard deviation (n − 1) and annualise with the square root of 252 trading days.

Reaction

Check what stuck

Question 1 · objective 2

An investment returns −10% with probability 40% and +15% with probability 60%. What is its expected return, in %?

%

Question 2 · objective 2

For the same investment, what is the standard deviation of its return, in %, to two decimal places?

%

Question 3 · objective 1

Why do shares, on average, have to offer higher expected returns than Treasury bills?

ABecause shares are taxed lessBBecause investors dislike uncertainty and must be paid to bear it, or they would hold bills insteadCBecause companies are required by law to pay moreDBecause higher volatility guarantees higher returns every year

Question 4 · objective 3

A fund returns −12% or +28% with equal chances. The risk-free rate is 3%. What is its Sharpe ratio, to two decimal places?

Question 5 · objective 3

A share's daily returns have a sample standard deviation of 1.2%. Using 252 trading days a year, what is its annual volatility, in %, to two decimal places?

%

Glossary

Expected return: The probability-weighted average of the possible returns: what you would earn on average over many tries.
Variance: The probability-weighted average of the squared gaps between each return and the expected return.
Standard deviation: The square root of the variance: the size of a typical swing around the expected return. Also called volatility.
Risk averse: Preferring a certain outcome to a gamble with the same expected value.
Risk-free rate: The return on an investment with no uncertainty over the period, such as a short-term government bill held to maturity.
Risk premium: The expected return above the risk-free rate: the extra that investors demand for bearing risk.
Sharpe ratio: The risk premium divided by the volatility: extra expected return for each point of volatility.

Deep Dives & Origins

Deep Dives

Brealey, R. A., Myers, S. C. and Allen, F., Principles of Corporate Finance, McGraw-Hill, the chapters on risk and return. Beginner.
Dimson, E., Marsh, P. and Staunton, M., Triumph of the Optimists: 101 Years of Global Investment Returns, Princeton University Press, 2002. Intermediate.

Origins

Markowitz, H., “Portfolio Selection”, Journal of Finance 7(1), 1952.
Sharpe, W. F., “Mutual Fund Performance”, Journal of Business 39(1), 1966.
Mehra, R. and Prescott, E. C., “The Equity Premium: A Puzzle”, Journal of Monetary Economics 15(2), 1985.
Bernoulli, D., “Exposition of a New Theory on the Measurement of Risk” (1738), English translation in Econometrica 22(1), 1954.

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