A sure 4%, or a shot at 30%? How investors price risk.
The story so far The shop had a good year, and you have $10,000 to invest for one year. You have three choices. Treasury bills pay 4% for sure. A bond fund will earn 8% or 2%, a coin flip. A stock fund will earn 30% or lose 10%, also a coin flip. Which is best? It depends on how you measure two things: what you expect to earn, and how wild the ride is. That's what this lesson, and all of Wall Street, runs on.
Return is what an investment earns. Risk is how much that return can swing away from what you expect.
Investors only take more risk if they expect more return. That trade-off sets the price of every stock, bond and fund.
Average the possible outcomes to get the expected return. Measure how far they spread to get the risk. Then compare the two.
Portfolio managers balance risk and return for funds. Wealth advisors match them to each client. Risk analysts measure how much a bank or fund could lose.
What you're looking at: the three choices drawn as rides. The gray T-bill line is flat because 4% is certain. The blue bond line wiggles a little. The green stock line swings the most, up to +30% or down to −10%.
Two numbers for every investment.
In Lesson 5.6 you saw that markets drop and recover. Now you put numbers on it, the way a finance class and a Wall Street desk do.
Expected return
The expected return is what you'd earn on average. Multiply each possible return by its chance and add them up. It's not what will happen this year, just the best single guess.
- Stock fund
- 0.5 × 30% + 0.5 × (−10%) = 15% − 5% = 10%
- Bond fund
- 0.5 × 8% + 0.5 × 2% = 4% + 1% = 5%
- T-bills
- 4%, no matter what.
Standard deviation: how far it swings
Risk is measured with standard deviation: how far a typical year lands from the expected return. It's the number behind the word volatility. When two outcomes are equally likely, it's simply half the gap between them.
- Stock fund
- 30% − (−10%)2 = 40%2 = 20%. Each outcome lands 20 points from the 10% average.
- Bond fund
- 8% − 2%2 = 3%
- T-bills
- 0%. No swing at all.
The two-in-three rule
Real returns aren't coin flips; they spread out like a bell curve. For a bell curve, about 2 years out of 3 land within one standard deviation of the expected return. That turns two numbers into a range you can picture.
- A stock fund like this
- 10% − 20% = −10%, up to 10% + 20% = 30%. On $10,000, most years end between $9,000 and $13,000. The other year in three lands outside that range.
The risk premium
Why would anyone pick the stock fund? Because they're paid for the risk. The risk premium is the extra expected return above the risk-free rate (Lesson 13.3). This is the risk-return trade-off: more expected return always comes with more risk.
- Stock fund
- 10% − 4% = 6% extra, for taking 20% of swing.
- Bond fund
- 5% − 4% = 1% extra, for taking 3% of swing.
The Sharpe ratio: reward per unit of risk
To compare investments with different risks, divide the risk premium by the standard deviation. That's the Sharpe ratio, named after the economist William Sharpe. Higher means more reward for each point of risk.
- Stock fund
- 10% − 4%20% = 0.30
- Bond fund
- 5% − 4%3% = about 0.33. Per point of risk, the bonds pay slightly more. The stocks just offer more of everything.
Mixing safe and risky
You don't have to pick just one. Put part of the $10,000 in stocks and the rest in T-bills. Each mix's expected return and standard deviation move in a straight line, so you dial in exactly the risk you can live with. Try it.
- Half and half
- Expected: 0.5 × 10% + 0.5 × 4% = 7%. Swing: 0.5 × 20% = 10%, since T-bills add none. Typical range: −3% to 17%.
What you're looking at: a number line of one-year returns from −15% to +35%. The bar is the typical range, where about 2 years in 3 land; the dot is the expected return. The rest of the money sits in T-bills at 4%.
Two minutes. Just listen.
Tap any line to jump to it.
Pick your risk.
Eight questions about risk and return. Each one gives you the numbers. One try each.
What you're looking at: made-up investments. Each lists the possible returns, then asks one question.
The logic, step by step.
Sizing up any investment takes four steps.
Step 1: Expected return
- Stock fund
- 0.5 × 30% + 0.5 × (−10%) = 10%.
Step 2: Risk
- Stock fund
- 20%. Typical range: −10% to 30%.
Step 3: Reward for the risk
- Stock fund
- 10% − 4% = 6%; 6%20% = 0.30.
Step 4: Match it to the person
- One year, money you'll need
- A $1,000 loss could hurt, so a smaller stock share fits.
Worked example: when do you need the money?
The shop needs $9,800 for a new lease deposit in exactly one year. Here are the facts:
- The money: $10,000, invested for one year.
- The stock fund: expected 10%, standard deviation 20%.
- T-bills: 4% for sure.
- All stocks. The low end of the typical range is −10%: $10,000 × 0.90 = $9,000. That misses the $9,800 deposit, and that's not even the worst case.
- The most you can afford to lose. $10,000 − $9,800 = $200, or 2%.
- Try 30% in stocks. Expected: 0.3 × 10% + 0.7 × 4% = 5.8%. Swing: 0.3 × 20% = 6%. Low end: 5.8% − 6% = −0.2%, or $9,980.
- Check it. $9,980 covers the $9,800 in a typical bad year.
- The answer. Keep about 30% in stocks or less. Money you need soon belongs mostly in safe assets; money you won't touch for decades can ride out the swings (Lesson 5.6).
Small birds gamble when they must.
In a classic experiment, scientists gave small birds called juncos a choice between two feeders: one that always gave the same number of seeds, and one that gave a lot or almost none. When the steady feeder gave enough to get through a cold night, the birds chose it. When the steady amount wasn't enough to survive, they switched to the gamble, because only the risky feeder gave them a chance of eating enough.
What you're looking at: an illustration, not data. A bird choosing between a feeder that always drops a few seeds and one that drops a big pile or nothing.
Why this sticks. The birds didn't always pick safe or always pick risky. They picked the risk that fit what they needed. That's exactly how an advisor picks a portfolio.
The page you'd copy down.
- Expected return = each return × its chance, added up. Stocks: 0.5 × 30% + 0.5 × (−10%) = 10%.
- Standard deviation (volatility): how far a typical year lands from the average. Two equal outcomes: high − low2. Stocks: 20%.
- Two-in-three rule: about 2 years in 3 land within one standard deviation. Stocks: −10% to 30%.
- Risk premium = expected return − risk-free rate. 10% − 4% = 6%.
- Sharpe ratio = risk premiumstandard deviation. Stocks 0.30, bonds about 0.33.
- Mixing with T-bills: expected and risk move in a straight line. 50/50: 7% expected, 10% swing.
Fill in the blanks
Grab a friend or a study group.
Coin-flip portfolio
Each person picks a stock share from the slider. Flip a coin 10 times (heads +30%, tails −10% for the stock part). Compare everyone's ending balance and range.
Who should hold what?
Make up three people: a 20-year-old, someone saving for a house in 2 years, and a retiree. Agree on a stock share for each and explain why.
Look up a fund
Find an S&P 500 index fund's page. Look for its average return and its standard deviation, and work out its typical range.
Just you and your money.
These stay on this device. Nobody else sees them.
Exam-style questions.
Written the way a college exam would ask them. Each right answer on the first try is worth 10 xp.
On the job: you're a wealth management associate
Two clients each have $100,000 to invest. Here's the situation:
- Maria, 30: saving for retirement, won't touch the money for 35 years. She wants 80% in a stock fund.
- Mr. Okafor, 63: retiring in 2 years and will start living on this money. He wants 30% in the stock fund.
- The stock fund: expected 10% a year, standard deviation 20%.
- The rest: T-bills at 4%.
Your task: for each client, find the expected return, the standard deviation and the typical one-year range in dollars. Do both choices make sense?
See a model answer
Maria (80% stocks): expected 0.8 × 10% + 0.2 × 4% = 8% + 0.8% = 8.8%. Swing: 0.8 × 20% = 16%. Typical range: 8.8% − 16% = −7.2% up to 8.8% + 16% = 24.8%, so $92,800 to $124,800 after one year.
Mr. Okafor (30% stocks): expected 0.3 × 10% + 0.7 × 4% = 3% + 2.8% = 5.8%. Swing: 0.3 × 20% = 6%. Typical range: −0.2% to 11.8%, so $99,800 to $111,800.
Do they make sense? Yes. Maria has 35 years to recover from bad years, so she can take the bigger swings for 3 more points of expected return. Mr. Okafor will need the money soon, so a smaller, steadier range protects him. Remember that 1 year in 3 lands outside the typical range, so both should be ready for worse.
Every price has a risk inside it.
What you're looking at: a map of where this lesson fits in the course. Read it left to right: the lessons it builds on, this lesson in the middle, and the lesson that comes next.
- Back · Unit 5Market drops and risk
- Back · Unit 5Index funds and the S&P 500
- Back · Unit 13The cost of capital
- You are hereRisk and return
- Coming soon · Unit 14Diversification