# Capital Asset Pricing Model (CAPM)

> CAPM says the return you should expect from an investment is a safe rate plus extra return for the market risk it carries, scaled by its beta.

Page: https://formulabro.yunusemrevurgun.com/finance/capital-asset-pricing-model/  
Category: Finance  
Published: 2026-10-23  
Updated: 2026-10-23

## Formula

$$
E(R_i) = R_f + \beta_i \left(E(R_m) - R_f\right)
$$

Read it out loud: "E of R i equals R f plus beta i times the quantity E of R m minus R f. E of R i is the expected return on the investment. R f is the risk-free rate, the return on a very safe investment. Beta i is how strongly the investment moves with the market. E of R m is the expected return on the whole market, and the bracket holds the market risk premium."

## Before you start

- **Rate and percent**: A **rate** is a share of the money, per period. Percent is hundredths, so 11% is 11 divided by 100, which is the decimal 0.11. The formulas use decimals, so 2% is written 0.02. A rate of 0.11 per year means 11 dollars of return for every 100 dollars invested over one year.
- **Return**: A **return** is the gain or loss on an investment as a share of the money put in. If 100 dollars grows to 111 dollars in one year, the gain is 11 dollars, and the return is 11 divided by 100, which is 0.11 or 11%. A return can be negative when the investment loses value.
- **Expected value**: The **expected** value is the average outcome you would get if the same situation could be repeated many times, with each outcome weighted by how likely it is. The letter E in front of a return, as in $E(R_i)$, means its expected value. It is a forecast, and the actual result may be different.
- **Risk and risk premium**: **Risk** means the actual return may differ from the expected return. A **risk premium** is the extra return that investors ask for in order to accept a risk. Investors usually expect more return from riskier investments, and the premium is that extra reward.
- **Greek letter beta and subscripts**: The Greek letter $\beta$, pronounced beta, is the usual name for a stock's sensitivity to the market. A small letter or number under a symbol is a **subscript**. In $\beta_i$ the subscript $i$ names one investment, and in $R_m$ the subscript $m$ stands for the market.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $E(R_i)$: The **expected return** on the investment, such as one stock, over one period, usually one year. It is a rate, so 0.11 means 11% a year. It is a forecast from the model, not a promise of what will happen.
- $R_f$: The **risk-free rate**, the return on an investment with almost no chance of not being paid, such as a short government bond. It is a rate per year. In the example it is 0.02, which is 2% a year.
- $\beta_i$: **Beta** of the investment, a plain number with no unit that measures how much it moves with the market. A beta of 1 moves with the market, a beta above 1 moves more, a beta below 1 moves less, and a negative beta moves against it.
- $E(R_m)$: The **expected return on the market**, the average return expected from a broad basket of stocks, such as a whole stock index. It is a rate per year, written as a decimal in the formula.

## The formula piece by piece

- $R_f$: Start with the **risk-free rate**, the safe base return. Every investor can earn this without taking market risk, so the formula starts here.
- $E(R_m) - R_f$: The **market risk premium**: the extra return the whole market is expected to pay above the safe rate. It is the reward for carrying market risk.
- $\beta_i \left(E(R_m) - R_f\right)$: The premium **scaled by beta**: the market premium multiplied by how strongly this investment moves with the market. It is the extra return that matches the investment's own market risk.
- $E(R_i) = R_f + \beta_i \left(E(R_m) - R_f\right)$: The expected return on the investment equals the risk-free rate plus beta times the market risk premium.

## What the formula is doing, and why

CAPM answers one question: **what return should an investor expect from one investment, given how risky it is?** The formula takes three inputs: the risk-free rate $R_f$, the expected market return $E(R_m)$ and the investment's beta $\beta_i$. It gives one output, the expected return $E(R_i)$, written as a rate per year. The model was developed in the 1960s by William Sharpe and others. Sharpe later shared the 1990 Nobel Memorial Prize in Economic Sciences for his work on financial economics, which includes this model.

**Why start from the risk-free rate?** A safe investment pays $R_f$ with no extra risk, so an investor has no reason to accept less than that from something risky. The formula starts from this safe base and adds a reward for taking risk. That is why the first term is $R_f$ on its own, with nothing subtracted from it.

**Why the market risk premium?** The bracket $E(R_m) - R_f$ is the extra return the whole market is expected to pay above the safe rate, for carrying market risk instead of holding the safe asset. It is called the market risk premium. It is the reward for one unit of market risk, and every investment is priced with the same premium.

**Why multiply by beta?** Beta scales the premium to the investment's own market risk. A beta of 1.5 means the investment carries 1.5 times the market's risk, so its premium is 1.5 times the market premium. A beta of 0 means the investment does not move with the market, so it earns only the risk-free rate. CAPM counts only this market-related risk, because investors can remove the other kind by holding many different investments, and the market does not pay for risk that can be removed.

## An everyday picture

Think of a toll road. The smooth, safe road costs nothing extra, and it is the risk-free rate. The main road is bumpier and a little more tiring, so drivers expect a toll for using it, and that toll is the market risk premium. A bus that bounces twice as hard as the main road should pay twice the toll, which is what beta does. CAPM adds the smooth-road return to the toll, multiplied by how much bumpier this particular vehicle is than the main road.

## Worked example: Expected return on a stock with a beta of 1.5

A stock has a beta of 1.5, so it tends to move about one and a half times as much as the market. A safe government bond pays 2% a year, which is $R_f = 0.02$. The whole market is expected to return 8% a year, which is $E(R_m) = 0.08$. All these rates are made up for the example. You are thinking of putting 1000 dollars into this stock for one year. What return does CAPM say to expect, and what expected gain does that mean in dollars?

1. **Find the market risk premium.** The market is expected to return more than the safe bond, and the difference is the reward for carrying market risk. We subtract the safe rate from the market return: $E(R_m) - R_f = 0.08 - 0.02 = 0.06$. That is a premium of 6% a year.
2. **Multiply the premium by beta.** The stock has a beta of 1.5, so it carries 1.5 times the market's risk and should earn 1.5 times the premium. We multiply: $1.5 \times 0.06 = 0.09$. This 9% a year is the extra return that matches the stock's extra market risk.
3. **Add the safe rate back.** Every investment starts from the risk-free return, so the total expected return is the safe rate plus the extra: $E(R_i) = 0.02 + 0.09 = 0.11$. A return of 0.11 means 11% a year.
4. **Turn the rate into dollars.** A rate is a share of the money put in, so we multiply the rate by the 1000 dollars invested: $1000 \times 0.11 = 110\ \text{dollars}$. The expected gain over one year is 110 dollars, so the money would be expected to grow to 1110 dollars.

**Answer:** CAPM says a stock with a beta of 1.5 should be expected to return **11% a year**. On 1000 dollars that is an expected gain of 110 dollars over one year. This is a forecast from the model, and the real result can be higher or lower.

## Common mistakes

- Using the market return itself as the premium. The premium is the market return minus the risk-free rate, so the subtraction must come before the multiplication by beta.
- Leaving out the risk-free rate at the start. Without $R_f$ the answer is only the premium part, and the expected return is understated.
- Reading the result as a guaranteed return. CAPM gives an expected return, and the real return of one year can be higher or lower.
- Confusing beta with volatility. Beta measures movement together with the market, while volatility measures total ups and downs, so a stock can be very volatile and still have a low beta.

## Where you will meet it

- Estimating the **cost of equity**, the return shareholders require from a company, which is a key input to its discount rate.
- Choosing a **discount rate** for valuing a stock or a company project, so that future cash is compared with money today.
- Judging a fund manager: the return beyond what CAPM predicts for the fund's beta is called alpha.
- Teaching and research in finance, where it is the standard first model of how risk and expected return are linked.

## Questions people ask

### What is the risk-free rate?

It is the return on an investment treated as having no default risk, such as a government bond. Analysts often use the yield of a government bond. The number is a rate per year, such as 0.02 for 2%.

### Does CAPM tell me what a stock costs?

Not directly. It gives an expected return. Combined with a forecast of future cash, such as in the Gordon growth model, that return can be used to work out a price.

### Is CAPM proven to be correct?

No. It is a simple model with clear assumptions, and tests have found that real returns depend on more than beta alone. It is still widely taught and used as a starting point.

## Related formulas

- [Beta](https://formulabro.yunusemrevurgun.com/finance/beta/)
- [Expected Return of a Portfolio](https://formulabro.yunusemrevurgun.com/finance/portfolio-expected-return/)
- [Gordon Growth Model](https://formulabro.yunusemrevurgun.com/finance/gordon-growth-model/)

## Sources

- [Capital asset pricing model, Wikipedia](https://en.wikipedia.org/wiki/Capital_asset_pricing_model)
- [Beta (finance), Wikipedia](https://en.wikipedia.org/wiki/Beta_(finance))
- [Modern portfolio theory, Wikipedia](https://en.wikipedia.org/wiki/Modern_portfolio_theory)

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