# Sharpe Ratio Definition

The sharpe ratio definition is the excess return or risk premium of a well diversified portfolio or investment per unit of risk. Measure sharpe ratio using standard deviation. You may also know this ratio as the reward to variability ratio or the reward to volatility ratio.

## Sharpe Ratio Explained

The sharpe ratio is a good measure for investors because it allows them to distinguish the amount of reward needed per unit of risk. This allows for risk averse investors to stay away from low reward high risk situation that they are uncomfortable with. The higher the ratio the better for an investor. It is also useful in establishing the ratio efficient frontier in which an investor can build a model for several different investments and build a portfolio that is exactly equal to the desired ratio. These efficient frontier models can distinguish down to the specific weights what an investor needs to do to build the desired portfolio.

## Sharpe Ratio Formula

Use the following sharpe ratio formula:

```SR = E(R-Rf)
σ```

Where:

R = asset return
Rf = Risk free return
E(R-Rf) = Expected return of the risk premium
σ = standard deviation of the risk premium

## Example

Tim is looking to invest in a stock that has an expected return of 12%. The risk free rate is 4%, and the standard deviation of the risk premium is 10%. Thus, the calculation is as follows:
Sharpe = (.12-.04)/.10 = .8

The .8 can be interpreted as meaning that for every unit of risk that you accept as an investor you will be taking on an additional one and a quarter amount of risk.  Risk premium is any return above the risk-free rate. The risk-free rate refers to the rate of return on a theoretically riskless asset or investment, such as a government bond. All other financial investments entail some degree of risk, and the return on the investment above the risk-free rate is called the risk premium.

When an investor purchases a financial instrument, such as stock or bonds, that investor is putting his capital at risk. The company that issued the stock could perform poorly and its stock could plummet in value; or the company issuing the bonds could default and its bonds could become worthless. Both of these potential scenarios represent risk for the investor or speculator. The return on an investment, which corresponds to the riskiness of the investment, is supposed to compensate the investor for that risk.

Different financial instruments have different degrees of riskiness and the returns on these instruments typically correspond with the level of risk. More risky assets have higher returns; less risky assets have lower returns. An asset with no risk, such as a U.S. government bond, has a comparatively low rate of return because there is little or no risk of the U.S. government defaulting. Therefore, the rate of return on that type of riskless asset is referred to as the risk-free rate. Any return above that rate is a risk premium which compensates the investor for the riskiness of the asset.

Assume the risk-free rate is 5%. This means a riskless U.S. government treasury bond offers an annual return of 5%. Let’s say an investor invests in the stock of a company and that stock has an annual return of 7%. The risk premium for that company’s stock is the difference between the risk-free rate of 5% and the expected return of the stock of 7%. So the risk premium is 2%.

Risk Premium = Asset Return – Risk-Free Rate

2% = 7% – 5%

## Risk Premium and the CAPM

The risk premium is also used in calculating the expected return on asset when using the capital asset pricing model (CAPM). In that case, the risk premium combines the market risk premium, or the overall stock market’s return above the risk-free rate, with the beta of the individual stock. This gives the risk premium for the particular stock over the risk-free rate. # Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is an equilibrium model that measures the relationship between risk and expected return of an asset based on the asset’s sensitivity to movements in the overall stock market.

CAPM is used to price the risk of an asset or a portfolio of assets. The model is based on the idea that there are two types of risk, systematic risk and idiosyncratic risk, and that the investor should be compensated for both types of risk, as well as, the time value of money. Systematic risk refers to market risk. Idiosyncratic risk refers to the risk of an individual asset. Time value of money refers to the difference between the present value of money and the future value of money. Also, use the model to measure the required rate of return for capital budgeting projects.

The CAPM states that an asset’s expected return equals the risk-free rate plus a risk premium. The risk-free rate refers to the return on an investment without risk, such as a US Treasury Bond, and represents the time value of money. The risk premium represents the incremental return for investing in a risky asset. In the CAPM, it is defined as the market premium, or the overall stock market return less the risk-free rate, multiplied by the beta of the asset. Beta is a factor that measures an asset’s sensitivity to movements in the overall stock market. According to the CAPM, riskier assets should yield higher returns.

## The CAPM Formula

Expected Return = Risk-Free Rate + Beta (Market Return – Risk-Free Rate)

For example, if the risk free rate is 5%, the market return is 10%, and the stock’s beta is 2, then the expected return on the stock would be 15%.

15% = 5% + 2 (10% – 5%)

### Problems with Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is based on assumptions. First, the model assumes that a riskier asset will yield a higher return. But this is not necessarily true. A risky asset could decline in value. Second, historical data determines beta. The model assumes this historical data an accurate predictor of future results. But the asset’s future volatility may not necessarily reflect its past volatility.

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