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# How is the slope of a capital market line (Sharpe Ratio) defined?

The slope of a capital market line of a portfolio is its Sharpe Ratio. We know that the greater the returns of a portfolio, the greater the risk. The optimal and the best portfolio is often described as the one that earns the maximum return taking the least amount of risk.

One method used by professionals to increase returns taking minimal risks is the eponymous "Sharpe Ratio". The Sharpe ratio is a calculation of risk-adjusted returns of how good is the investment return vis-a-vis the amount of risk taken. An increased Sharpe ratio for an investment means a better risk-adjusted return.

## How to Calculate?

The Sharpe Ratio is easy to calculate, as it takes only three variables −

- Risk-free rate,
- Expected return, and
- Standard deviation (SD).

SD is the most popular way to calculate the risk of a portfolio, as it shows the variation of returns from the average. Risk usually goes up with increasing SD.

The risk-free rate is the rate of a theoretical investment with no risk and a typical proxy is a short-duration government bond yield.

The Sharpe Ratio is calculated using the formula −

$$\mathrm{\frac{Expected\:Return\:of\:Portfolio − Risk\:free\:Rate}{SD\:of\:Portfolio}}$$

## Different Assets in a Portfolio Matters

Assume that portfolio A had a 17 percent rate of return last year, while the overall market returns were only 11 percent. The initial thought will be that portfolio A is better than the overall market because of the added return. However, although the return of A was greater than the overall market, taking into consideration the risk of your portfolio, calculated using the Sharpe Ratio, portfolio A has actually assumed much more risk. Hence, portfolio A was not optimal.

Let’s assume that your portfolio had an SD of 14 percent versus 6 percent for the overall market, and the risk-free rate was 2 percent.

Sharpe Ratio for your portfolio −

$$\mathrm{\frac{(17 − 2)}{14}= 1.07}$$

Sharpe Ratio for the overall market −

$$\mathrm{\frac{(11 − 2)}{6}= 1.5}$$

In this example, we see that the Sharpe ratio is less even though portfolio A earned more than the market. The market portfolio with a better Sharpe Ratio was more optimal even though the return was less than portfolio A. Therefore, portfolio A assumed excess risk without any additional compensation. Alternately, the overall market, with a higher Sharpe Ratio, had a better risk-adjusted return.

## Not Everything Is Normal

The Sharpe Ratio relies on the SD as a measure of risk, however, the standard deviation assumes a normal distribution where the mode, mean, and median are all equal. Recent history has shown that market returns are not usually normally distributed in the short term. In fact, market returns are actually skewed.

In a skewed distribution, the SD becomes useless because the mean can go greater than or less than the other measures of central dependency. In addition, short-term volatility spikes with large swings in both directions, the SD rises and causes the Sharpe Ratio to go lower.

## Why Diversification is Useful

Standard Deviation of a portfolio of multiple assets is calculated using each asset’s standard deviation. The correlation coefficient among the assets and the weight of the asset in the portfolio is considered before calculating the SD of the portfolio.

When a number of assets have low correlations and are mixed to form a portfolio, the portfolio SD goes lower than the sum of the two SDs. As a result, the Sharpe Ratio goes higher since the denominator of the ratio is lower.

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