Question

Consider the risk-free rate over your investment period was 6%, and the market's average return was 14% with a standard deviation of 20%. What was your Sharpe ratio?

What if you leverage your position? For example, what if you borrow USD 1,000 and invest USD 2,000 (USD 1,000 which you borrowed and USD 1,000 of your own money) in the market, what is your Sharpe ratio for this leveraged position?

Explain (or proof) how the Sharpe ratio changes if you leverage up your position.

Answer 1

Sharpe ratio is an investment measurement which is widely used to evaluate the risk-adjusted performance of an stock/fund (asset)- that is, this ratio describes how much extra return an investor will receive on holding a risky asset. It can be ascertained with the following formula:

Sharpe ratio = (Market Return-Risk free rate) / Standard deviation

Applying the values given in the question to be above formula,

Sharpe ratio = (14-6)/20 = 0.40

Leverage of position

Sharpe ratio remains the same even if we leverage up our position. In the given case,

Market Return = 14%

Borrowed a multiple of portfolio value ($1000+$1000), thus we leveraged the portfolio '2' times. Thus the new market return = 2*14% ror 14%+14%=28%

Risk-free rate = 6% (which is the rate that can be applied for using the own funds as it can be safely assumed that this will be the rate one can both borrow and lend)

Thus, the new rate for the leveraged position = 28%-6% = 22%

The standard devitaion for the portfolio increases to 2*20% = 40%

Applying the above value in the formula,

Sharpe ratio = (((14+(14-6))-6)/(2*20))= 22-6 / 40 = 0.40 which remains unchanged .