Let's consider a stock market index, such as the S&P 500. It had a historical average rate of return of about 12% per annum, and a historical standard deviation of about 20% per annum. Assume for the moment:

(a) Known statistical distributions: You know the expected reward and risk. In our example, we assume that they are the historical averages and risks. This is convenient.

(b) Independent stock returns: Stock returns are (mostly) uncorrelated over time periods. This is reasonable because if this were not so, you could earn money purchasing stocks based on their prior performance in a perfect market. (This will be the subject of Chapter 11.)

(c) No compounding: The rate of return over X years is the simple sum of X annual rates of return. (That is, we ignore the cross-product terms that are rates of return on rates of return.) This is problematic over decades, but not over just a few months or even yearOur goal is to work out how asset risk grows with time under these assumptions. The variance shortcut formula will help us.

(a) Write down the formula for the total rate of return over 2 years.

(b) What is the expected total rate of return over 2 years?

(c) Write down the formula for the variance over 2 years.

(d) What is the specific risk here (variance and standard deviation) over 2 years?

(e) The Sharpe ratio is a common (though flawed) measure of portfolio performance. It is usually computed as the expected rate of return above the risk-free rate, then divided by the standard deviation. Assume that the risk-free rate is 6%. Thus, the 1-year Sharpe ratio is (12% - 6%)/20% ≈ 0.3. What is the 2-year Sharpe ratio?

(f) What are the expected rate of return and risk (variance and standard deviation) over 4 years? What is the 4-year Sharpe ratio?

(g) What are the expected rate of return and risk (variance and standard deviation) over 16 years? What is the 16-year Sharpe ratio?

(h) What are the expected rate of return and risk (variance and standard deviation) over T years? What is the T-year Sharpe ratio?

(i) What are the expected rate of return and risk (variance and standard deviation) over 1 month? What is the 1-month Sharpe ratio?

(j) What are the expected rate of return and risk (variance and standard deviation) over 1 trading day? What is the 1-day Sharpe ratio? Assume 250 trading days per year.