This is an important question. In fact, you should memorize Formula 8.15 that describes how risk grows over time. The assumption that there is no compounding (that you can ignore the cross-product) and that risk is roughly constant per period is reasonable over periods that are not more than a few years long.

(a) If we can ignore the cross-products, then we are using a simple weighted-average formula with weights of 1 on each term: ˜r0, 2

≈ 1 . ˜r0, 1

+ 1 . ˜r1, 2. (The exact formula would have been ˜r0, 2

=

˜r0, 1

+ ˜r1, 2

+ ˜r0, 1 . ˜r1, 2.)

(b) The expected rate of return over 2 years is E(˜r0, 2) ≈ E(˜r0, 1) + E(˜r1, 2) = 12% + 12% = 24%.

(c) The variance of the rate of return over 2 years is Var(˜r0, 2) ≈ 1 . Var(˜r0, 1) + 1 . Var(˜r1, 2) + 2 . 1 . 1 .

Cov(˜r0, 1, ˜r0, 2). In a perfect market, the last term should be approximately zero.

(d) The variance over 2 years for our specific example is Var(˜r0, 2) ≈ 1 . Var(˜r0, 1) + 1 . Var(˜r1, 2) + 0

= (20%)2 + (20%)2 = 2 . (20%)2 = 800%%

Therefore, the standard deviation is

√

2 . 20% ≈ 28%.

(e) The Sharpe ratio is 2 . (12% − 6%)/28% ≈ 0.43.

(f) The variance is 4 . (20%)2 = 1600%%. The standard deviation is 20% .

√

4 = 40%. The Sharpe ratio is (6% . 4)/(20% .

√

4) = 0.3 .

√

4 = 0.6.

(g) The variance is 16 . (20%)2 = 6400%. The standard deviation is 20% .

√

16 = 80%. The Sharpe ratio is 0.3 .

√

16 ≈ 1.2.

(h) The variance is T . (20%)2. The standard deviation is 20% .

√

T. In other words, the standard deviation grows with the square root of the number of time periods:

IMPORTANT: How asset risk grows with time: Sdv(˜r0,T) ≈

√

T . Sdv(˜r0, 1) (8.15)

If the rates of return on an asset are approximately uncorrelated over time (a perfect market consequence), if the risk in different time periods remains constant, and ignoring all cross-product terms.

The Sharpe ratio is 0.3 .

√

T.

(i) The formulas also work with fractions. The variance is therefore 1/12 . (20%)2 ≈ 33.3%%. The standard deviation is therefore

√

1/12 . 20% ≈ 5.8%. The monthly Sharpe ratio is

√

1/12 . 30% ≈ 0.09.

(j) The variance is 1/250 . (20%)2 = 1.6%%. The standard deviation is

√

1/250 . 20% ≈ 1.3%. The daily Sharpe ratio is about 0.019.