Standard Deviation

With an 8% return, a 2% risk-free rate, and a 10% standard deviation, the Sharpe ratio is:

1. A.0.6Correct - 6 / 10.
2. B.0.2This divides the risk-free rate by the standard deviation (2/10) instead of the excess return. The numerator must be the return earned above the risk-free rate, which is 6%.
3. C.0.8This divides the total return by standard deviation (8/10) and skips subtracting the risk-free rate. Sharpe measures reward per unit of risk for the portion of return that required taking risk at all, so the 2% earned risk-free must come out first.
4. D.1.0This adds the risk-free rate rather than subtracting it, giving 10/10. Adding it inflates the ratio, but the risk-free return is the baseline an investor could have had without any volatility.

Why: Sharpe = (8 - 2) / 10 = 0.6.

A client holds a long call option and the underlying stock trades at essentially the same price for several weeks. Over that period the premium on his option will generally

1. A.decline, because time value erodes as the expiration date draws nearer.Correct. Time is an input in its own right, and it is consumed whether or not the stock moves.
2. B.remain unchanged, since the price of the underlying stock has not moved.Wrong. The premium has two components, and the passage of time consumes one of them regardless.
3. C.rise, because a longer record of stability makes the option safer to hold.Wrong. Stability reduces the likelihood of the very move the option needs in order to pay.
4. D.decline, but only where the option is currently in the money.Wrong. An out-of-the-money option is nothing but time value, so it decays fastest of all.

Why: An option is a wasting asset because one of the two components of its price is the time remaining before expiration. Holding the underlying still, the passage of time removes optionality without replacing it with anything, so the premium erodes and the erosion accelerates as expiration approaches. This is why a long option position needs the underlying to move, and to move soon enough, before it can be profitable. A rise in implied volatility could offset the decay for a time, but nothing in a period of price stability supplies one.

An adviser distinguishes the risk capacity of a client from the risk tolerance of that client. Risk capacity refers to

1. A.the amount of volatility the client says he is emotionally comfortable holding through.Wrong. That is tolerance, an attitude the client reports rather than a fact about his finances.
2. B.the rate of return the client must earn in order to reach the goal on schedule.Wrong. That is required return, which describes what the goal demands rather than what the client can withstand.
3. C.the amount of loss the client can absorb without putting a stated goal out of reach.Correct. It ties the measure to the balance sheet and the goal, which is what makes capacity objective.
4. D.the extent to which the client believes a particular investment is dangerous.Wrong. That is risk perception, which can be corrected with information and is not a limit on anything.

Why: Capacity is a financial fact drawn from the balance sheet, the cash flow and the timing of the goals, and it measures how much loss the client can absorb before a stated objective is put out of reach. Tolerance is an attitude, describing how much volatility the client is willing to live with, and the two frequently disagree. A complete profile records both, together with the return the goals actually require, because a recommendation has to respect the lower of what the client can bear and what the client will bear. Where capacity is low the allocation must reflect that however comfortable the client says he is.

A portfolio returns 10% with a 12% standard deviation; the risk-free rate is 2%. The Sharpe ratio is:

1. A.0.67Correct - 8 / 12 = 0.67.
2. B.0.20This comes from putting the risk-free rate over the return (2/10) rather than dividing excess return by risk. Standard deviation, the measure of risk, must be the denominator in a reward-per-unit-of-risk ratio.
3. C.0.83This is 10/12, the total return over standard deviation with the risk-free rate never subtracted. Because 2% was available with no volatility at all, only the 8% earned above it deserves credit for the risk taken.
4. D.1.20This inverts the fraction, dividing 12 by 10. A Sharpe ratio above 1.0 would mean excess return exceeded volatility, which cannot be true when the excess return here is 8% against 12% of risk.

Why: Sharpe = (return - risk-free) / standard deviation = (10 - 2) / 12 = 0.67.