The rule that optimal portfolios will maximize the Sharpe ratio only applies when which of the following conditions is satisfied:
1. It is possible to borrow or lend any amounts at the risk free rate.
II. Investors' risk preferences are fully described by expected returns and standard deviation.
III. Investors are risk neutral
The Sharpe ratio does not require investors to be risk neutral, only that for a given level of returns they prefer less risk to more risk. (Risk neutral means that investors are indifferent to the level of risk, and are only driven by a desire to maximizing expected value, regardless of risk levels.)
The ability to borrow and lend any amounts of money at the risk free rate is a fundamental assumption for the rule that optimal portfolios will maximize the Sharpe ratio.
This rule also assumes that risk preferences are completely described by return and standard deviation of returns.
Therefore Choice 'd' is the correct answer as statements I and II are correct.
Given identical prices, a bond trader prefers dealing with Bank A over Bank B. Given a choice between Bank B and Bank C, he prefers Bank B. Yet, when given a choice between Bank A and Bank C, he prefers dealing with Bank C. What axiom underlying the utility theory is he violating?
Remember the four basic axioms underlying the principal of maximum expected utility:
- Transitivity, ie if A is preferred over B, and B is preferred over C, then A must be preferred over C;
- Continuity, ie if A is preferred over B, and B is preferred over C, then B is on a continuum between A and C such that we can be indifferent between receiving B, or a lottery offering either A or C with probabilities p & 1-p respectively.
- Independence, ie choices are not affected by the way they are presented
- Stochastic dominance, ie a gamble that offers a greater probability of a preferred out come will be preferred.
In this case, the first axiom is being violated. Therefore Choice 'c' is the correct answer.
Which of the following statements are true:
1. The Kappa family of indices take only downside risk into account
II. The Treynor ratio provides information on the excess return per unit of specific risk
III. All else remaining constant, the Sharpe ratio for a portfolio will increase as we increase leverage by borrowing and investing in the risky bundle
IV. In the market portfolio, we can expect Jensen's alpha to equal zero.
Kappa indices, which include the Sortino ratio and the Omega statistic, consider semi-variance, ie variance calculated only in respect of the downside risk instead of variance (which includes both upside and downside). This is because one criticism of other risk adjusted performance measures is that they take both upside and downside risk into account, even though a portfolio manager or investor is only concerned with managing the downside. The Kappa indices consider returns that are below a threshold to measure performance by. Therefore statement I is correct.
The Treynor ratio is calculated as [(Portfolio return - Risk free return)/Portfolio's beta]. It therefore calculates the excess return per unit of systematic risk and not specific risk. Therefore statement II is not correct.
Increasing portfolio leverage, ie borrowing to invest more in the portfolio, will increase the excess return of the portfolio in a linear fashion. However, it will also increase the volatility of such a portfolio in an identical linear way. Since Sharpe ratio is the ratio of 'excess returns' to standard deviation, the Sharpe ratio will stay constant. In other words, leverage does not impact Sharpe ratio, which is an attribute of this measure that allows us to compare performance across managers and funds regardless of leverage used. Statement III is not correct.
Jensen's alpha results from security selection, ie from specific risk the manager has taken. In a market portfolio, all specific risk is diversified away and only systematic risk remains. Therefore, statement IV is correct.
Calculate the settlement amount for a buyer of a 3 x 6 FRA with a notional of $1m and contract rate of 5%. Assume settlement rate is 6%.
An m x n FRA is an agreement to borrow money for a period starting at time m and ending at time n at the contracted rate. Therefore, the buyer of the 3 x 6 FRA has committed to borrow $1m at the beginning of 3 months and return it at the end of 6 months, ie a total borrowing period of 3 months at a rate of 5%. Of course, the $1m is never actually exchanged, and at the beginning of the 3 month period when the next three months' interest rate is known (6%), the parties merely exchange the difference in the interest. SInce this interest was only due at the end of the 6 months and is being exchanged at the 3 month time point, it will have to be discounted to its present value.
The correct answer to this question is =(1,000,000 * (6% - 5%) * 3/12)/(1 + (6%*3/12))=$2463.05. Since interest rates rose, the borrower gained as he has the right to borrow at a lower rate, and therefore the seller will pay the borrower.
(Here:
- $1m is the notional
- 6% - 5% represents the difference between the contracted and the realized interest rates
- 3/12 is the 3 month period from month 3 to 6
- Finally, we divide by the current interest rate for 3 months to present value the payment from month 6 to month 3)
Which of the following statements are true:
1. An yield curve plots zero coupon spot rates for different maturities for bonds with different credit ratings
II. An yield curve represents the term structure of interest rates for similar instruments across a range of maturities
III. The liquidity preference theory explains why the yield curve can be downward sloping
IV. The term structure refers to the relationship between bond yields and bond maturities
An yield curve would be meaningless if it combines, for example, the yield on a AAA bond with the yield on a bond near default on the same curve. Therefore statement I is incorrect as any typical yield curve generally refers to similar instruments, and similarity includes similarity of issuer type and credit risk profiles.
Statement II accurately describes an yield curve, and statement IV explains what the term structure is.
Statement III is incorrect as the liquidity preference theory offers an explanation for an upward sloping yield curve and not a downward sloping one.
