QUESTION 1 - BONDS
You have just purchased $5 million par amount of the 5yr and 30yr US Treasury bonds shown below, at the prices shown. (Assume all bonds are semi-annual compounding.)
Par Amt Coupon Maturity Yield Price $Duration
$millions (semi-ann) (yrs) (semi-ann) (= DV01)
5.00 6.00% 5 6.000% 100.00 4.00
5.87% 10 6.000% 99.00 7.50
5.00 6.04% 30 6.000% 100.50 13.00
(1) Calculate the market values of the 5yr and 30yr bonds you purchased.
(a) 5yr: $5 Million; 30yr: 5.025 Million
(b) 5yr: $50 Million; 30yr: 50.25 Million
(c) 5yr: $500 Million; 30yr: $502.5 Million
(d) None of the above
(2) Calculate the "DV01 risk" for both of the bonds that you purchased. DV01 risk, for a given bond, is the dollar amount that you would make/lose if the bond's yield moved by 1% instantaneously.
(a) 5yr: $200,000; 30yr: $650,000
(b) 5yr: $20 Million; $30yr: 65 Million
(c) 5yr: $20 Million; 30yr: $65.325 Million
(d) 5yr: $200,000; 30yr: $653,250
(e) None of the above
(3) What par amount do you need to buy or sell of the 10yr US Treasury bond, so that the DV01 risk of the entire portfolio is negligable? (Negligable in this context means "less than $100 of DV01 risk over the entire portfolio".) You must calculate the par amount of the 10yr bond to the 3rd decimal place (i.e., to the nearest thousand dollars).
(a) Sell $11.333 million par amount
(b) Sell $11.492 million par amount
(c) Sell $11.22 million par amount
(d) Buy $11,333 million par amount
(e) Buy $11.492 million par amount
(f) Buy $11.22 million par amount
QUESTION 1 - BONDS (CONTINUED)
(4) What are the coupons on the three bonds?
(a) 5yr: 6.00%; 10yr: 5.87%; 30yr: 6.04%
(b) 5yr: 5.93%; 10yr: 5.80%; 30yr: 6.00%
(c) 5yr: 6.00%; 10yr: 6.00%; 30yr: 6.00%
(d) none of the above
(e) Cannot be determined
(5) In addition to the 30yr bond listed above, there is a zero coupon bond in the 30yr sector whose maturity date is 3 months later. Call the 30yr bond in the table above Bond A, and the zero coupon 30yr Bond B. Which of the following statements is (are) true/false/cannot say?
(i) The price of A is greater than the price of B True False Cannot say
(ii) The magnitude of bond A's DV01 is greater than that of B's DV01 True False Cannot say
(iii) The ModDur of A is greater than the ModDur of B True False Cannot say
(iv) For bond B, DV01/100 is smaller than its ModDur True False Cannot say
(6) Which of the following most closely approximates the yield curve view implied by this portfolio?
(a) Curve flattening
(b) Curve steepening
(c) A parallel shift in the yield curve
(d) Curve steepens for short to medium maturity bonds and flattens for medium to long maturity bonds
(e) Curve flattens for short maturity bonds and steepens for long maturity bonds
(f) None of the above
(7) Which of the following statements most accurately reflects the motivation for setting up the trade so that the DV01 risk of the portfolio is zero?
(a) protects against small parallel shifts in the curve
(b) protects against large parallel shifts in the curve
(c) protects against an upward parallel shift in the curve
(d) protects against a downward parallel shift in the curve
(e) protects against the curve becoming more "humped"
(f) protects against the curve becoming less "humped"
QUESTION 1 - BONDS (CONTINUED)
(8) Suppose that, rather than just hedging the DV01 risk, you wanted to allow for the fact that different sectors of the curve tend to experience different levels of volatility. Which of the following statements is correct?
(a) Short maturity bonds have greater price volatility, so you should purchase more of these bonds than the amount implied by duration-neutrality
(b) Long maturity bonds have greater price volatility, so you should purchase more of these bonds than the amount implied by duration-neutrality
(c) Short maturity bonds have greater yield volatility, so you should purchase more of these bonds than the amount implied by duration-neutrality
(d) Long maturity bonds have greater yield volatility, so you should purchase more of these bonds than the amount implied by duration-neutrality
(e) Short maturity bonds have greater price volatility, so you should purchase fewer of these bonds than the amount implied by duration-neutrality
(f) Long maturity bonds have greater price volatility, so you should purchase fewer of these bonds than the amount implied by duration-neutrality
(g) Short maturity bonds have greater yield volatility, so you should purchase fewer of these bonds than the amount implied by duration-neutrality
(h) Long maturity bonds have greater yield volatility, so you should purchase fewer of these bonds than the amount implied by duration-neutrality
(9) Imagine that you are the risk manager at a commercial bank, whose primary assets are 30 year fixed rate mortgage loans. You finance these loans by accepting deposits from consumers, usually for a guaranteed interest rate for a duration of 1 - 2 years. Which of the following trades might you make to hedge your bank's interest rate exposure?
(a) "Buying the curve" (buying short maturity bonds, and selling longer maturity bonds)
(b) "Selling the curve" (selling short maturity bonds, and buying longer maturity bonds)
(c) Buying both short and long maturity bonds
(d) Selling both short and long maturity bonds
(e) None of the above.
QUESTION 2 - PORTFOLIOS
You have been asked to evaluate the performance of several portfolios managed by Blue Devil Hedge Fund (BDHF), to see whether these appear to be good investments. The spreadsheet exam_F2011.xls contains monthly returns on three Blue Devil portfolios from April 1993 to December 2008 in the Funds worksheet. The spreadsheet also contains monthly returns on a series of Russell Indices of US equities, as well as the monthly riskless rate (US TBills) in the Indices worksheet.
Step 1
Select appropriate index benchmarks for each of the three BDHF portfolios. It should be clear which is the right benchmark for each portfolio. However, to confirm your selection, estimate the correlations between each of the three BDHF portfolios against all of the indices, using the Step 1 template in the Indices worksheet. For each portfolio, the most highly correlated index should be used as the benchmark for that portfolio. Please complete this matrix, and use bold font to indicate the highest correlation index for each portfolio.
Step 2
Calculate the average annual return on each of the three BDHF portfolios, as well as on their
benchmarks, using the Step 2 template in the Funds spreadsheet . On this basis, which (if any) of the three portfolios appear to outperform their benchmarks?
Step 3
Calculate the Sharpe ratios of each of the BDHF portfolios. Remember that Sharpe ratios require excess return on the numerator. A portfolio's Sharpe ratio can be compared against the Sharpe ratio of its benchmark, to see whether it is a fund that justifies investment (relative to a passive investment in the index benchmark). On this basis, which portfolio(s) look like good investments?
Step 4
Suppose now that you are considering investing in one of the BDHF funds, as a diversification tool against a number of other hedge fund investments. Which performance measurement metric (Sharpe Ratio, Treynor's Measure, or Jensen's alpha should you use to determine which BDHF portfolio (if any) you should select?
Step 5
Calculate the betas of the three DBHF funds, and hence their (annual) alphas, as well as the values of the performance measurement metric that you identified in Step 4. What are the implications of the alphas on your decision about whether to invest with BDHF?
Step 6
Now consider the fact that investing in one of the hedge fund portfolios will be expensive, relative to a passive investment in an index. Suppose that you can invest in an index for zero cost (in reality, it will cost you a few basis points). Bear in mind that hedge funds typically charge a 2% management fee (i.e., 2% of assets under management, on an annual basis, regardless of performance). Qualitatively, how would this affect your decision about whether to invest in any of the BDHF portfolios? Are there typically any other fees charged by hedge funds that would further discourage you from investing with BDHF?
QUESTION 3 - CAPM
You are a mutual fund manager in charge of the TwoStock growth fund. This fund, which consists of two stocks, has the following return characteristics, based on historical data.
Expected Return Standard Deviation
Stock 1 6% 25%
Stock 2 12.4% 35%
The fund is split equally between the two assets (i.e. 50% in each stock), and the correlation between the two stocks is 0.3.
The current market environment is as follows. The riskless rate is 4% and the expected return on the S&P500 is 12% with a standard deviation of 20%. Stock 1 and stock 2 have estimated correlations of 0.2 and 0.6 with the S&P500, respectively.
(1) Estimate the beta of each asset.
(a) 0.05, 0.21
(b) 0.16, 0.34
(c) 0.25, 1.05
(d) 5.00, 15.00
(2) Using your results from part (1), calculate the beta of the fund assuming the S&P500 is a reasonable proxy for the market portfolio.
(a) 0.13
(b) 0.65
(c) 0.25
(d) 10.00
(3) Find the portfolio offering the same expected return as the TwoStock Fund but with the least amount of risk (i.e. smallest standard deviation) on the Capital Market Line ("CML"). What assets are in this portfolio?
(a) The riskless asset and both of the two stocks
(b) The riskless asset and the market portfolio
(c) The riskless asset and stock 1 only
(d) The riskless asset and stock 2 only
QUESTION 3 - CAPM (CONTINUED)
(4) What proportion of this portfolio should be allocated to the riskless asset?
(a) 0.35
(b) 0.45
(c) 0.55
(d) 0.65
(e) 0.00
(f) 1.00
(5) What is the standard deviation of this portfolio?
(a) 0.00
(b) 0.03
(c) 0.04
(d) 0.07
(e) 0.13
(f) 0.20
(g) Cannot be determined
QUESTION 4 - EQUITIES
For this question, you will use the Comps worksheet in the Exam_F2011.xls spreadsheet. Complete all of the cells highlighted yellow, using the appropriate formulas in each case.
QUESTION 5 - DIVERSIFICATION
You are considering creating an investment portfolio using the following 2 stocks:
Stock 1 has an expected return of 9.5% and volatility 25%
Stock 2 has an expected return of 14.1% and a volatility of 20%
Assume that the percentage weight in Stock 1 is designated w1 , similarly w2.
(1) What is the minimum variance portfolio that you can create with just Stock 1 and Stock 2, assuming that the two stocks have a correlation of 50%?
(a) w1 = 0.21, w2 = 0.79
(b) w1 = 0.29, w2 = 0.71
(c) w1 = 0.43, w2 = 0.57
(d) w1 = 0.71, w2 = 0.29
(2) What is the minimum variance portfolio that you can create with just Stock 1 and Stock 2, assuming that the stocks have a covariance of 2%?
(a) w1 = 0.32, w2 = 0.68
(b) w1 = 0.39, w2 = 0.61
(c) w1 = 0.61, w2 = 0.39
(d) w1 = 0.79, w2 = 0.21
(3) What is the maximum correlation between Stocks 1 and 2 such that there are still benefits from diversification, assuming that short sales are not permitted? [HINT: "benefits to diversification" may be defined as the opportunity to create a portfolio with lower volatility than either stock by itself.]
(a) ρ12 = 0.49
(b) ρ12 = 0.59
(c) ρ12 = 0.69
(d) ρ12 = 0.79
(e) ρ12 = 0.89
QUESTION 6 - TRUE OR FALSE - CAPM
Use the answer_template.doc to indicate True or False to the following statements.
1) In the CAPM model of the world, markets are assumed to be efficient
2) In a CAPM world, individuals have similar risk preferences
3) In the CAPM world, individuals can both borrow and lend
4) In the CAPM world, individuals will all select the same risky portfolio (assuming borrowing and lending rates are the same)
5) The CAPM model states that the expected excess return on any individual asset is proportional to the excess return on the market portfolio
Formulas
1. Bond price formula:
P = C/y [1 - (1 + y )-n] + 100/(1+y)n where P = bond price; C = coupon cash flow; y = yield; n = number of periods
For a semi-annual bond:
C = (quoted coupon)/2,
y = (quoted yield)/2
n = 2*maturity
2. Constant dividend growth formula for Stocks:
P0 = D1/(r-g) where P = stock price; D = dividend; r = discount rate; g = constant growth rate note also that Ed = D where E = earnings; d = dividend payout ratio.
Note that the discount rate, r, may also be called the "cost of capital" or "cost of equity capital"
3. Expected return on a portfolio of two assets
E[rp] = w1 E[r1] + w2 E[r2] where w1 and w2 are portfolio weights in assets 1 and 2; w1 + w2 = 1
4. Variance on a portfolio of two risky assets
σ p
2 = w1
2 σ1
2 + w2
2 σ2
2 + 2 w1 w2 σ1,2 where σ1 = standard deviation of asset 1 (similarly σ2 )
w1 = portfolio weight in asset 1 (similarly w2)
σ1,2 = covariance between assets 1 and 2; σ1,2 = ρ1,2 σ1 σ2 ; ρ = correlation.
5. Minimum variance portfolio
To obtain the minimum variance portfolio from a portfolio of two risky assets, set w1 so that:
w1 = (σ2
2 - σ1,2) / (σ1
2 + σ2
2 - 2 σ1,2) where w and σ are defined as above (in #5)
6. The beta β1 of Asset 1 is given by:
β1 = σ1,m / σm 2 where M is the market portfolio.
7. The Capital Asset Pricing Model (CAPM) assumes that the following formula holds for the expected return on asset 1 when it is added (in small proportion) to a well-diversified portfolio:
E[r1 ] = rf + β1 (E[rm] - rf ) Where rf is the risk free rate, and E[rm] is the expected return on the market portfolio.
8. Futures
The formula relating commodities futures to the underlying asset price is given by: F = S0 e(r + q) t
Where F is the futures price, S0 is the underlying asset price at time 0, r is the continuously
compounded riskless rate, q is the continuously compounded carry cost on commodities and t is the time to maturity on the futures contract (quoted in years).