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GU4261/GR5261 Exam 1—Statistical Methods for Finance 2022
发布时间：2022-06-30
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Exam 1—Statistical Methods for Finance (GU4261/GR5261)
2022
Name:
1. (10pt) Suppose that a security A in an efficient market is such that σA = 0.50, ρAM = 0.70. If σM = 0.25,
(a) (5pt) compute βA .
(b) (5pt) If µA = 12%, µM = 8% and µf = 5%, is this asset appropriately priced? If not, is it underpriced or overpriced?
2. (15pt) Suppose that the continuously compounded forward rate is
r(t) = 0.030 + 0.004t.
(a) (5pt) What is the price of a 5-year zero coupon bond with face value (PAR) $1000 ?
(b) (10pt) What is the price of a 5-year coupon bond with face value (PAR) $1000 and 5 annual coupon payments of $50 each?
3. (10pt) Assume that an efficient market consists of three risky assets A, B and C with the following β’s and σej(2)
|
βj |
σej |
|
|
Asset A Asset B Asset C |
1.25 1.10 0.90 |
20% 18% 15% |
Assume that the pairwise correlations among the ∈js are all equal to zero and that σM = 10% and that you hold a portfolio with weights ωA = 0.50, ωB = 0.30 and ωC = 0.20 on these three assets.
(a) (5pt) What are the values of Cov(RA, RB ), Cov(RA, RC ) and Cov(RB, RC )? (b) (5pt) What proportion of the total risk of this portfolio is due to the market risk?
4. (10pt) A 4 year coupon bond with -annual coupon payments has a par value of $ 1000. Suppose that the coupon rate is 5% and the yield to maturity is 4%,
(a) (5pt) What is the price of this bond?
(b) (5pt) What is the current yield equal to?
5. (10pt) A zero coupon bond with PAR = $1000 matures in 10 years and sells for $900.
(a) (5pt) What is the value of the interest (discount) rate, r, if compounding is done con- tinuously?
(b) (5pt) Suppose that this zero coupon bond may default and that its time to default T has an exponential distribution with parameter β = 0.03 (the cdf of T is FT(t) = 0 if t - 0 and FT(t) = 1 _ e −βt if t s 0.). The present value of this bond is
PV = 
Find the expected present value E(PV) of this bond (use for r the value in a)), that is, what is the fair price for this bond?
6. (5pt) A firm is planning to invest $ 2,000,000 and has enough capital reserves to cover a loss of $ 500,000 but not more. The firm plans to invest a portion ω in a risky asset A with return RA and the rest in a risk free asset with expected return µf = 5%. If RA has a normal distribution with mean µA = 14% and σA = 30%, find the value of ω if the firms desires to have the probability of loosing $200,000 or more equal to 0.01. (Hint: Φ − 1 (0.01) = _2.33)
7. (5pt) If the current price of a stock is P0 (not random) and its price a year from now is P1 , then P1 = (1 + R)P0 where R is the net return over next year. If R has a normal distribution N (µR = 0.10, σR(2) = 0.25), what are µP1 and σP(2)1 equal to? What is the distribution of P1?
8. (35pt) Suppose that a market consists of two risky assets A and B with returns RA and RB , respectively. Assume that µA = 10%, µB = 14%, and σA = 20%, σB = 25%, ρAB = _0.50 and that the expected return on the risk free asset is µf = 5%.
(a) (10pt) Find the value of (ωA(min), ωB(min)), the weights of the minimum variance portfolio (MVP). What are the values of (µMVP , σMVP ), its corresponding mean and risk.
(b) (5pt) Find the efficient portfolio of the risky assets only with µp = 12%. Compute its risk.
(c) (10pt) Find the tangency portfolio. What are its mean and its risk equal to?
(d) (5pt) Compute the Sharpe ratio of each asset.
(e) (5pt) What is the expected return of the efficient portfolio (made of risky assets and risk free asset) P with risk is σP = 18%?
