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Summer 2013 examination

Finance I

Suitable for all candidates

Instructions to candidates TIME ALLOWED:​​ 2 hours

There are six questions.

Answer​​ four​​ out of the six questions.

All questions will be given an equal weight (25%).

Questions with parts have marks allocated as shown.

You may use an electronic calculator (financial calculators are NOT allowed).

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• This question is about bonds, forwards, and the immunization strategy

Consider two bonds. Bond 1 is a three-year zero with face value £100 and maturity in June 2016, and bond 2 is a two-year 10% bond with face value £100 and maturity in June 2015. Current prices of bonds 1 and 2 in June 2013 are £72 and £103, respectively. Moreover, there are two annuities, which are traded at equal prices. The first one is a three-year annuity that pays £10 and matures in June 2016, and the second pays £14 and matures in June 2015.

• [5 marks] What are the discount factors​​ d1,​​ d2,​​ d3​​ in the economy? What are the prices of the annuities?

• [5 marks] Determine the forward rates​​ f1, 2,​​ f2, 3, and​​ f1, 3.

• [5 marks] Consider a perpetuity with growth that makes annual payments starting in year 4. The first payment is £10, and then the payment grows at a rate of 1% forever. Consider a forward contract that delivers this perpetuity in year 3. What is the forward price at date 0 if the term structure is flat at​​ r=2%? Suppose, in year 1 the term structure shifts to​​ 1.8%​​ and remains flat. What is the value of the forward in year 1? [Hint: do not confuse forward price with forward value.]​​ Finance tutors, lse Finance tutors, Finance tutors kings kcl, Finance tutor ucl, Finance tutors Bristol, Finance tutors Manchester, Finance tutors Leicester, Finance tutors Lancaste

• [5 marks] An insurance company needs to pay £1 million in both years 3 and 4 from now. The term structure is flat at​​ r=2%. What is the present value of the company's liabilities? What is the duration?​​ Finance tutors surrey, Finance tutors warwick, Finance tutor Cambridge, Finance tutor oxford

• [5 marks] Suppose, the firm in part (d) decides to immunize its liabilities. Its current assets are in cash, and the net worth is zero. The firm decides to invest half of its cash in 2-year zeros and the other half in T-year zeros, both with face value £100. What should be the maturity​​ T​​ for this immunization strategy to work?

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• This question is about portfolio choices

Consider an investor who can invest in two stocks with expected returns​​ Er1=10%  and the immunization strategyficiency and behavorial tion strategy33333333333333333333333333333333333333333333333333333333333​​ and​​ Er2=20%​​ respectively, and in a risk-free asset that pays interest rate​​ r. Consider two portfolios of stocks and bonds A and B that have expected returns and standard deviations given by​​ ErA=8.85%,​​ σA=5%​​ and​​ ErB=12.7%12.7 dgma 333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333,​​ σB=10%​​ respectively. It is known that portfolios A and B lie on the efficient frontier.

• [5 marks] What is the interest rate r? What are the Sharpe ratios of portfolios A and B? [Hint: use the properties of efficient portfolios.]

• ​​ [5 marks] Consider also portfolios C and D with expected returns and standard deviations given by​​ ErC=12.6%,​​ σC=11%​​ and​​ ErD=24.25%12.7 dgma 333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333,​​ σD=25%. Determine whether these portfolios lie on the efficient frontier.

• [5 marks] Suppose, an investor wants to construct portfolio P with expected return of​​ 15%​​ and minimal possible risk. What is the standard deviation of portfolio P?

• [5 marks] Suppose, we know that the standard deviation of the tangency portfolio is equal to​​ 14.9%. What are the portfolio weights of this portfolio? That is, what fractions of wealth​​ are invested in stocks 1 and 2?

• [5 marks] What is diversification, and what are the limits to diversification? What types of risk can be diversified away, and what types cannot?

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• This question is about option pricing

The current price of stock XYZ is £100, and each month it either goes up by 10% or goes down by 20%. The monthly interest rate is 2%. Assume that the stock does not pay dividends.

• [5 marks] Construct a three-period binomial tree for stock price movements (i.e., with three dates 0, 1, 2), and calculate risk neutral probabilities of the up and down moves.​​ Finance tutor Brighton, Finance tutor Cambridge, Finance tutor Bristol, Finance tutor Cardiff

• [5 marks] Calculate the prices of American puts and calls, both with a strike price of £88 and maturity at date 2.​​

• [5 marks] Using only put-call parity and the results in part b), calculate the prices of European puts and calls with the same strike and maturity as in part b).​​ Finance tutor Chester, Finance tutor hull, Finance tutor Edinburgh, Finance tutor Essex, Finance tutor Fulham,

• [5 marks] How many units of stocks and bonds do you need to buy at date 0 to replicate date-I payoffs of the American put in part b)? Assume that the bond is a one-month zero with face value £100.​​ Finance tutor Barkingside, Finance tutor Liverpool, Finance tutor Lancaster, Finance tutor Manchester, Finance tutor Leicester, Finance tutor Nottingham, Finance tutor Albany,

• [5 marks] Consider a European 1-year bull spread, which is constructed as a portfolio that buys one call option with strike price £20 and sells one call option with strike price £30. The one-year riskless interest rate is 2%, and the bull price is £5. What is the price of a European 1-year bear spread, which is constructed as a portfolio that buys one put with strike price £30 and sells one put with strike price £20?​​ Finance tutor Southampton, Finance tutor surrey, Finance tutor Sheffield, Finance tutor Coventry, Finance tutor Edinburgh, Finance tutor Hampshire.

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• This question is about the CAPM

Consider two stocks, X and Y, in the economy, and suppose that the CAPM holds in this economy. The stock characteristics are listed below (all returns are annualized):

 Stock X Y Expected Return 12% 15% Standard Deviation 6% 8% Return Correlation 1.0

• [5 marks] What is the risk-free interest rate in this economy?​​ Finance tutor Brighton, Finance tutor Cambridge, Finance tutor Bristol, Finance tutor Cardiff, Finance tutor Chester

• [5 marks] Assume stock X has a market beta of 1.5, what is the market expected return in this economy? What is the market beta of stock Y?​​ Finance tutor hull, Finance tutor Edinburgh, Finance tutor Essex, Finance tutor Fulham, Finance tutor Barkingside, Finance tutor Liverpool, Finance tutor Lancaster, Finance tutor Manchester

• [5 marks] Two mutual funds in this economy are comparing performance. One averaged a 19% return with a portfolio beta of 2, while the other managed 16% with a portfolio beta of 1.4. What are the CAPM alphas of the two managers?​​ Finance tutor Leicester, Finance tutor Nottingham, Finance tutor Albany, Finance tutor Southampton, Finance tutor surrey, Finance tutor Sheffield, Finance tutor Coventry, Finance tutor Edinburgh, Finance tutor Hampshire.

• [5 marks] As we discussed in class, the CAPM is derived based on a number of assumptions. List two of the key assumptions.​​ Finance tutors London, London Finance tutors, online Finance tutors London, university Finance tutors, degree level Finance tutors, phd Finance tutors, lse Finance tutors, ucl Finance tutors,

• [5 marks] In the past five decades (i.e., in the post-1960 period), do stocks with higher betas have higher average returns, and do stocks with lower betas have lower average returns? That is, does the CAPM hold in the data?​​ lse Finance tutors, Finance tutors kings kcl, Finance tutor ucl, Finance tutors Bristol, Finance tutors Manchester, Finance tutors Leicester,

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• This question is about equity valuation

• [5 marks] Assume that a firm has a constant return on equity (ROE) and a constant plowback ratio (PBR), show that the firm has a constant dividend growth ratio, which is equal to ROE * PBR ​​​​ Finance tutor Brighton, Finance tutor Cambridge, Finance tutor Bristol, Finance tutor Cardiff, Finance tutor Chester, Finance tutor hull,

• [5 marks] Show that the present value of a firm with a constant ROE and a constant PBR can be written as​​ ​​ Hint: You need to derive the formula for a perpetuity with a constant growth rate.​​ Finance tutor Edinburgh, Finance tutor Essex, Finance tutor Fulham, Finance tutor Barkingside, Finance tutor Liverpool, Finance tutor Lancaster,

• [5 marks] Consider stock AAA in an economy that has two potential outcomes. AAA's payoff in the good economic outcome is 20% and the payoff in the bad economy is -10%. Assume that the good economic outcome has a probability of 0.6 and the bad economic outcome has a probability of 0.4.​​ What is stock AAA's expected return?​​ Finance tutor Manchester, Finance tutor Leicester, Finance tutor Nottingham, Finance tutor Albany, Finance tutor Southampton,

• [10 marks] Further assume that firm AAA has an ROE of 20% in the next 5 years, and the ROE will drop to 10% and remain at 10% from year 6 onward​​ In response, the firm will have a plowback ratio of 100% in the first 5 years, and a plowback ratio of 0 from year 6 onward. Calculate the present value of stock AAA, assuming that the EPS in year 1 is \$1.​​ Finance tutor surrey, Finance tutor Sheffield, Finance tutor Coventry, Finance tutor Edinburgh, Finance tutor​​ Hampshire.

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• This question is about market efficiency and behavioral finance

• [5 marks] Define the three forms of the Efficient Market Hypothesis (EMI-I). If an investor can make abnormal profits from trading on private information, which form(s) of market efficiency is violated?

• [5 marks] We discussed in class two necessary conditions under which markets may be inefficient — i.e., the Efficient Market Hypothesis may break down. Describe these two conditions.

• [5 marks] List two empirical findings that are generally consistent with the weak form of market efficiency. Are they also consistent with the semi-strong form of market efficient?

• [5 marks] As we discussed in class, two securities with identical future cash flows may at times be traded at different prices (e.g., the twin stock puzzle). What are the possible explanations for this phenomenon?

• [5 marks] Describe the long-term stock return reversal effect documented in De Bondt and Thaler (1985). A recent study shows that most of the return reversal is concentrated around subsequent earnings announcement dates. For example, a firm with unexpectedly high past returns as of April 2013 tends to have low stock returns around the subsequent earnings announcement (e.g., July 2013). What is a plausible interpretation of this evidence?

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#### This question is about bonds, swaps, and immunization

1. [5 marks] Consider three bonds, all with face value £100. The coupons are paid annually. Bond 1 is a one-year zero; bond 2 is a two-year 10% bond; bond 3 is a three-year 15% bond. Bonds 1, 2, and 3 have yields to maturity 5%, 10%, and 15%, respectively. What are the prices of these bonds? What are the discount factors , , and ?
2. [5 marks] Determine the forward rates f1,2, f2,3, f1,3 and from the discount factors in part a).
3. [5 marks] What is an interest rate swap? What are its main characteristics?
4. [5marks] Consider a three-year interest rate swap contract with annual payments in years 1, 2, 3. The notional amount is £1 million, and the term structure is , , . What is the appropriate fixed rate to set on this contract so that the value of the swap is zero at time 0?
5. [5 marks] A pension fund needs to pay £1 million in years 4 and 5 from now. The term structure is flat at r=5%. What is the present value of the company’s liabilities? What is the duration? How many 3-year zeros with £100 face value does the company need to buy to immunize the liabilities?

#### This question is about portfolio choices

For parts a), b) and c) consider the following two stocks. Stock 1 has expected return  and standard deviation of returns . Stock 2 has expected return  and standard deviation of returns . The correlation between these two stocks is 0.5. The riskless interest rate is 5%.

1. [5 marks] Consider an investor who invests in portfolio P of stocks 1 and 2 in such a way that the expected portfolio return is 11.6%. What are the investor’s portfolio weights? What is the standard deviation of portfolio returns?
2. [5 marks] Suppose, the investor wants to construct a portfolio with standard deviation of returns equal to 5% by investing in stock 2 and the riskless asset. What should be the portfolio weights? What is the expected return of this portfolio?
3. [5 marks] Explain what is tangency portfolio. Suppose, it is known that portfolio P in part a) lies on the efficient frontier, and the frontier is constructed assuming that the riskless asset is traded. Determine whether this portfolio is the tangency portfolio or not.
4. [5 marks] Consider portfolios A, B, C with the following expected returns and standard deviations: . The riskless asset is traded, and the riskless rate is 5%. It is known that one of the portfolios lies on the efficient frontier. Which of the portfolios is efficient? What is its Sharpe ratio?
5. [5 marks] Suppose, in addition to portfolios A, B, and C in part d), the investors can invest in portfolio D with standard deviation It is known that portfolio D belongs to the same efficient frontier as in part d). What should be this portfolio’s expected return?

#### This question is about option pricing

The current price of stock XYZ is £100, and each month it either goes up by 15% or goes down by 10%. The monthly riskless interest rate is 5%. Assume that the stock does not pay dividends.

1. [5 marks] Construct a binomial tree with three dates 0, 1, and 2 for stock price movements and calculate risk neutral probabilities of the up and down moves.
2. [5 marks] Calculate the time-0 price of a European call with strike price of £100 and maturity at date 2.
3. [5 marks] Determine the delta of the call option in part b) at time zero. Using put-call parity determine the time-0 price of a put option with strike price £100, which matures at date 2.
4. [5 marks] What is the price of a bull spread with strikes and  maturing at date 2? Assume that the price movements of the stock are described by the binomial tree in part a). (Hint: a bull spread is constructed as a long position in one European call with strike Kl and a short position in one European call with strike ).
5. [5 marks] Consider a European security, which pays at the terminal date . That is, the payoff is the minimum of two numbers: stock price  and . Assume that Plot the payoff diagram. Construct a portfolio of stocks and call options, which replicates the payoff of this security at date .

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