Financial Modeling 2nd Edition Simon Benninga - Solutions

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Show that the distributions of ( yF) and (vyF) obtained for the AR(1) reference analysis using the conditional likelihood are those given in Example 1.6.
Show that the expressions for p( yF) and p(yF) under the reference analysis in Section 1.5.3 are those given on page 21.
Consider the AR(1) model yt = yt 1 + t with t N(0v).(a) Find the MLE of ( v) for the conditional likelihood.(b) Find the MLE of ( v) for the unconditional likelihood (1.17).(c) Assume that v is known. Find the MAP estimator of under a uni form prior p( ) = U( 11) for the conditional and
Show that the autocorrelation function of an autoregression of order one with AR parameter such that< 1 and zero mean uncorrelated Gaussian innovations with variance v is (h) = h for h = 0 1 2
10. Fibonacci numbers (named after Leonardo Fibonacci, 1170–1230, an outstanding European mathematician of the medieval period) are defined as follows:and so on In general, F(n) = F(n − 2) + F(n − 1).
9. Using the information provided in exercise 8, write a two-argument function DEP(DFV, Years) that will return the monthly contribution necessary to get a certain sum in the future (two, three, four, or five years). Note:This problem is more interesting; remember that the interest rate is
7. Implement a VBA function ChooseBond(Deposit, Years, FT0, FT1). The function will return the value 1 if the superior investment is the bank in exercise 5 or the value 2 if it is the company in exercise 6.
6. An investment company offers a bond linked to the FT100 index. On redemption the bond pays the face value plus the larger of (a) the face value times the change in the index or (b) 5 percent yearly interest compounded monthly. For example, 100 invested when the index was 110 and redeemed a year
5. Using the function in exercise 4 implement a function NewDFV(Deposit, Years). The function will return the future value of a deposit with the bank assuming that the deposit and accrued interest are reinvested for a given number of years. Thus, for example NewDFV(10000,10) will return
4. A bank offers different yearly interest rates to its customers based on the size of the deposit in the following way:For deposits up to $1,000 the interest rate is 5.5 percent.For deposits from $1,000 up to $10,000 the interest rate is 6.3 percent.For deposits from $10,000 up to $100,000 the
3. Suppose a share was priced at price P0 at time 0, and suppose that at time 1 it will be priced P1. Then the continuously compounded return is defined as In . Implement this function in VBA.There are two ways to perform this calculation. You can use Application.Ln or the VBA function Log.
2. Write a VBA function for f(x)= + 2x. Note that there are two ways to carry out this assignment. The first is to use the VBA function Sqr. The second is to use the VBA operator ^. We suggest you try both.
1. Write a VBA function for f(x)=x2 − 3.
Solve the following system using the Gauss-Seidel method:Note that in order to get a solution, you may have to hit the F9 (recalculate spreadsheet) key a few times. You will have gotten a solution if the numbers on the screen stop changing.
3. Solve the equations AX = Y, where
2. Find the inverses of the following matrices:a.b.c.
1. Use Excel to perform the following matrix operations:a.b.c.
6. It is well known that if Z is a standard normal random variable (i.e., with mean μ = 0 and standard deviation σ= 1) then X = aZ + b is normally distributed with μ = b and σ =a. Modify normalSimulation to produce normal, nonstandard distributions, with the mean and the standard deviation
5. Program normalSimulation, but put the output into more bins. (Can you make the number of bins and their size controllable from the spreadsheet?) Does this method get rid of the "fat tails" in the distribution graph?
4. Many states have daily lotteries, which are played as follows: Sometime during the day, you buy a lottery ticket, on which the seller inscribes a number you choose, between 000 and 999. That night there is a drawing on television in which a three-digit number is drawn. If the number on your
3. Define AmodB as the remainder when A is divided by B. For example, 36mod25 = 11. Excel has this function;it is written Mod(A,B). Now here is another random-number generator:a. Let X0 = 1.b. Let Xn+1 = (7*Xn)mod108c. Let Un+1 = Xn+1/108.The list of numbers U1, U2, ... contains the pseudo-random
2. Here is a random-number generator you can make yourself:a. Start with some number, Seed.b. Let X1 = Seed + π. LetX2 = e5+In(X1).c. The first random number is Random = X2 − Integer (X)2), where Integer (X1) is the integer part of X1.d. Repeat the process, letting See`d = Random.Implement this
1. Use the program randomlist from section 25.2 to produce a list of 200 random numbers. Use the Excel function frequency to produce a histogram of the results.
3. In contradistinction to the prevailing (and published) belief, the CTD is—in many cases—not a bond with extremal duration.
2. The maturity of the cheapest to deliver is the shortest maturity of all deliverable bonds if the market interest rate is less than 8 percent. If the market interest rate is greater than 8 percent, the maturity of the CTD is the largest deliverable maturity if the optimal coupon is greater than 8
1. The cheapest-to-deliver bond always has either the highest (if the market interest rate is less than 8 percent)or lowest coupon of all deliverable bonds. This result assumes that the set of deliverable bonds is convex, but it is independent of the term structure.
5. An underwriter issues a new seven-year CCC bond. The anticipated recovery rate in default of the bond is expected to be 55 percent. What should be the coupon rate on the bond so that its expected return is 9 percent? Assume the transition matrix given in section 23.5.
4. An underwriter issues a new seven-year B bond with a coupon rate of 9 percent. If the expected rate of return on the bond is 8 percent, what is the bond's implied recovery percentage l? Assume the transition matrix given in section 23.5.
3. A bond of XYZ Corporation has the following characteristics:Market price: 108.32 percent of par Coupon rate: 15 percent Number of annual payments (including return of principal) left on bond: 15 Time to first payment: 8 months XYZ Corporation's debt is currently rated CCC. Use the model of
2. Consider the case of five possible rating states, A, B, C, D, and E. The states A, B, and C are initial bond ratings; D symbolizes first-time default; and E indicates default in the previous period. Assume that the transition matrix π is given by:A 10-year bond issued today at par with an A
5. In exercise 3, recalculate the portfolio proportions assuming that you need a target duration of 12. Which portfolio would you prefer now?
4. In exercise 3, which portfolio (a orb) would you prefer to immunize an obligation with a duration of 8?
3.a. Using the example of section 21.3, find a combination of bonds 1 and 3 with a duration of 8.b. Find a combination of bonds 1 and 2 with a duration of 8.
2. Set up a spreadsheet that enables you to duplicate the calculations of section 21.5.
8. Rewrite the formula DDuration in section 20.5.1, so that if the timeToFirstPayment α is not inserted, then αautomatically defaults to 1.[3]In Chapter 22 we discuss polynomial approximations to the term structure. For a further reference on general term structure models, see Hull (2000,
7. This exercise relates to the file TermStruc.XLS. You are asked to do the following:a. Produce at least three graphs of six term structures each for 10 typical subperiods. For example, the term structures from January to June 1953, July to December 1980, and so on.b. For January 1980, what would
6. On January 23, 1987, the market price of a West Jefferson Development bond was $1,122.32. The bond pays$59 in interest on March 1 and September 1 of each of the years 1987–93. On September 1, 1993, the bond is redeemed at its face value of $1,000. Calculate the yield to maturity of the bond,
5. Replicate the two graphs in section 20.4.
4. A pure discount bond with maturity N is a bond with no payments at times t = 1, …, N − 1; at time t = N, a pure discount bond has a single terminal payment of both principal and interest. What is the duration of such a bond?
3. "Duration can be viewed as a proxy for the riskiness of a bond. All other things being equal, the riskier of two bonds should have lower duration." Check this claim with an example. What is its economic logic?
2. What is the effect on a bond's duration of increasing the bond's maturity? As in exercise 1, use a numerical example and plot the answer. Note that as N→∞. the bond becomes a consol (a bond that has no repayment of principal but an infinite stream of coupon payments). The duration of a
1. What is the effect of raising the coupon payment on the duration of a bond? Assume that the bond's yield to maturity does not change. Use a numerical example and plot the answer.
2. Consider a series of American puts on a stock whose current price is S = 100. Suppose that all the puts have exercise price X = 120 and maturity T = 1. Divide this interval into sub-intervals of length Δt = 0.1 (i.e., let n =10 in the spreadsheet). Compare the early exercise boundary for σ =
1. Consider an at-the-money American put on a stock whose current price is S = 50. Find the early exercise boundary point for t = 0.5 for a series of these puts, when T = 1, 2, 3, 4, ..., 10. Assume that the risk-free interest rate is 8 percent and that σ= 30 percent. Can you draw any conclusions
5. Consider the following cash flows:a. If the cost of capital is 30 percent and the risk-free rate is 5 percent, find the state prices that match the project's NPV.b. If there exists an abandonment option so that we can change all negative cash flows to zero, value the project.
4. Suppose that the market portfolio has mean μ = 15 percent and standard deviation σ = 20 percent.a. If the risk-free rate of interest is 8 percent, calculate the one-period state prices for an "up" and a"down" state.b. Show the effect (in a data table) of the risk-free rate on the state
3. Consider a project whose cash flows are as follows:Chapter 18 - Real Options Financial Modeling, Second Edition Simon Benninga Copyright © 2000 Massachusetts Institute of Technology Year Cash Flow of Single Machine 0 −550 1 100 2 200 3 300 4 400a. Using the state prices, value the project.b.
2. Your company is considering the purchase of a new piece of equipment. The equipment costs $50,000, and your analysis indicates that the PV of the future cash flows from the equipment is $45,000. Thus the NPV of the equipment is −$5,000. This estimated NPV is based on some initial numbers
1. Your company is considering purchasing 10 machines, each of which has the following expected cash flows(the entry −$550 is the cost of the machine):You estimate the appropriate discount rate for the machines as 25 percent.a. Would you recommend buying just one machine, if there are no options
4. You have been offered the chance to purchase stock in a firm. The seller wants $55 per share, but offers to repurchase the stock at the end of one-half year for $50 per share. If the σ of the share's log returns is 80 percent, determine the true value per share. Assume that the interest rate is
3. Go back to the numerical example of section 17.6. Write a VBA function that solves for the implied asset value Va. (Hint: Use the bisection method.) Then use this function to create a graph showing the trade-off between the implied asset value and the asset volatility.
2. Simulate the strategy of exercise 1, assuming weekly rebalancing of the portfolio.
1. You are a portfolio manager, and you want to invest in an asset having σ = 40 percent. You want to create a put on the investment so that at the end of the year you have losses no greater than 5 percent. Since there is no put on this specific asset, you plan to create a synthetic put by
10. Note that you can also calculate the Black-Scholes put option premium as a percentage of the exercise price in terms of S/X:where Implement this in a spreadsheet. Find the ratio of S/X for which C/X and P/X cross when T = 0.5, σ 25%, r =10%. (You can use a graph or you can use Excel's Solver.)
9. Note that you can use the Black-Scholes formula to calculate the call option premium as a percentage of the exercise price in terms of S/X:where Implement this in a spreadsheet.
8. The Merton model in exercise 7 can also be used to price options on currencies. Suppose for example, we are pricing a call option to buy 10,000 for a rate of $1.10 per in 3 months. Suppose that the U.S.interest rate is 5.5 percent and that the Euro interest rate is 3.8 percent. The following
7. As shown in proposition 7, Chapter 13, a European option on a stock should be priced by netting out from the underlying asset price all the dividends to be paid before the option matures. Merton (1973) shows that for the case of an asset with price S paying a continuously compounded dividend
6. Use the Excel Solver to find the stock price for which there is the maximum difference between the Black-Scholes call option price and the option's intrinsic value. Use the following values: S = 45, X = 45, T = 1, σ =40 percent, r = 8 percent.
5. Refer back to the American Express options in Chapter 13. Assume that the current date is Friday, January 26, 1996, and that the expiration dates of the options are as follows:"Feb" = February 16, 1996 "Mar" = March 15, 1996 "Apr" = April 19, 1996 "Jul" = July 19, 1996 Generate two tables:a. One
4. Produce a graph comparing a put's intrinsic value [max (X - S, 0)] ad its Black-Scholes price. From this graph you should be able to deduce that it may be optimal to exercise early a put priced by the Black-Scholes formula.
3. Produce a graph comparing a call's intrinsic value [defined as max (S − X, 0)] and its Black-Scholes price.From this graph you should be able to deduce that it may be optimal to exercise early a call priced by the Black-Scholes formula.
2. Use the data from exercise 1 and Data|Table to produce graphs that showa. The sensitivity of the Black-Scholes call price to changes in the initial stock price S.b. The sensitivity of the Black-Scholes put price to changes in σ.c. The sensitivity of the Black-Scholes call price to changes in
1. Use the Black-Scholes model to price the following:a. A call option on a stock whose current price is 50, with exercise price X = 50, T = 0.5, r = 10 percent, σ= 25 percent.b. A put option with the same parameters.
13. Reconsider exercise 12. Show that if the date-1 exercise price is X, the date-2 exercise price is X*(1 + r), and the date-3 exercise price is X*(1 + r)2, you will not exercise the option early.[5][5]It can also be shown that this property holds if the exercise prices grow more slowly than the
12. A call option is written on a stock whose current price is $50. The option has maturity of three years, and during this time the annual stock price is expected to increase by 25 percent or to decrease by 10 percent.The annual interest rate is constant at 6 percent. The option is exercisable at
11. Here's an advanced version of exercise 10. Consider an alternative parameterization of the binomial:Construct binomial European call and put option-pricing functions in VBA for this parameterization and show that they also converge to the Black-Scholes formula. (The message here is that the
10. This problem is a continuation of the discussion of section 14.6. Show that as n→∞, the binomial European put price converges to the Black-Scholes put price. (Note that, as part of the spreadsheet Chapter14.xls, we have included a function called BSPut that computes the Black-Scholes put
9. A prominent securities firm recently introduced a new financial product. This product, called the Best of Both Worlds (BOBOW for short), costs $10. It matures in five years, at which point it repays the investor the $10 cost plus 120 percent of any positive return in the S&P 500 index. There are
8. Consider the following three-date binomial model, in which the stock price either goes up by 30 percent or decreases by 10 percent in each period, and in which the one-period interest rate is 25 percent:a. Consider a European call with X = 30 and T = 2. Fill in the blanks in the tree:b. Price a
7. Consider the following 3-date binomial model, in which the annual interest rate is 9 percent and in which the stock price goes up by 15 percent or down by 10 percent per period:a. Price a European call on the stock with exercise price 60.b. Price a European put on the stock with exercise price
6. Fill in all the cells labeled ??? in the following spreadsheet:
5. A stock is currently selling for 60. The price of the stock at the end of the year is expected either to increase by 25 percent or to decrease by 20 percent. The riskless interest rate is 5 percent. Calculate the price of a European put on the stock with exercise price 55. Use the binomial
4. All reliable analysts agree that a share of ABC Corporation, selling today for $50, will be priced at either $65 or $45 tomorrow. They further agree that the probabilities of these events are 0.6 and 0.4, respectively. The market risk-free rate is 6 percent. What is the value of a call option on
3. In a binomial model a call option and a put option are both written on the same stock. The exercise price of the call option is 30, and the exercise price of the put option is 40. The call option's payoffs are 0 and 5 and the put option's payoffs are 20 and 5. The price of the call is 2.25 and
2. In exercise 1, compute the state prices qu and qd, and use these prices to calculate the value today of a oneyear put option on the stock with exercise price $30. Show that put-call parity holds: That is, using your answer from this problem and the previous problem, show that
1. A stock selling for $25 today will, in one year, be worth either $35 or $20. If the interest rate is 8 percent, what is the value today of a one-year call option on the stock with exercise price $30? Use the simultaneousequation approach of section 14.1 to price the option.
10. Consider the following option strategy, which consists only of calls:a. Draw the profit diagram for this strategy.b. The prices given include one violation of an arbitrage condition. Identify this violation and explain.[9]This is a very fundamental fact of finance: If a financial strategy has
9. The current stock price of ABC Corporation is 50. Prices for six-month calls on ABC are given in the following table:Draw a profit diagram of the following strategy: Buy one 40 call, write two 50 calls, buy one 60 call, and write two 70 calls.
8. Prove Proposition 6. Then solve the following problem:Three puts on shares of XYZ with the same expiration date are selling at the following prices:Exercise price 40: 6 Exercise price 50: 4 Exercise price 60: 1 Show an arbitrage strategy that allows you to profit from these prices and prove that
7. A European call with a maturity of six months and exercise price X = 80 written on a stock with a current price of 85 is selling for $12.00; a European put written on the same stock with the same maturity and with the same exercise price is selling for $5.00. If the annual interest rate
6. A share of ABC Corporation sells for $95. A call on the share with exercise price $90 sells for $8.a. Graph the profit pattern from buying one share and one call on the share.b. Graph the profit pattern from buying one share and two calls.c. Consider the profit pattern from buying one share and
5. Consider the following two calls:Both calls are written on shares of ABC Corporation, whose current share price is $100. ABC does not pay any dividends.Both calls have one year to maturity.One call has X1 = 90 and has price of 30; the second call has X2 = 100 and has price of 20.The riskless,
4. A put with an exercise price of 50 has a price of 6, and a call on the same stock with an exercise price of 60 has a price of 10. Both put and call have the same expiration date. On the same set of axes, draw the "profit"diagram fora. One put bought and one call bought.b. Two puts bought and one
3. A European call option is written on a stock whose current price S = 80. The exercise price is X = 80, the interest rate is r = 8 percent, and the time to option exercise is T = 1. The stock is assumed to pay a dividend of 3 at time t = 1/2. Use Proposition 7 to determine the minimum price of
2. An American call option is written on a stock whose price today is S = 50. The exercise price of the call is X =45. If the call price is 2, explain how you would use arbitrage to make an immediate profit.If the option is exercisable at time T = 1 year and if the interest rate is 10 percent, what
1. Look at the prices of the American Express options in the chapter. The February call option with X = 37.5 is priced at 6 3/8, whereas the April option with the same exercise price is priced at 6. Can you devise an arbitrage out of these prices? Do you have an explanation for the newspaper quotes?
2. On the same set of axes, graph the efficient frontier for these six stocks with and without short sales.
1. Given these data, calculate the efficient frontier assuming no short sales are allowed.
4. This problem returns to the four-asset problem considered in section 7.5:Calculate the envelope set for these four assets and show that the individual assets all lie within this envelope set. You should get a graph that looks something like the following:
3. Consider the example given in section 9.4 Use Excel to find an envelope portfolio whose β with respect to the efficient portfolio y is zero. Hint: Notice that because the covariance is linear, so is β: Suppose that z = λx +(1−λ)y is a convex combination of x and y, and that we are trying
2. A sufficient condition to produce positively weighted efficient portfolios is that the variance-covariance matrix be diagonal, that is, that σij = 0, for i≠j. By continuity, positively weighted portfolios will result if the offdiagonal elements of the variance-covariance matrix are
1. In Chapter 8 you were asked to calculate the variance-covariance matrix of returns for six furniture companies. The calculated variance-covariance matrix and mean returns for these firms are as follows:a. Given this matrix, and assuming that the risk-free rate is 0 percent, calculate the
3. In the spreadsheet Problems O8.xls you will find monthly return data for the 30 stocks in the Dow-Jones Index of 30 Industrials. Use the data to calculate the variance-covariance matrix of the Index (in the next chapter you will use this matrix to calculate the efficient frontier for the index).
2. For the firms from exercise 1: Suppose that the standard deviation of the market index is 18 percent.Calculate the variance-covariance matrix using the single-index model.
1. In the following table you will find annual return data for six furniture companies between the years 1982 and 1992. Use these data to calculate the variance-covariance matrix of the returns.
8. Consider a three-asset world with the following parameters:Suppose you have two portfolios with the following portfolio weights:Portfolio 1 = (0.3 0.2 0.5)Portfolio 2 = (0.5 0.4 0.1)a. Calculate the following:n The mean and variance of each portfolio's returns n The covariance and correlation
7. Consider two assets, A and B, which have the following means and variances:Now consider three cases:Graph the combinations of portfolio means and variances for each case. (Graphs like these appear in virtually all elementary finance books. The standard deviation usually appears on the x-axis and
6. Pfizer and Merck are two American pharmaceutical firms. The following table gives the end-of-year stock prices for each of the firms for the years 1981–92 as well as the dividends paid in the years 1982–92.a. For the decade 1982–92 calculate the following:n Annual returns from each of the
5. Suppose that X and Y are two random variables and that Y = X2. Let X have values −5, −4, −2, 2, 4, 5 with equal probabilities. Show that the correlation coefficient between X and Y is zero. Does this mean that X and Y are independent random variables?
4. Using the data base of the DJ Industrials—For American Airlines (AA), Procter & Gamble (PG), and General Electric (GE)—compute:a. The average monthly returnsb. The covariances of the monthly returns:n σAA,AA = Covariance(RAA, RAA), σPG,PG = Covariance(RPG, RPG), σGE,GE =

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Financial Modeling 2nd Edition Simon Benninga - Solutions

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