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essentials of investments

Questions and Answers of Essentials of Investments

16. (Iree harvesting) You are considering an investment in a tree farm. Trees grow each year by the following factors: Year 12 3 4 5 6 7 8 9 10 Growth 1.6 15 14 13 12 115 11 105 1.02 :01 The price of
1. (Numerical evaluation of normal distribution) The cumulative normal distribution can be approximated (to within about six decimal places) by the modified polynomial relation - N'(x) (ask + ask +
2. (Perpetual puto) Consider a perpetual American put option (with 7 = co) For small stock prices it will be advantageous to exercise the put. Let G be the largest such stock price. The
3. (Sigma estimation) Traders in major financial institutions use the Black-Scholes formula in a backward fashion to infer other traders' estimates of o from option prices In fact, traders frequently
4. (Black-Scholes approximation o) Note that to first order N(d) = +d;2 Use this to derive the value of a call option when the stock price is at the present value of the strike price; that is, S =
5. (Delta) Using the same parameters as in Example 13.2, find the value of the 5-month call if the initial value of the stock is $63 Hence estimate the quantity AAC/AS Estimate @ = AC/AI
6. (A special identity) Gavin Jones believes that for a derivative security with price P(S), the values of A, F, and are related Show that in fact N' (d) SaT
7. (Gamma and theta o) Show that for a European call or put on a non-dividend-paying stock = SN' (d) = - Ke N(d) 2T [Hint Use Exercise 6]
8. (Great Western CD) Great Western Bank has offered a special certificate of deposit (CD) tied to the S&P 500. Funds are deposited into the account at the beginning of a month and are held in the
9. (The control variate method) Suppose that it is desired to estimate the expected value of a random variable x. (This random variable might be the discounted terminal value of a call option on a
10. (Control variate application) Use the control variate method of Exercise 9 to determine the value of a 5-month Asian call option on a stock with Sp= $62, = 20%, and 10% and a strike price of $60
11. (Pay-later options) Pay-later options are options for which the buyer is not required to pay the premium up front (ie, at the time that the contract is entered into). At expiration. the holder of
12. (California housing put ) Suppose you buy a new home and finance 90% of the price with a mortgage from a bank. Suppose that a few years later the value of your home falls below your mortgage
13. (Forest value) Solve Exercise 16 in Chapter 12 assuming that the annual storage cost of cut lumber is 5% of its value
14. (Mr. Smith's put) Find the value of the put for Mr Smith described in Example 13.9.
1. (A callable bond) Construct a short rate lattice for periods (years) 0 through 9 with an initial rate of 6% and with successive rates determined by a multiplicative factor of either = 12 or d = 9.
2. (General adjustable formula) Let V, be the value of an adjustable-rate loan initiated at period k and states with initial principal of 100 The loan is to be fully paid at period. The interest rate
3. (Bond futures option) Explain how you would find the value of a bond futures option.
4. (Adjustable-rate CAP) Suppose that the adjustable-rate auto loan of Example 145 is modified by the provision of a CAP that guarantees the borrower that the interest rate to be applied will never
5. (Forward construction) Use the forward equation to find the spot rate curve for the lattice constructed in Exercise 1.
6. (Ho-Lee volatility) Show that for the Ho-Lee model the (risk-neutral) standard deviation of the one-period rate is exactly by/2
7. (Term match) Use the Black-Derman-Toy model with b = 01 to match the term structure of Example 147
8. (Swaps) Consider a plain vanilla interest rate swap where party A agrees to make six yearly payments to party B of a fixed rate of interest on a notional principal of $10 million and in exchange
9. (Swaption pricing) A swaption is an option to enter a swap arrangement in the future Suppose that company B has a debt of $10 million financed over 6 years at a fixed rate of interest of 8.64%
10. (Change of variable) Suppose a short rate process in a risk-neutral world is defined by dr (r,t) dr +(1, 1) de where (r) is a standardized Wiener process. A standard way to approximate this
11. (Ho-Lee term structure) Refer to Example 14.11 Let F(r) be the forward rate from 0 to By the basic definition of the forward rate, we have the identity e-Fun P(1,0,1) Find an explicit formula for
12. (Continuous zero) Gavin wants to dig deep into pricing theory, so he decides to work out an application of Eq (14.11). He suggests to himself that a simple model of interest rates in the
1. (Simple wheel strategy) Consider a strategy of the form (y,0,0) for the investment wheel Show that the overall factor multiplying your money after steps is likely to be (1+2y)(1). Find the value
2. (How to play the state lottery) In a certain state lottery, people select eight numbers in advance of a random drawing of six numbers. If someone's selections include the six drawn, they receive a
3. (Easy policy) Show that (, ) is the optimal policy for Example 152
4. (A general betting wheel o) Consider a wheel with sectors. If the wheel pointer lands on sector i, the payoff obtained is 7, for every unit bet on that sector The chance of landing on sector i is
5. (More on the wheel o) Using the notation of Exercise 4, assume that 1/1, but try to find a solution where one of the 's is zero In particular, suppose the segments are ordered in such a way that
6. (Volatility pumping) Suppose there are n stocks. Each of them has a price that is governed by geometric Brownian motion Each has v; = 15% and = 40%. However, these stocks are correlated, and for
7. (The Dow Jones Average puzzle) The Dow Jones Industrial Average is an average of the prices of 30 industrial stocks with equal weights applied to all 30 stocks (but the sum of the weights is
8. (Power utility) A stock price is governed by ds S where is a standardized Wiener process. Interest is constant at rate. An investor wishes to construct a constantly rebalanced portfolio of these
9. (Discrete-time, log-optimal pricing formula) Suppose there are assets Asset i, i = 1,2., has rate of return r, over a single period There is also a risk-free asset with rate of return ry The
1. (A state tree) A certain underlying state graph is a tree where each node has three succes- sor nodes, indexeda, b,c. There are two assets defined on this tree which pay no dividends except at the
2. (Node separation) Consider a short rate binomial lattice where the risk-free rate at r = 0 is 10% At the rate is either 10% (for the upper node) or 0% (for the lower node) Trace out the growth of
3. (Bond valuation) Assuming the short rate process of Exercise 2 and risk-neutral proba- bilities of 5, consider a zero-coupon bond that pays $1 at time = 2. Find the value at time 0 of this bond in
4. (Optimal option valuation) Find the values of the 5-month call option of Example 162 using the same trinomial lattice used in that example but employing the utility function U(x). What is a?
5. (Gold correlation) Suppose that in the double stochastic Simplico gold mine example the real probability of an up move in gold is 6 and the real probability of an up move in the short rate is 7.
6. (Complexico mine) Use the information about the Complexico mine of Example 128, Chapter 12, but assume that gold prices and interest rates are governed by the models of Example 164 Find the value
7. (Simultaneous solution) Calculate the volatility and the current price of oil futures implied by the call 1600 August and the call 1700 August of Figure 168 by using the Black-Scholes formula with
8. (Default risk) A company issues a 10% coupon bond that matures in 5 years However, this company is in trouble, and it is estimated that each year there is a probability of I that it will default
9. (Automobile choice) Mr. Smith wants to buy a car and is deciding between brands A and B. Car A costs $20,000, and Mr. Smith estimates that at the rate he drives he will sell it after 2 years and
10. (Continuco mine simulation) Evaluate the Continuco gold mine lease by simulation, using Ar = 25
11. (Gavin's final) Mr. Jones was considering a new grapefruit venture that would generate a random sequence of yearly cash flows. He asked his son, Gavin, "People tell me I should use a cost of
On January 1, you sold one March maturity S&P 500 Index futures contract at a futures price of 800. If the futures price is 850 on February 1, what is your profit? The contract multiplier is $250.
The current level of the S&P 500 is 800. The dividend yield on the S&P 500 is 2%. The risk-free interest rate is 1%. What should a futures contract with a one-year maturity be selling for? LO.1
A one-year gold futures contract is selling for $941. Spot gold prices are $900 and the one-year risk-free rate is 4%. What arbitrage opportunity is available to investors? What strategy should they
a. Turn to Figure 17.1 and locate the contract on the Standard & Poor’s 500 Index. If the margin requirement is 10% of the futures price times the multiplier of $250, how much must you deposit with
Why might individuals purchase futures contracts rather than the underlying asset? LO.1
What is the difference in cash flow between short-selling an asset and entering a short futures position?Intermediate LO.1
Suppose the value of the S&P 500 Stock Index is currently $800. If the one-year T-bill rate is 3% and the expected dividend yield on the S&P 500 is 2%, what should the oneyear maturity futures price
It is now January. The current interest rate is 4%. The June futures price for gold is$946.30, while the December futures price is $952. Is there an arbitrage opportunity here? If so, how would you
The Excel Application box in the chapter (available at www.mhhe.com/bkm; link to Chapter 17 material) shows how to use the spot-futures parity relationship to find a“term structure of futures
One Chicago has just introduced a new single-stock futures contract on the stock of Brandex, a company that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in
The multiplier for a futures contract on the stock market index is $250. The maturity of the contract is one year, the current level of the index is 800, and the risk-free interest rate is .5% per
a. How should the parity condition (Equation 17.2) for stocks be modified for futures contracts on Treasury bonds? What should play the role of the dividend yield in that equation?b. In an
Desert Trading Company has issued $100 million worth of long-term bonds at a fixed rate of 7%. The firm then enters into an interest rate swap where it pays LIBOR and receives a fixed 6% on notional
What type of interest rate swap would be appropriate for a speculator who believes interest rates soon will fall? LO.1
The margin requirement on the S&P 500 futures contract is 10%, and the stock index is currently 800. Each contract has a multiplier of $250. How much margin must be put up for each contract sold? If
The multiplier for a futures contract on a certain stock market index is $250. The maturity of the contract is one year, the current level of the index is 800, and the risk-free interest rate is .2%
You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in three months. You believe rates soon will fall and would like to repurchase the company’s sinking fund
A manager is holding a $1 million bond portfolio with a modified duration of eight years. She would like to hedge the risk of the portfolio by short-selling Treasury bonds.The modified duration of
A corporation plans to issue $10 million of 10-year bonds in three months. At current yields the bonds would have modified duration of eight years. The T-note futures contract is selling at F0 100
The one-year futures price on a particular stock-index portfolio is 406, the stock index currently is 400, the one-year risk-free interest rate is 3%, and the year-end dividend that will be paid on a
A silver futures contract requires the seller to deliver 5,000 Troy ounces of silver. Jerry Harris sells one July silver futures contract at a price of $14 per ounce, posting a $4,000 initial margin.
In each of the following cases, discuss how you, as a portfolio manager, could use financial futures to protect a portfolio.a. You own a large position in a relatively illiquid bond that you want to
Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated by the information given in the following exhibit.Commodity Price and Interest Rate Information Spot
A stock index is currently trading at 50. Paul Tripp, CFA, wants to value two-year index options using the binomial model. In any year, the stock will either increase in value by 20% or fall in value
Ken Webster manages a $200 million equity portfolio benchmarked to the S&P 500 Index. Webster believes the market is overvalued when measured by several traditional fundamental/economic indicators.
Janice Delsing, a U.S.-based portfolio manager, manages an $800 million portfolio($600 million in stocks and $200 million in bonds). In reaction to anticipated shortterm market events, Delsing wishes
Imagine you are a provider of portfolio insurance. You are establishing a four-year program. The portfolio you manage is currently worth $100 million, and you promise to provide a minimum return of
Return to Example 16.1 . Use the binomial model to value a one-year European put option with exercise price $110 on the stock in that example. Does your solution for the put price satisfy put-call
Consider an increase in the volatility of the stock in the previous problem. Suppose that if the stock increases in price, it will increase to $130, and that if it falls, it will fall to$70. Show
A collar is established by buying a share of stock for $50, buying a six-month put option with exercise price $45, and writing a six-month call option with exercise price $55.Based on the volatility
In this problem, we derive the put-call parity relationship for European options on stocks that pay dividends before option expiration. For simplicity, assume that the stock makes one dividend
Maria VanHusen, CFA, suggests that forward contracts on fixed income securities can be used to protect the value of the Star Hospital Pension Plan’s bond portfolio against the possibility of rising
These three put options all are written on the same stock. One has a delta of .9, one a delta of .5, and one a delta of .1. Assign deltas to the three puts by filling in the table below.Put X Delta A
Consider a six-month expiration European call option with exercise price $105. The underlying stock sells for $100 a share, and pays no dividends. The risk-free rate is 5%. What is the implied
The hedge ratio of an at-the-money call option on IBM is .4. The hedge ratio of an at-the-money put option is .6. What is the hedge ratio of an at-the-money straddle position on IBM? LO.1
According to the Black-Scholes formula, what will be the value of the hedge ratio of a call option as the stock price becomes infinitely large? Explain briefly.22 . According to the Black-Scholes
If the time to expiration falls and the put price rises, then what has happened to the put option’s implied volatility? LO.1
If the stock price falls and the call price rises, then what has happened to the call option’s implied volatility? LO.1
Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive to changes in interest rates than the rate of return of the underlying bond? LO.1
All else being equal, will a call option with a high exercise price have a higher or lower hedge ratio than one with a low exercise price? LO.1
Suppose that you want to create a spread position using S&P 500 futures contracts. Go to www.cmegroup.com and listed under Equity Index Products, select the S&P 500.Click on “Contract
All else being equal, is a call option on a stock with a lot of firm-specific risk worth more than one on a stock with little firm-specific risk? The betas of the stocks are equal. LO.1
All else being equal, is a put option on a high beta stock worth more than one on a low beta stock? The firms have identical firm-specific risk. LO.1
Would you expect a $1 increase in a call option’s exercise price to lead to a decrease in the option’s value of more or less than $1? LO.1
Assume that you bought the three month contract and sold the nine month contract about two months ago. The Wall Street Journal provides a comprehensive list of current and historical prices at
What would be the Excel formula in Spreadsheet 16.1 for the Black-Scholes value of a straddle position? LO.1
Find the Black-Scholes value of a put option on the stock in the previous problem with the same exercise price and expiration as the call option.12 . Recalculate the value of the option in Problem
Return to the CME listings and find the closing prices of the two contracts on today’s date. Assume that you close your positions in both contracts. LO.1
Calculate the value of a call option on the stock in the previous problem with an exercise price of 110. Verify that the put-call parity relationship is satisfied by your answers to both Problems 8
Using the beginning and ending prices for the futures contracts that you bought and sold, calculate the return you earned on the spread position during the two-month period, before commissions. LO.1
We will derive a two-state put option value in this problem. Data: S0 100; X 110;1 r 1.10. The two possibilities for ST are 130 and 80.a. Show that the range of S is 50 while that of P is 30 across
Show that Black-Scholes call option hedge ratios increase as the stock price increases.Consider a one-year option with exercise price $50 on a stock with annual standard deviation 20%. The T-bill
Reconsider the determination of the hedge ratio in the two-state model (Section 16.2), where we showed that one-third share of stock would hedge one option. What would be the hedge ratio for each of

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