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essentials of investments

Questions and Answers of Essentials of Investments

12. (Pure duration o) It is sometimes useful to introduce variations of the spot rates that are different from an additive variation Let s === (s, ss) be an initial spot rate sequence (based on m
13. (Stream immunizatione) A company faces a stream of obligations over the next 8 years as shown: where the numbers denote thousands of dollars The spot rate curve is that ofExample 48. Find a
14. (Mortgage division) Often a mortgage payment stream is divided into a principal payment stream and an interest payment stream, and the two streams are sold separately We shall examine the
15. (Short rate sensitivity) Gavin Jones sometimes has flashes of brilliance. He asked his instructor if duration would measure the sensitivity of price to a parallel shift in the short rate curve.
1. (Capital budgeting) A firm is considering funding several proposed projects that have the financial properties shown in Table 56. The available budget is $600,000. What set of projects would be
2. (The road) Refer to the transportation alternatives problem of Example 5 2. The bridge at Cay Road is actually part of the road between Augen and Burger Therefore it is not reasonable for the
3. (Two-period budget ) A company has identified a number of promising projects, as indicated in Table 57 The cash flows for the first 2 years are shown (they are all negative)The cash flows in later
4. (Bond matrix ) The cash matching and other problems can be conveniently represented in matrix form. Suppose there are m bonds We define for each bond j its associated yearly cash flow stream
5. (Trinomial lattice) A trinomial lattice is a special case of a trinomial tree From each node three moves are possible: up, middle, and down The special feature of the lattice is that certain pairs
6. (A bond projecte) You are the manager of XYZ Pension Fund On November 5, 2011. XYZ must purchase a portfolio of US Treasury bonds to meet the fund's projected liabili- ties in the future The bonds
7. (The fishing problem) Find the solution to the fishing problem of Example 54 when the interest rate is 33%. Are the decisions different than when the interest rate is 25%? At what critical value
8. (Complexico mine) Consider the Complexico mine and assume a 10% constant interest rate; also assume the price of gold is constant at $400/oz. (a) Find the value of the mine (not a 10-year lease)
10. (Multiperiod harmony theorem o) its possible cash flow streams. This The value of a firm is the maximum present value of can be expressed as x1 X V = max + + + + (1+52)2 (+)" where the
11. (Growing annuity) Show that for g
12. (Two-stage growth) It is common practice in security analysis to modify the basic div- idend growth model by allowing more than one stage of growth, with the growth factors being different in the
1. (Shorting with margin) Suppose that to short a stock you are required to deposit an amount equal to the initial price Xo of the stock. At the end of 1 year the stock price is X and you liquidate
2. (Dice product) Two dice are rolled and the two resulting values are multiplied together to form the quantity. What are the expected value and the variance of the random variable z? [Hint. Use the
3. (Two correlated assets) The correlation between assets A and B is 1, and other data are given in Table 6.3. [Note p = AB/(AOB)](a) Find the proportions a of A and (I-a) of B that define a
4. (Two stocks) Two stocks are available. The corresponding expected rates of return are Fi and F; the corresponding variances and covariances are of, o, and oz What percentages of total investment
5. (Rain insurance) Gavin Jones's friend is planning to invest $1 million in a rock concert to be held 1 year from now The friend figures that he will obtain $3 million revenue from his $1 million
6. (Wild cats) Suppose there are n assets which are uncorrelated. (They might be n different "wild cat" oil well prospects) You may invest in any one, or in any combination of them The mean rate of
7. (Markowitz fun) There are just three assets with rates of return 1, 2, and 3, respectively The covariance matrix and the expected rates of return are [2 107 V = 1 2 1 F = 8 12. 8 (a) Find the
8. (Tracking) Suppose that it is impractical to use all the assets that are incorporated into a specified portfolio (such as a given efficient portfolio) One alternative is to find the portfolio,
9. (Betting wheel) Consider a general betting wheel with segments The payoff for a $1 bet on a segment is A, Suppose you bet an amount B, 1/A, on segment i for each i Show that the amount you win is
10. (Efficient portfolio o) Derive (69). [Hint Note that a 1/2 -1/2 =()-(-)"] 10/10)
1. (Capital market line) Assume that the expected rate of return on the market portfolio is 23% and the rate of return on T-bills (the risk-free rate) is 7%. The standard deviation of the market is
2. (A small world) Consider a world in which there are only two risky assets, A and B, and a risk-free asset F The two risky assets are in equal supply in the market; that is, M = (A+B) The following
3. (Bounds on returns) Consider a universe of just three securities They have expected rates of return of 10%, 20%, and 10%, respectively Two portfolios are known to lie on the minimum-variance set.
4. (Quick CAPM derivation) Derive the CAPM formula for F-7 by using Equation (69) in Chapter 6 [Hint: Note that ] W= Cov(FM) Apply (6.9) both to asset k and to the market itself.
5. (Uncorrelated assets) Suppose there are n mutually uncorrelated assets The return on asset i has variance of The expected rates of return are unspecified at this point. The total amount of asset i
6. (Simpleland) In Simpleland there are only two risky stocks, A and B, whose details are listed in Table 7.4.Furthermore, the correlation coefficient between the returns of stocks A and B is PAB = t
7. (Zero-beta assets) Let wo be the portfolio (weights) of risky assets corresponding the minimum-variance point in the feasible region Let w, be any other portfolio on the efficient frontier Define
8. (Wizards o) Electron Wizards, Inc. (EWI) has a new idea for producing TV sets, and it is planning to enter the development stage. Once the product is developed (which will be at the end of 1
9. (Gavin's problem) Prove to Gavin Jones that the results he obtained in Examples 7.5 and 7.7 were not accidents Specifically, for a fund with return ar; +(1), show that both CAPM pricing formulas
1. (A simple portfolio) Someone who believes that the collection of all stocks satisfies a single-factor model with the market portfolio serving as the factor gives you information on three stocks
2. (APT factors) Two stocks are believed to satisfy the two-factor model 12+3/+4/2 In addition, there is a risk-free asset with a rate of return of 10%. It is known that 7 = 15% and F 20%. What are
3. (Principal components ) Suppose there are a random variables.x.x2. ., and let V be the corresponding covariance matrix. An eigenvector of V is a vector v=(. . . ) such that Vvv for some 2 (called
4. (Variance estimate) Let r,, for i = 1, 2, ., be independent samples of a return of mean and variancea. Define the estimates Show that E(s) = a 31 = i=1 1 => 71 i=1
5. (Are more data helpful?) Suppose a stock's rate of return has annual mean and variance of F and a To estimate these quantities, we divide 1 year into a equal periods and record the return for each
6. (A record) A record of annual percentage rates of return of the stock S is shown in Table 8.7 (a) Estimate the arithmetic mean rate of return, expressed in percent per year (b) Estimate the
7. (Clever, but no cigar o) Gavin Jones figured out a clever way to get 24 samples of monthly returns in just over one year instead of only 12 samples; he takes overlapping samples; that is, the
8. (General tilting o) A general model for information about expected returns can be ex- pressed in vector-matrix form as p = Pr+e In the model P is an mxn matrix, F is an n-dimensional vector, and p
1. (Certainty equivalent) An investor has utility function U(r) x for salary He has a new job offer which pays $80,000 with a bonus The bonus will be $0, $10,000, $20,000, $30,000, $40,000, $50,000,
2. (Wealth independence) Suppose an investor has exponential utility function U(x) = -e and an initial wealth level of W. The investor is faced with an opportunity to invest an amount w W and obtain
3. (Risk aversion invariance) Suppose U(x) is a utility function with Arrow-Pratt risk aver- sion coefficient a(x) Let V(x)=c+bU (x) What is the risk aversion coefficient of V?
4. (Relative risk aversion) The Arrow-Pratt relative risk aversion coefficient is f(x)= xU"(x) U'(x) Show that the utility functions U(x) = Inx and U(x) = yr" have constant relative risk aversion
5. (Equivalency) A young woman uses the first procedure described in Section 9.4 to deduce her utility function U(x) over the range AxB She uses the normalization U(A) = A, U(B) B To check her
6. (HARA) The HARA (for hyperbolic absolute risk aversion) class of utility functions is defined by U(x)=- ax +b b> 0 The functions are defined for those values of x where the term in parentheses is
7. (The venture capitalist) A venture capitalist with a utility function U(r) = carried out the procedure of Example 9.3. Find an analytical expression for C as a function ofe, and for e as a
8. (Certainty approximation ) There is a useful approximation to the certainty equivalent that is easy to derive A second-order expansion near = E(x) gives U(x) UT)+U')(x)+U")(x) - Hence,
9. (Quadratic mean-variance) An investor with unit wealth maximizes the expected value of the utility function U(x) = ax-bx/2 and obtains a mean-variance efficient portfolio A friend of his with
10. (Portfolio optimization) Suppose an investor has utility function U There are n risky assets with rates of return, i = 1,2, n, and one risk-free asset with rate of return ry. The investor has
11. (Money-back guarantee) The promoter of the film venture offers a new investment de- signed to attract reluctant investors One unit of this new investment has a payoff of $3,000 if the venture is
12. (General positive state prices result o) The following is a general result from matrix. theory: Let A be an m x matrix Suppose that the equation Ax = p can achieve no p 0 except p=0. Then there
13. (Quadratic pricing o) Suppose an investor uses the quadratic utility function U(x): xx Suppose there are n risky assets and one risk-free asset with total return R Let Ry be the total return on
14. (At the track) At the horse races one Saturday afternoon Gavin Jones studies the racing form and concludes that the horse No Arbitrage has a 25% chance to win and is posted at 4 to 1 odds (For
15. (General risk-neutral pricing) We can transform the log-optimal pricing formula into a risk-neutral pricing equation From the log-optimal pricing equation we have P = E () where R is the return
1. (Gold futures) The current price of gold is $412 per ounce The storage cost is $2 per ounce per year, payable quarterly in advance. Assuming a constant interest rate of 9% com- pounded quarterly,
2. (Proportional carrying charges o) Suppose that a forward contract on an asset is written at time zero and there are M periods until delivery Suppose that the carrying charge in period k is qS(k),
3. (Silver contract) At the beginning of April one year, the silver forward prices (in cents per troy ounce) were as follows: Apr 406.50 July 416.64 Sept 423.48 Dec 433.84 (Assume that contracts
4. (Continuous-time carrying charges) Suppose that a continuous-time compounding frame- work is used with a fixed interest rate Suppose that the carrying charge per unit of time is proportional to
5. (Carrying cost proof) Complete the second half of the proof of the "forward price formula with carrying cost" in Section 10.3. To construct the arbitrage, go long one unit of a forward and short
6. (Foreign currency alternative) Consider the situation of Example 10 10. Rather than short- ing a futures contract, the U.S. firm could borrow 500/(1+1) Deutsche mark (where G is the 90-day
7. (A bond forward) A certain 10-year bond is currently selling for $920 A friend of yours owns a forward contract on this bond that has a delivery date in 1 year and a delivery price of $940 The
8. (Simple formula) Derive the formula (106) by converting a cash flow of a bond to that of the fixed portion of the swap.
9. (Equity swap o) Mr. A. Gaylord manages a pension fund and believes that his stock selection ability is excellent However, he is worried because the market could go down He considers entering an
10. (Forward vanilla) The floating rate portion of a plain vanilla interest rate swap with yearly payments and a notional principal of one unit has cash flows at the end of each year defining a
11. (Specific vanilla) Suppose the current term structure of interest rates is (070, 073, 077, 081. 084, 088) A plain vanilla interest rate swap will make payments at the end of each year equal to
12. (Derivation) Derive the mean-variance hedge formula given by (10.12)
13. (Grapefruit hedge) Farmer D. Jones has a crop of grapefruit that will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months. Jones is worried about possible price
14. (Opposite hedge variance) Assume that cash flow is given by y=S, W + (Fr - Fo)h Let avar (Sr), var(Fr), and as cov(St. Fr). = (a) In an equal and opposite hedge, I is taken to be an opposite
15. (Immunization as hedging o) A pension fund has just paid some of its liabilities, and as a result of this transaction the fund is no longer fully immunized The fund manager decides that instead
16. (Symmetric probability o) Suppose the wealth that is to be received at a time 7 in the future has the form W = a+hr+cx where a is a constant and x is a random variable The value of the variable
17. (Double symmetric probability o) Suppose that revenue has the form RAxy Bx hy where can be chosen and x and y are random variables. The distribution of x and y is symmetric about (0, 0); that is,
18. (A general farm problemo) Suppose that, as in the corn farm example, the farm has random production and the final spot price is governed by the same demand function. However, the crop of the farm
1. (Stock lattice) A stock with current value S(0) 100 has an expected growth rate of its logarithm of 12% and a volatility of that growth rate of 20% Find suitable parameters of a binomial lattice
2. (Time scaling) A stock price S is governed by the model In S(k + 1) = In 5(k)+w(k) where the period length is 1 month. Let v = E[w(k)] and var[w(k)] for all k. Now suppose the basic period length
3. (Arithmetic and geometric means) Suppose that u, ,, are positive numbers The arithmetic mean and the geometric mean of these numbers are, respectively, 21 " 1/0 VA= VI and VG = Vi (a) It is always
4. (Complete the square o) Suppose that e", where w is normal with expected value in and variance o Then 2O L -0 -- dw Show that Use the fact that (W)-) 207- 20 202 - {w (w + o)] + w - 2 e-(x-3 1/20
5. (Log variance) Use the method of Exercise 4 to find the variance of a lognormal variable in terms of the parameters of the underlying normal variable
6. (Expectations) A stock price is governed by geometric Brownian motion with 4 = .20 and = 40 The initial price is S(0) = 1 Evaluate the four quantities E[In S(1)], E[S(1)], stdev[In S(1)]
7. (Application of Ito's lemma) A stock price $ is governed by dS as dr+bs dz where is a standardized Wiener process Find the process that governs G(t) = S (1)
8. (Reverse check) Gavin Jones was mystified by Ito's lemma when he first studied it, so he tested it He started with S governed by and found that = In S satisfies SS + dz dQ (o) do dz He then
9. (Two simulations o) A useful expansion is * = 1+x+x+ Use this to express the exponential in equation (11 20) in linear terms of powers of Ar up to first order Note that this differs from the
10. (A simulation experimente) Consider a stock price 5 governed by the geometric Brow- nian motion process ds 10 dr + 30 d: 5(1)(a) Using A 1/12 and S(0) = 1, simulate several (ie, many) years of
1. (Bull spread) An investor who is bullish about a stock (believing that it will rise) may wish to construct a bull spread for that stock One way to construct such a spread is to buy a call with
2. (Put-call parity) Suppose over the period 10, 7] a certain stock pays a dividend whose present value at interest rate is D Show that the put-call parity relation for European options at 0,
3. (Parity formula) To derive the put-call parity formula, the payoff associated with buy- ing one call option, selling one put option, and lending dk is Q = max(0, S - K) - max(0, KS) + Show that
4. (Call strikes) Consider a family of call options on a non-dividend-paying stock, each option being identical except for its strike price The value of the call with strike price K is denoted by
5. (Fixed dividende) Suppose that a stock will pay a dividend of amount D at time 7. We wish to determine the price of a European call option on this stock using the lattice method. Accordingly, the
6. (Call inequality) Consider a European call option on a non-dividend-paying stock. The strike price is K, the time to expiration is 7. and the price of one unit of a zero-coupon bond maturing at T
7. (Perpetual call) A perpetual option is one that never expires (Such an option must be of American style) Use Exercise 6 to show that the value of a perpetual call on a non- dividend-paying stock
8. (A surprise) Consider a deterministic cash flow stream (...) with all pos- itive flows. Let PV(r) denote the present value of this stream at an interest rate (a) If decreases, does PV(7) increase
9. (My coin) There are two propositions: (a) 1 flip a coin. If it is heads, you are paid $3; if it is tails, you are paid $0. It costs you $1 to participate in this proposition. You may do so at any
10. (The happy call) A New York firm is offering a new financial instrument called a "happy call" It has a payoff function at time 7 equal to max( 55, SK), where S is the price of a stock and K is a
11. (You are a president) It is August 6 You are the president of a small electronics company The company has some cash reserves that will not be needed for about 3 months, but interest rates are
12. (Simplico invariance) If the Simplico mine is solved with all parameters remaining the same except that u = 12 is changed to u = 1.3, the value of the lease remains unchanged to within three
13. (Change of period length) A stock has volatility = 30 and a current value of $36 A put option on this stock has a strike price of $40 and expiration is in 5 months The interest rate is 8% Find
14. (Average value Complexico) Suppose that the price received for gold extracted from time k to k+I is the average of the price of gold at these two times; that is, (gx+8k+1)/2 However, costs are
15. ("As you like it" option) Consider the stock of Examples 12.3 and 124, which has = 20 and an initial price of $62 The interest rate is 10%, compounded monthly Consider a 5-month option with a

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