Questions and Answers of Accounting For Financial Instruments

Exercise 5.6.8 Consider the following four zero-coupon bonds:Type Maturity Price Yield Type Maturity Price Yield Treasury 1 year 94 6.28% Treasury 2 year 87 7.09%Corporate 1 year 92 8.51% Corporate 2
Exercise 5.6.7 (1) Confirm that a 50-year bond selling at par ($1,000) with a semiannual coupon rate of 2.55% is equivalent to a 50-year bond selling for $1,000 with a semiannual coupon rate of 2.7%
Exercise 5.6.6 (1) The fact that the forward rate can be locked in today means that future spot rates must equal today’s forward rates, or S(a,b) = f (a, b), in a certain economy. Why? How about an
Exercise 5.6.5 Prove that the forward rate curve lies above the spot rate curve when the spot rate curve is normal, below it when the spot rate curve is inverted, and that they cross where the spot
Exercise 5.6.4 Let the price of a 10-year zero-coupon bond be quoted at 60 and that of a 9.5-year zero-coupon bond be quoted at 62. Calculate the percentage changes in the 10-year spot rate and the
Exercise 5.6.3 Show that f (T, T +1) = d(T)/d(T +1)−1 (to be generalized in Eq. (24.2)).
Exercise 5.6.2 Argue that [ 1+ f (a, a +b+c) ]b+c = [ 1+ f (a, a +b) ]b[ 1+ f (a +b, a +b+c) ]c.
Exercise 5.6.1 Assume that all coupon bonds are par bonds. Extract the spot rates and the forward rates from the following yields to maturity: y1 = 0.03, y2 = 0.04, and y3 = 0.045.
Exercise 5.5.3 Suppose that the bonds making up the yield curve are ordinary annuities instead of coupon bonds. (1) Prove that a yield curve is normal if the spot rate curve is normal. (2) Still, a
Exercise 5.5.2 Contrive an example of a normal yield curve that implies a spot rate curve that is not normal.
Exercise 5.5.1 Prove the following statements: (1) The spot rate dominates the yield to maturity when the yield curve is normal, and (2) the spot rate dominates the yield to maturity if the spot rate
Programming Assignment 5.3.2 Implement the algorithm in Fig. 5.4 plus an option to return the annualized spot rates by using the user-supplied annual compounding frequency.
Exercise 5.3.1 Suppose that S(i) = 0.10 for 1 ≤ i < 20 and a 20-period coupon bond is selling at par, with a coupon rate of 8% paid semiannually. Calculate S(20).
Exercise 5.2.1 Prove that the yield to maturity y is approximatelyi [ ∂Ci (y)/∂y ] S(i)∂ P/∂y to the first order, where Ci (y) ≡ Ci /(1+ y)i denotes the ith cash flow discounted at the
Exercise 4.3.6 Generalize Exercise 4.3.5: Prove that a barbell portfolio achieves immunization with maximum convexity given n > 3 kinds of zero-coupon bonds.
Exercise 4.3.5 Prove that the barbell portfolio has the highest convexity for n = 3.
Exercise 4.3.4 Verify that convexity (4.15) increases as the coupon rate decreases.
Exercise 4.3.3 Show that the convexity of a zero-coupon bond is n(n+1)/(1+ y)2.
Exercise 4.3.2 Prove that ∂(duration)/∂ y = (duration)2 −convexity.
Exercise 4.3.1 In practice, convexity should be expressed in percentage terms, call it C%, for quick mental calculation. The percentage price change in percentage terms is then approximated by −D%
Exercise 4.2.12 Show that theMDof a floating-rate instrument cannot exceed the first reset date.
Exercise 4.2.11 (1) To achieve full immunization, we set up cash inflows at more points in time than liabilities as follows. Consider a single-liability cash outflow Lt at
Exercise 4.2.10 The liability has an MD of 3 years, but the money manager has access to only two kinds of bonds with MDs of 1 year and 4 years. What is the right proportion of each bond in the
Exercise 4.2.9 Consider a liability currently immunized by a coupon bond. Suppose that the interest rate changes instantaneously. Prove that profits will be generated when rebalancing is performed at
Exercise 4.2.8 Start with a bond whose PV is equal to the PV of a future liability and whose MD exceeds the horizon. Show that, at the horizon, the bond will fall short of the liability if interest
Exercise 4.2.7 Show that, in the absence of interest rate changes, it suffices to match the PVs of the liability and the asset.
Exercise 4.2.6 In setting up the two-bond immunization in Eqs. (4.11), we did not bother to check the convexity condition. Justify this omission.
Exercise 4.2.5 Show that the duration of an n-period zero-coupon bond is n.
Exercise 4.2.4 Verify that the MD of a traditional mortgage is (1+ y)/y−n/((1+ y)n −1).
Exercise 4.2.3 Consider a coupon bond and a traditional mortgage with the same maturity and payment frequency. Show that the mortgage has a smaller MD than the bond when both provide the same yield
Exercise 4.2.2 Duration is usually expressed in percentage terms for quick mental calculation: Given duration D%, the percentage price change expressed in percentage terms is approximated by −D%
Exercise 4.2.1 Assume that 9% is the annual yield to maturity compounded semiannually.Calculate theMDof a 3-year bond paying semiannual coupons at an annual coupon rate of 10%.
Exercise 4.1.3 (1) Prove that price volatility always decreases as the yield increases when the yield equals the coupon rate. (2) Prove that price volatility always decreases as the yield increases,
Exercise 4.1.2 Show that price volatility never decreases as the coupon rate decreases when yields are positive.
Exercise 4.1.1 Verify Eq. (4.1).− ∂ P/P ∂y =−(C/y) n−(C/y2)((1+ y)n+1 −(1+ y))−nF (C/y) [ (1+ y)n+1 −(1+ y) ]+ F(1+ y), (4.1)
Exercise 3.5.9 Prove that the holding period yield of a level-coupon bond is exactly y when the horizon is one period from now.
Programming Assignment 3.5.8 Write a program that computes (1) the accrued interest as a percentage of par and (2) the BEY of coupon bonds. The inputs are the coupon rate as a percentage of par, the
Exercise 3.5.7 Consider a bond with a 10% coupon rate and paying interest semiannually.The maturity date is March 1, 1995, and the settlement date is July 1, 1993.The day count used is actual/actual.
Exercise 3.5.6 It has been mentioned that a bond selling at par will continue to sell at par as long as the yield to maturity is equal to the coupon rate. This conclusion rests on the assumption that
Exercise 3.5.5 Prove that a level-coupon bond will be sold at par if its coupon rate is the same as the market interest rate..
Exercise 3.5.4 (1) Derive ∂ P/∂n and ∂ P/∂r for zero-coupon bonds. (2) For r = 0.04 and n = 40 as in Example 3.5.1, verify that the price will go down by approximately d×8.011% of par value
Exercise 3.5.3 How should pricing formula (3.18) be modified if the interest is taxed at a rate of T and capital gains are taxed at a rate of TG?
Exercise 3.5.2 A company issues a 10-year bond with a coupon rate of 10%, paid semiannually. The bond is callable at par after 5 years. Find the price that guarantees a return of 12% compounded
Exercise 3.5.1 A consol paying out continuously at a rate of c dollars per annum has value ∞0 ce−rt dt, where r is the continuously compounded annual yield. Justify the preceding formula.
Exercise 3.4.8 Describe a bisection method for solving systems of nonlinear equations in the two-dimensional case. (The bisection method may be applied in cases in which the Newton–Raphson method
Exercise 3.4.7 Write the analogous n-dimensional formula for Eqs. (3.16).
Exercise 3.4.6 Let ξ be a root of f and J be an interval containing ξ . Suppose that f (x) = 0 and f (x) ≥0 or f (x) ≤ 0 for x ∈ J . Explain why the Newton–Raphson method converges
Exercise 3.4.5 Suppose that f (ξ ) = 0 and f (ξ ) is bounded.Verify that condition(3.14) holds for the Newton–Raphson method.
Exercise 3.4.4 Let f (x) ≡ x3 −x2 and start with the guess x0 = 2.0 to the equation f (x) = 0. Iterate the Newton–Raphson method five times.
Exercise 3.4.3 Repeat the calculation for Example 3.4.6 for an expected return of 4%.
Exercise 3.4.2 A financial instrument pays C dollars per year for n years. The investor interested in the instrument expects the cash flows to be reinvested at an annual rate of r and is asking for a
Exercise 3.4.1 A security selling for $3,000 promises to pay $1,000 for the next 2 years and $1,500 for the third year. Verify that its annual yield is 7.55%.
ProgrammingAssignment 3.3.3 Write a program that prints out the monthly amortization schedule. The inputs are the annual interest rate and the number of payments.
Exercise 3.3.2 Start with the cash flow of a level-payment mortgage with the lower monthly fixed interest rate r −x. From the monthly payment D, construct a cash flow that grows at a rate of x per
Exercise 3.3.1 Explain why PV1+ r mk−k i=1 C1+ r mi−1 where the PV from Eq. (3.6) equals that of Eq. (3.8).
Exercise 3.2.1 Derive the PV formula for the general annuity due.
Exercise 3.1.4 Prove the correctness of the FV algorithm mentioned in the text.
Exercise 3.1.3 (1) It was mentioned in Section 1.4 that workstations improved their performance by54%per year between 1987 and 1992 and that theDRAMtechnology has quadrupled its capacity every 3
Exercise 3.1.2 Below is a typical credit card statement:NOMINAL ANNUAL PERCENTAGE RATE (%) 18.70 MONTHLY PERIODIC RATE (%) 1.5583 Figure out the frequency of compounding.
Exercise 3.1.1 Verify that, given an annual rate, the effective annual rate is higher the higher the frequency of compounding.
Exercise 2.2.2 Prove the following relations: (1)n i=1 i = O(n2), (2)n i=1 i 2 =O(n3), (3)log2 n i=0 2i = O(n), (4)α log2 n i=0 2i = O(nα), (5) nn i=0 i−1 = O(n ln n).
Exercise 2.2.1 Show that f +g = O( f ) if g = O( f ).
Exercise B.1 The MATLAB file DataCauchy contains observations from a Cauchy distribution(see Appendix A.6.16) with densityAssume that the data are independent. We want to estimate μ and σ >
Exercise A.3 Prove Lemma A.6.1, i.e., for any function f ≥ 0, f(||||2) dx Rd = 2nd/2 (d/2)rd-1f (r) dr = 1/2)r(d-2)/2 f(r)dr. Td/2
Exercise A.2 Suppose that Λ ∼ Gamma(α, β) and suppose that conditionally on Λ = λ, X has a Poisson distribution with parameter λ.(a) Show that for any integer k ≥ 0,(b) Show that the
Exercise A.1 Suppose N ∼ Poisson(λ) and that conditionally on N = n, X has a binomial distribution with parameters (n, p). Find the (unconditional) distribution of X.
5. Construct a MATLAB function to perform the test of goodness-of-fit based on the Rosenblatt transform for one-dimensional regime-switching models. Use the MATLAB function EstHMM1d.
4. Construct a MATLAB function to simulate a regime-switching Brownian motion observed a periods h, 2h, . . . , nh.
3. Construct a MATLAB function to simulate a regime-switching model with Gaussian regimes.
2. Construct a MATLAB function for the Kalman equations of an ARMA(p,q) model.
1. Construct a MATLAB function to compute the coefficients ψ1, . . . , ψr−1 arising from a MA representation of an ARMA(p,q) model. Here r =max(p, q + 1).
Exercise 10.9 For the replication of hedge funds indices, suppose that there are constraints on the values of the positions β, for example p j=1β(j)i≤ 2. Can you still use the Kalman
Exercise 10.8 Using parametric bootstrap to compute P-values, you get an estimated P-value of 3%, using N = 100 replications.(a) Can you safely conclude that the real P-value is less than 5%?(b) What
Exercise 10.7 Two models for the log-returns of Apple were not rejected: GARCH(1,1)with GED innovations, and a regime-switching model with three Gaussian regimes. Which one would you choose? Motivate
Exercise 10.6 Consider the regime-switching model of Example 10.2.2.(a) What is distribution of the regimes for the next trading day, i.e., January 18th, 2011?(b) Generate 100000 observations from
Exercise 10.5 Consider an ARMA(p,q) and set r = max(p, q). Show that the coefficients ψ1, . . . , ψr, from the MA representation of an ARMA(p,q) satisfy the following equations: j-1 V = 0; j +
Exercise 10.4 Consider an ARMA(2,2) model with parameters φ1 = 0.4, φ2 = −0.1 andθ1 = −0.3, θ2 = 0.1.(a) Using Proposition 10.1.1, find F, g, and a.(b) Compute U0|0.(c) Using (10.3), compute
Exercise 10.3 For an ARMA(1, 1) model, show that for all i ≥ 1, Zi|i = ( 02 - Yi-H - and ei+1 + 2 and Uili = = Yi+1 262 - - with f+1 == 1+ - (Y ) + 0. Also Uolo e = Y- and f = 1+ - 12 ===
Exercise 10.2 Consider an ARMA(1,1) model with parameters φ1 = 0.4 and θ1 = −0.5.(a) Using Proposition 10.1.1, find F, g, and a.(b) Compute U0|0.(c) Using (10.3), compute f2 i and Ui|i for i ∈
Exercise 10.1 For a MA(1) model, show that for all i ≥ 1, Zi|i =Yi − μ−θ ei f2 iand Ui|i =0 0 0 θ2 (f2 i−1)f2 i, with f2 i+1 = 1+θ2 − θ2 f2 i, e1 = Y1 − μ, and ei+1 =Yi+1 − μ +
4. Construct a MATLAB function for computing the Kalman equations in the general model.
3. Construct a MATLAB function to estimate the parameters of the Schwartz model when observing d futures. The n × d matrix of maturities must be an input.
2. Construct a MATLAB function to simulate trajectories for Schwartz model.
1. Construct a MATLAB function for computing the value of a futures under Schwartz model.
Exercise 9.7 Suppose you implement a particle filter. How can you compute a 95%prediction interval for each component of the signal? Can you estimate the(conditional) density of the signal?
Exercise 9.6 Suppose that for the model Xi = α + φXi−1 + wi, Yi = Xi + εi, the correlation between the Gaussian error terms εi and wi is ρ. Transform this model into one satisfying the
Exercise 9.5 Consider the model in Example 9.2.2. Suppose that at some point the minimization algorithm is trapped in a local minimum giving Q = 0, which is like starting the minimization with Q = 0.
Exercise 9.4 Consider the Schwartz model for the spot price of a commodity and its convenience yield.(a) Simulate a trajectory of n = 503 daily values for the spot price and its convenience yield,
Exercise 9.3 Write the Kalman equations corresponding to the Schwartz model where the observations consists in two futures.
Exercise 9.2 Consider the Schwartz model for the spot price of a commodity and its convenience yield. Assume that S(0) = 16, δ(0) = 0.025, r = 5%, and that the time scale is in years.(a) Compute the
Exercise 9.1 Consider the following model: Zi = μ + wi, wi ∼ N(0,Q), and Yi =Zi +εi, with εi ∼ N(0,R). Write the Kalman equations. Can you estimate all parameters using maximum likelihood?
4. Construct a MATLAB function to perform a goodness-of-fit test based on the Rosenblatt transform for the Gumbel, Frank, and Ali-Mikhail-Haq copulas.
3. Construct a MATLAB function to compute the Kendall function for the Gumbel, Frank, and Ali-Mikhail-Haq copulas.
2. Construct a MATLAB function to perform a goodness-of-fit test for the Clayton copula by comparing the empirical Kendall tau withˆθnˆθn+2, where ˆθn is the maximum pseudo likelihood estimator
1. Construct a MATLAB function for finding the parameter of a Plackett copula corresponding to a given Kendall tau.
Exercise 8.16 Find a way to generate observations from the Plackett copula, for all θ > 0.
Exercise 8.15 Show that the Plackett family is positively ordered with respect to the PQD order. This exercise is difficult.
Exercise 8.14 Suppose that we have a portfolio composed of 3 risky bonds with face value$100000, and the time to default is exponentially distributed with a mean of 15. The issuer offers no guaranty,
Exercise 8.13 Find a way to generate observations from the bivariate Frank copula, for all θ > 0.
Exercise 8.12 One obtains the following output from the MATLAB function EstDep.pseudo: [500x3 double]stat: [0.3851 0.5533 0.5498 0.5537]std: [0.5439 0.7148 0.6723 0.6934]error: [0.0477 0.0627 0.0589

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