Financial Asset Pricing Theory 1st Edition Claus MUNK - Solutions

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Show that if each investor has shifted CRRA utility with the same coefficientρ > 0 and shift ζh, then, as asserted in Section 4.4, any Pareto-optimal allocation involves an affine sharing rule.
Suppose all investors have shifted CRRA utility with the same coefficientρ > 0. Suppose w˜ m > ζ . Consider an allocation w˜ h = ζh + bh(w˜ m −ζ )where H h=1 bh = 1. Show that the allocation is Pareto optimal. Hint: Show that it solves the social planner’s problem with weights λh
Suppose all investors have CARA utility. Consider an allocation w˜ h = ah +bhw˜ m where bh = τh/τ and H h=1 ah = 0. Show that the allocation is Pareto optimal.Hint: Show that it solves the social planner’s problem with weights λh defined asλh = τheah/τh .
Suppose each investor h has a concave utility function, and suppose an allocation (w˜ 1,...,w˜ m) of market wealth w˜ m satisfies the first-order condition uh(w˜ h) = γhm˜for each investor h, where m˜ is an SDF and is the same for each investor. Show that the allocation solves the social
Suppose there are two investors, the first having constant relative risk aversion ρ > 0 and the second having constant relative risk aversion 2ρ.(a) Show that the Pareto-optimal sharing rules are w˜ 1 = ˜wm +η − η2 +2ηw˜ m , and w˜ 2 = η2 +2ηw˜ m − η, for η > 0. Hint: Use the
Adopt the assumptions of the previous exercise, but assume there is no risk-free asset and there is consumption only at date 1. Show that the vector p = γμ− αθis an equilibrium price vector for any γ > 0. Note: When γ < 0, this is also an equilibrium price vector, but each investor has a
Assume there is a risk-free asset in zero net supply. Let θ = (θ 1 ··· θ n) denote the vector of supplies of the n risky assets. Let μ denote the mean and the covariance matrix of the vector X˜ = (x˜1 ···˜xn) of asset payoffs. Assume is nonsingular. Suppose the utility functions
Suppose there are n risky assets with normally distributed payoffs x˜i. Assume all investors have CARA utility and no labor income y˜h. Define α to be the aggregate absolute risk aversion as in Section
Assume the payoff of each asset has a finite variance and the law of one price holds. Apply facts stated in Section 3.8 to show that there is a unique SDF m˜ p in the span of the asset payoffs. Show that the orthogonal projection of any other SDF onto the span of the asset payoffs equals m˜ p.
Suppose there is a risk-free asset. Adopt the notation of Exercise 3.3, and assume the risky asset returns have a joint normal distribution. Show that the optimal portfolio of risky assets for an investor with no labor income is π =δ−1(μ−Rfι)for some real number δ by applying the
Suppose there is an SDF m˜ with the property that for every function g there exists a portfolio θ (depending on g) such that ni=1θix˜i = g(m˜ ).Consider an investor with no labor income y˜. Show that her optimal wealth is a function of m˜ . Hint: For any feasible w˜, define w˜ ∗ = E[ ˜w
Show by example that the law of one price can hold but there can still be arbitrage opportunities.
Show that, if there is a strictly positive SDF, then there are no arbitrage opportunities.
Supposetwo random vectorsX˜ and Y˜ are joint normally distributed. Explain why the orthogonal projection (3.32) equals E[Y˜ | X˜].
Assume there is a risk-free asset. Let R˜ denote the vector of risky asset returns, let μ denote the mean of R˜ , and let denote the covariance matrix of R˜ . Let ι denote a vector of 1’s. Derive the following formula for the SDF m˜ p from the projection formula (3.32):m˜ p = 1 Rf+ι
Assume there are three possible states of the world: ω1, ω2, and ω3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with return R1 in state ω1, R2 in state ω2, and R3 in state ω3. Assume the probabilities are 1/4 for state ω1, 1/2 for state ω2,
Assume there are two possible states of the world: ω1 and ω2. There are two assets, a risk-free asset returning Rf in each state, and a risky asset with initial price equal to 1 and date–1 payoff x˜. Normalize the unit of measurement of the risk-free asset so that its price is 1 at date 0 and
Letting c∗0 denote optimal consumption in the previous exercise, define the precautionary premium π by u((w0 −π − c∗0)Rf) = E[u((w0 − c∗0)Rf + ˜y)].(a) Show that c∗0 would be the optimal consumption of the investor if she had no labor income and had initial wealth w0 −π.(b)
Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Suppose the investor has time-additive utility with u0 = u and u1 = δu for a common function u and discountfactor δ. Supposethe investor has labor income y˜ atthe
Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Assume the investor has time-additive power utility, so she solves max 1 1− ρc 1−ρ0 + δ1 1 −ρc 1−ρ1 subject to c0 +1 Rf c1 = w0 .As shown in Exercise 2.6,
Consider a utility function v(c0,c1). The marginal rate of substitution (MRS)is defined to be the negative of the slope of an indifference curve and is equal to MRS(c0,c1) = ∂v(c0,c1)/∂c0∂v(c0,c1)/∂c1.The elasticity of intertemporal substitution (EIS) is defined as
Suppose there is a risk-free asset and n risky assets. Consider an investor with quadratic utilityζE[ ˜w] −1 2E[ ˜w]2 − 1 2var(w˜)and no labor income. Show that the optimal portfolio for the investor isφ = 1 1+ κ 2 (ζ −w0Rf)−1(μ− Rfι), whereκ 2 = (μ− Rfι)−1(μ−
Consider a CARA investor with n risky assets having normally distributed returns, as studied in Section 2.4, but suppose there is no risk-free asset, so the budget constraint is ιφ = w0. Show that the optimal portfolio isφ = 1α−1μ +αw0 − ι−1μαι−1ι−1ι.Note: As will be
Suppose there is a risk-free asset with return Rf and a risky asset with return R˜. Consider an investor who maximizes expected end-of-period utility of wealth and who has CARA utility and invests w0. Suppose the investor has labor income y˜ at the end of the period, so her end-of-period wealth
Suppose there is a risk-free asset and n risky assets with payoffs x˜i and prices pi. Assume the vector x˜ = (x˜1 ···˜xn) is normally distributed with mean μx and nonsingular covariance matrix x. Let p = (p1 ···pn). Suppose there is consumption at date 0 and consider an investor with
Supposethe covariance matrix of the risky asset returns is =⎛⎝0.09 0.06 0 0.06 0.09 0 0 0 0.09⎞⎠Suppose the returns are normally distributed. What is the optimal fraction of wealth to invest in each of the risky assets for a CARA investor with αw0 = 2?Why is the optimal investment higher
Suppose there is a risk-free asset with return Rf = 1.05 and three risky assets each of which has an expected return equalto
Show that risk neutrality [u(w) = w for all w] can be regarded as a limiting case of negative exponential utility asα → 0 by showingthatthere are monotone affine transforms of negative exponential utility that converges to w as α → 0.Hint: Take an exact first-order Taylor series expansion of
Show that any monotone LRT utility function is a monotone affine transform of one of the five utility functions: negative exponential, log, power, shifted log, or shifted power. Hint: Consider first the special cases (i) risk tolerance = A and (ii) risk tolerance = Bw. In case (i) use the fact that
Show that condition (ii) in the discussion of second-order stochastic dominance in the end-of-chapter notes implies condition (i); that is, assume y˜ = ˜x + ˜z + ˜ε where z˜ is a nonpositive random variable and E[˜ε | ˜x + ˜z] = 0 and show that E[u(x˜)] ≥ E[u(y˜)] for every monotone
Which LRT utility functions are DARA utility functions with increasing relative risk aversion, for some parameter values? Which of these utility functions are monotone increasing and bounded on the domain w ≥ 0?
Suppose an investor has log utility u(w) = logw for each w > 0.(a) Construct a gamble w˜ such that E[u(w˜)]=∞. Verify that E[ ˜w]=∞.(b) Construct a gamble w˜ such that w˜ > 0 in each state of the world and E[u(w˜)] = −∞.(c) Given a constant wealth w, construct a gamble ε˜ with w+
The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so ={ω1,ω2,ω3} with P({ω1}) = P({ω2}) = P({ω3}) = 1/3. Let G be the collection of all subsets of :G =
Let y˜ = ex˜, where x˜ is normally distributed with mean μ and variance σ2.Show that stdev(y˜)E[˜y] =eσ 2 −1 .
Use the law of iterated expectations to show that if E[˜ε | ˜y] = 0, then cov(y˜,ε)˜ = 0 (thus, mean independence implies uncorrelated).
Consider a person with constant relative risk aversion ρ.(a) Verify that the fraction of wealth she will pay to avoid a gamble that is proportional to wealth is independent of initial wealth (that is, show that π defined in (1.15) is independent of w for logarithmic and power utility).(b)
Calculate the mean, variance, and skewness of the following two random variables:w˜ 1 =2.45 with probability 0.5141 , 7.49 with probability 0.4859 , w˜ 2 =⎧⎪⎨⎪⎩0 with probability 0.12096 , 4.947 with probability 0.750085 , 10 with probability 0.128955 .You should see that w˜ 2 has a
This exercise is a very simple version of a model of the bid-ask spread presented by Stoll (1978). Consider an individual with constant absolute risk aversion α. Assume w˜ and x˜ are joint normally distributed with means μw andμx, variances σ2 w and σ2 x , and correlation coefficient ρ.(a)
What values of ρ (if any) seem reasonable?
Consider a person with constant relative risk aversion ρ who has wealth w.(a) Suppose she faces a gamble in which she wins or loses some amount x with equal probabilities. Derive a formula for the amount π that she would pay to avoid the gamble; that is, find π satisfying u(w− π ) = 1
Calculate the risk tolerance of each of the LRT utility functions (negative exponential, log, power, shifted log, and shifted power) to verify the formulas for risk tolerance given in Section 1.3.
Consider a swap with starting date T0 and a fixed rate K. For t ≤ T0, show that Vfl t /Vfix t = L˜ δ,T0 t /K, where L˜ δ,T0 t is the forward swap rate.
Show the parity (13.30). Show that a payer swaption and a receiver swaption(with identical terms) will have identical prices, if the exercise rate of the contracts is equal to the forward swap rate L˜ δ,T0 t .
Let ˜lδT0(k) be the equilibrium swap rate for a swap with payment dates T1, T2, ... , Tk, where Ti = T0 + iδ as usual. Suppose that ˜lδT0(1), ... ,˜lδT0(n) are known.Find a recursive procedure for deriving the associated discount factors BT1 T0, BT2 T0, ... , BTn T0.
Let FT,S t and T,S t denote the forward price and futures price at time t, respectively, for delivery at time T > t of a zero-coupon bond maturing at time S > T.Under the assumptions of the Vasicek model introduced in Section 11.5.1, show that FT,S t = exp{−[A(S − t) − A(T − t)] − [Bκ
Let S1 = (S1t) and S2 = (S2t) be the price processes of two assets. Consider the option to exchange (at zero cost) one unit of asset 2 for one unit of asset 1 at some prespecified date T. The payoff is thus max (S1T − S2T, 0). The assets have no dividends before time T.(a) Argue that the time t
Show by differentiation that the Black–Scholes–Merton call option price satisfies Eqs. (13.15) and (13.16). Hint: First show that Stn(d) = Ke−r[T−t]n(d − σ√T − t).
Consider a coupon bond with payment dates T1 < T2 < ··· < Tn. For each i = 1, 2, ... , n, let Yi be the sure payment at time Ti. For some t < T < Ti, let T,Ti t denote the futures price at time t for delivery at time T of the zero-coupon bond maturing at time Ti with a unit payment. Show that
In the same two-period economy considered in Examples 12.3 and 12.4, compute the price of an asset giving a time 1 dividend of 0 in the upper or middle node and 1 in the lower node and a time 2 dividend of 3, 2, 3, 3, 4, or 5 from the top node and down (this is asset 3 in Exercise 4.8).
Consider Example 12.4. Compute the conditional Q2-probabilities of the transitions over the second period of the tree. Compare with conditional Q-probabilities illustrated in Fig. 12.2 and explain why they are (not?) different.
Take a continuous-time framework and assume that ζ = (ζt)t∈[0,T] is a state-price deflator. What is the Q-dynamics of ζ ?
Show that Eqs. (12.6) and (12.7) are equivalent.
Provide a detailed proof of the bond price expression in the Cox–Ingersoll–Ross model as stated in Theorem 11.4.
Consider an economy with complete financial markets and a representative agent with CRRA utility, u(C) = C1−γ1−γ , where γ > 0, and a time preference rate of δ. The aggregate consumption level C is assumed to follow the stochastic process dct = ct(a1X2 t + a2Xt + a3)dt + σc dzt, where z
Assume a continuous-time economy where the state-price deflator ζ = (ζt)has dynamics dζt = −ζt [rt dt + λ dz1t], where z1 = (z1t) is a (one-dimensional) standard Brownian motion, λ is a constant, and r = (rt) follows the Ornstein–Uhlenbeck process drt = κ[r¯ − rt] dt + σr dz1t.This
Constantinides (1992) develops the so-called SAINTS model of the nominal term structure of interest rates by specifying exogenously the nominal state-price deflator ζ˜.In a slightly simplified version, his assumption is thatζ˜t = ke−gt+(Xt−α)2, where k, g, and α are constants, and X =
Go through the derivations in Example 11.1.
Show Eqs. (11.55) and (11.56).
The purpose of this exercise is to show that the claim of the gross return pure expectation hypothesis is inconsistent with interest rate uncertainty. In the following we consider time points t0 < t1 < t2.(a) Show that if the hypothesis holds, then 1Bt1 t0= 1 Bt2 t0 Et0Bt2 t1.Hint: Compare two
The term premium at time t for the future period [t, T] is the current forward rate for that period minus the expected spot rate, that is f t,T t − Et[yT t]. This exercise will give a link between the term premium and a state-price deflator ζ = (ζt).(a) Show that BT t = Btt EtBT t+
Show Eq. (11.35).
Show that if there is no arbitrage and the short rate can never go negative, then the discount function is non-increasing and all forward rates are non-negative.
In a continuous-time framework an individual with time-additive expected power utility induces the state-price deflatorζt = e−δt ct c0−γ, where γ is the constant relative risk aversion, δ is the subjective time preference rate, and c = (ct)t∈[0,T] is the optimal consumption process of
Consider a discrete-time economy in which asset prices are described by an unconditional linear factor modelζt+1ζt= a + b · xt+1, t = 0, 1, ... , T − 1, where the conditional mean and second moments of the factor are constant, that is Et[xt+1] = μ and Et[xt+1x t+1] = for all t.You want to
Using the orthogonal characterization of the mean-variance frontier, show that for any mean-variance efficient return Rπ different from the minimum-variance portfolio there is a unique mean-variance efficient return Rz(π) satisfying Cov[Rπ , Rz(π)] = 0.Show that E[Rz(π)] = E[R∗] − E[Re∗]
Consider a discrete-time economy with a one-dimensional conditional pricing factor x = (xt) so that, for some adapted processes α = (αt) and some η = (ηt), Et[Ri,t+1] = αt + βt[Ri,t+1, xt+1] ηt, for all assets i and all t = 0, 1, ... , T − 1.(a) Show thatζt+1ζt= 1αt1 − xt+1 −
Consider a one-period, finite-state economy where an individual has utility from the consumption of two goods. The consumption of the first good is denoted by c, and the consumption of the second good is denoted by q. The first good is the numeraire good, and Pq denotes the price of the second
Consider a continuous-time economy with a representative agent with timeadditive subsistence HARA utility, that is the objective of the agent is to maximize ET 0e−δt 1 1 − γ(ct − c¯)1−γ dt, where c¯ ≥ 0 is the subsistence consumption level. Assume that aggregate consumption c =(ct)
Consider a discrete-time representative individual economy equipped with preferences E[∞t=0 e−δt u(ct, qt)], where ct is the consumption of good 1 and qt is the consumption of good 2. Assume a Cobb–Douglas type utility function, u(c, q) = 1 1 − γ(cαq1−α)1−γwhere γ > 0 and α ∈
(This problem is based on Longstaff and Piazzesi (2004).) Consider a continuous-time model of an economy with a representative agent and a single non-durable good. The objective of the agent at any time t is to maximize the expected time-additive CRRA utility, Et∞t e−δ(s−t) C1−γ s 1 −
(This problem is based on Lochstoer (2009).) Suppose a representative individual has preferences of the form E⎡⎣T t=0 e−δt u(Bt, Lt; Xt)⎤⎦ , u(B, L; X) = 1 1 − γ(Lα[B − X]1−α)1−γ, where α ∈ (0, 1) and γ > 1. Here Bt denotes the time t consumption of basic goods and Lt
(This problem is based on Menzly, Santos, and Veronesi (2004).) Consider an economy with a representative agent with life-time utility given by U(C) = E∞0 e−ϕt ln(Ct − Xt) dt, where Xt is an external habit level and ϕ is a subjective discount rate. As in Campbell and Cochrane (1999)
with u(c, h) = 1 1−γ (c − h)1−γ .(a) Can optimal consumption follow a geometric Brownian motion under these assumptions?(b) Assume that the excess consumption rate cˆt = ct − ht follows a geometric Brownian motion. Show that the state-price deflator will be of the form ζt = f(t)e−δt
Consider the set-up of
Consider a continuous-time economy with complete markets and a representative individual having an ‘external habit formation’ or ‘keeping up with the Joneses’ utility function so that, at any time t, the individual wants to maximize Et T t e−δ(s−t)u(cs, Xs) ds, where u(c, X) = 1
In the Chan and Kogan model, show how Eq. (9.20) follows from Eq. (9.6).Note: with a continuum of individuals, the sum over individuals should be replaced by an integral, that is L l=1( 1 Al(clt ,Xt))is replaced by ∞1( 1 A(ct,Xt;γ ))dγ , where A(ct, Xt; γ ) is the absolute risk aversion of
Consider the Chan and Kogan model. Show the expressions in Eqs. (9.15),(9.16), (9.17), (9.18), and (9.19). Show Theorem 9.3.
Give a proof of Theorem 9.1.
Consider a two-period binomial economy where the state-price deflator ζ is related to the growth rate of aggregate consumption C, that is,ζt+1ζt= a + b Ct+1 Ct, t = 0, 1, where a and b are constants with b = −3. Aggregate consumption may grow by either 1%or 5% over each period and the two
In the simple consumption-based asset pricing model, the growth rate of aggregate consumption is assumed to have a constant expectation and standard deviation(volatility). For example, in the continuous-time version aggregate consumption is assumed to follow a geometric Brownian motion. Consider
Carl Smart is currently (at time t = 0) considering a couple of investment projects that will provide him with a dividend in one year from now (time t = 1) and a dividend in two years from now (time t = 2). He figures that the size of the dividends will depend on the growth rate of aggregate
Consider a one-period setting. Let g denote the growth rate of consumption over the period.(a) For any portfolio π, show that Corr[Rπ , g]2 =π Cov[R, g]2π Var[R]π Var[g].(b) Show that the first-order condition for the maximization of Corr[Rπ , g]2 implies that Cov[R, g]π Var[R]π = π Cov[R,
Consider a risky asset i in a one-period setting. Assume all individuals have time-additive expected utility.(a) Explain why the price of asset i and the consumption of individual l are related as follows: It is technically complicated to include transaction costs and trading restrictions in
Provide a proof of Theorem 7.5 in the case of CARA utility (corresponding toα = 0 in the notation introduced just above the statement of the theorem).
George and John live in a continuous-time economy in which the relevant uncertainty is generated by a one-dimensional standard Brownian motion z = (zt)t∈[0,T].Both have time-additive utility of their consumption process: George maximizes UG(c) = E T0 e−0.02t ln ct dt, while John maximizes
Show how the results for CARA utility and CRRA utility stated in Section 7.5 generalize to economies with more than two individuals.
Consider an economy with two individuals and Pareto-optimal equilibrium allocations. Both individuals have time-additive expected utility and their time preference rates are identical.(a) Assume that ul(c) = 1 1−γ (c − c¯l)1−γ for l = 1, 2. Find the preferences of the representative
Suppose that aggregate time 1 consumption can only take on the values 1, 2, ... , K for some finite integer K. Assume that European call options on aggregate consumption are traded for any exercise price 0, 1, 2, ... , K. Consider a portfolio with one unit of the option with exercise price k − 1,
(Use a spreadsheet or similar computational tool.) Consider a one-period economy with 5 possible states and 5 assets traded. The state-contingent dividends and prices of the assets and the state probabilities are as follows:State 1 State 2 State 3 State 4 State 5 Price asset 1 1 1 1 1 1 0.9 asset 2
Assume a discrete-time economy with L agents. Each agent maximizes timeadditive expected utility E T t=0 βtu(ct)where u is strictly increasing and concave.Show thatζt+1ζt=L=1 βu(c,t+1)L=1 u(ct)is a valid one-period state-price deflator. How do you define the full state-price
Consider a discrete-time economy with L individuals with identical preferences so that agent l = 1, ... , L at time 0 wants to maximize E⎡⎣T t=0βt 1 1 − γ c 1−γl,t⎤⎦where cl,t denotes the consumption rate of individual l at time t. Let c∗l,t be the optimal consumption rate of
Consider an individual with a time-additive expected utility characterized by a utility function u(c) and a time preference rate δ. The individual lives in a continuous-time economy. Let us write the dynamics of the optimal consumption process of the individual as dct = ctμct dt + σct dzt.(a)
Look at an individual with habit formation living in a continuous-time complete market economy. The individual wants to maximize his expected utility ET 0e−δt u(ct, ht) dtwhere the habit level ht is given by ht = h0e −αt + βt 0 e −α(t−u)cu du.We can write the budget constraint as E T 0
In a one-period model where the returns of all the risky assets are normally distributed, any greedy and risk-averse investor will place herself on the upward-sloping part of the mean-variance frontier. But where? Consider an agent that maximizes expected utility of end-of-period wealth with a
Think of the mean-variance framework in a one-period economy. Show that if there is a risk-free asset, then any two mean-variance efficient returns (different from the risk-free return) are either perfectly positively correlated or perfectly negatively correlated.Is that also true if there is no
Let R1 denote the return on a mean-variance efficient portfolio and let R2 denote the return on another not necessarily efficient portfolio with E[R2] = E[R1]. Show that Cov[R1, R2] = Var[R1] and conclude that R1 and R2 are positively correlated.
Let Rmin denote the return on the minimum-variance portfolio. Let R be any other return, efficient or not. Show that Cov[R, Rmin] = Var[Rmin].
Give a proof of Theorem 6.4.
Show Eq. (6.19).
Consider a one-period economy with four possible, equally likely, states at the end of the period. The agents in the economy consume at the beginning of the period(time 0) and at the end of the period (time 1). The agents can choose between three different consumption plans as shown in the

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Financial Asset Pricing Theory 1st Edition Claus MUNK - Solutions

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