Questions and Answers of Equity Asset Valuation

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 =
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
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) =
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
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
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
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
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
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
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
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
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
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
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})
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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]
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
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
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
(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
(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 −
(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
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
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
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
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
Consider a one-period economy and an individual with a time-additive but state-dependent expected utility so that the objective is maxθ u(c0, X0) + e−δ E[u(c, X)].The decisions of the individual
Use Eq. (5.7) to compute approximate relative risk premia for the consumption gamble underlying Table 5.5 and compare with the exact numbers given in the table.
Consider an individual with log-utility, u(c) = lnc. What is her certainty equivalent and consumption risk premium for the consumption plan which with probability 0.5 gives her (1 − α)c¯ and with
Provide a proof of Theorem 5.5.
Consider an atemporal setting in which an individual has a utility function u of consumption. His current consumption isc. As always, the absolute risk aversion is ARA(c) = −u(c)/u(c) and the
Consider a one-period choice problem with four equally likely states of the world at the end of the period. The consumer maximizes expected utility of end-of-period wealth. The current wealth must be
(Adapted from Chapter 3 in Kreps (1990).) Imagine a greedy, risk-averse, expected utility maximizing consumer whose end-of-period income level is subject to some uncertainty. The income will be Y
(Adapted from Problem 3.3 in Kreps (1990).) Consider the following two probability distributions of consumption. π1 gives 5, 15, and 30 (dollars) with probabilities 1/3, 5/9, and 1/9, respectively.
Give a proof of Theorem 5.3.
Consider a two-period arbitrage-free economy where the resolution of uncertainty is illustrated in the following binomial tree.t = 0 t = 1 t = 2 10.9 0.95 0.95(2; 8)(8; 2)(6; 10)(10; 4) (10; 6)(4;
In a one-period framework show that if x is a pricing factor and k1, k2 are constants with k2 = 0, then y = k1 + k2x is also a pricing factor.
Show that ζ ∗t = V∗0 /V∗t , whereV∗ and ζ ∗ are given by Eqs. (4.42) and (4.41).
Consider a continuous-time economy in which the state-price deflator follows a geometric Brownian motion:dζt = −ζtrf dt + λ dzt, where rf and λ are constant.(a) What is the price Bs t of a
Consider a two-period economy where the resolution of uncertainty can be represented by the tree in Fig. 2.2 in Chapter 2. Assume that three assets are traded. Their dividend processes are
Consider a one-period economy where two basic financial assets are traded without portfolio constraints or transaction costs. There are three equally likely end-ofperiod states of the economy and the
Consider a one-period economy where the assets are correctly priced by a state-price deflator M. A nutty professor believes that the assets are priced according to a model in which Y is a state-price
Assume a one-period framework with no redundant assets. Show that ζ ∗ can be rewritten as in (4.47).
Show Lemma 4.1.

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