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equity asset valuation
Questions and Answers of
Equity Asset Valuation
Consider the portfolio choice model with a single risky asset described in Section 25.5, in which there is no ambiguity about the marginal distribution aboutthe asset payoff butthere is ambiguity
Consider CRRA disappointment-averse utility and a random wealth w˜ = ez˜, where z˜ is normally distributed with mean μ and variance σ2. Let ξdenote the certainty equivalent of w˜, and set θ =
Consider CRRA weighted utility.(a) Show thatg in (25.13) is strictly monotone in y > 0—so the preferences are monotone with regard to stochastic dominance—if and only ifγ ≤ 0 and ρ ≤ γ + 1
Consider weighted utility. Let ε˜ have zero mean and unit variance. For a constant σ, denote the certainty equivalent of w + σε˜ by w − π(σ ). Assumeπ(·) is twice continuously
Consider the following pairs of gambles:A :90% chance of $3,000 10% chance of $0 versus B :45% chance of $6,000 55% chance of $0 C :0.2% chance of $3,000 99.8% chance of $0 versus D :0.1% chance of
Consider the following pairs of gambles:A : 100% chance of $3,000 versus B :80% chance of $4,000 20% chance of $0 C :25% chance of $3,000 75% chance of $0 versus D :20% chance of $4,000 80% chance of
In the continuous-time Kyle model, assume logv˜ is normally distributed instead of v˜ being normally distributed. Denote the mean of logv˜ by μ and the variance of log v˜ by σ2. Set λ =
Assume there are two buyers in an auction who have a common value.Assume the buyers receive signals that are independently uniformly distributed on [0,1], and assume the value is the sum of the
Assume there are two buyers in an auction who have independent private values. Assume the value of each buyer is uniformly distributed on [0,1]. Each buyer knows her own value but does not know the
In the single-period Kyle model, assume the informed investor has CARA utility. There is a linear equilibrium. Derive an expression for λ as a root of a fifth-order polynomial.
Suppose there is a representative market maker with constant absolute risk aversion α, and competition forces the bid and ask to the prices that make the market maker indifferent about trade.
Verify that (t) defined in (23.33) satisfies the ODE (23.31) with initial condition (0) = var(X0).
Consider the model of Section 23.5 but assume there are n possible states. Label them as {1,...,n}. Let Ni be independent Poisson processes with parameters λi, for i = 1,...,n. Assume the state Xt
Assume there is a single risky asset with dividend-reinvested price S satisfying dS S = μdt + σ dB1 , where dμ = κ(θ −μ)dt + γ dB2 with σ, κ, θ, and γ being constants and where B1 and B2
For the model of Section 23.4, derive the ODE that the market price-dividend ratio satisfies.
In the model of Section 23.4, assume ζ = 0 (that is, C and μ are locally uncorrelated).(a) Show that β < 1. This implies that μˆ has a lower standard deviation than does μ. Intuitively, why is
In the model of Section 23.4, define X = μ −θγ , Y =σ2 − θσt + logCσ .Write down the innovation process, filtering equation, and ODE for the conditional variance of X. Explain why these
Consider the model of Section 22.5, but assume there is a continuum of investors indexed by h ∈ [0,1] with possibly differing risk-aversion coefficientsαh and possibly differing error variances
In the economy of Section 22.4, assume the uninformed investors are risk neutral. Find a fully revealing equilibrium, a partially revealing equilibria in which the price reveals ˜s + bz˜ for anyb,
Consider an infinite-horizon version of the model in Section 21.5 in which both investors agree the dividend process is a two-state Markov chain, with states D = 0 and D = 1. Suppose the investors’
Assume all investors have constant relative risk aversion ρ andthe same discount factor δ. Solve the social planning problem in a finite-horizon discrete-time model to show that the social
Suppose each investor h has CARA utility with absolute risk aversion αh.Assume the information in the economy is generated by w˜ m. Assume investor h believes w˜ m is normally distributed with
Derive the PDE in Exercise 15.11 by working under the risk-neutral probability corresponding to M. Use Girsanov’s theorem and the fact that the expected rate of return of the asset under the
In the Markov model of Section 15.1, consider valuing an asset that pays an infinite stream of dividends D, where dDt Dt= γ (Xt)dt +θ (Xt)dBt for functions γ and θ. Assume there is no bubble.
Specifically,(a) Derive (15.1) from the fact that E[df]f = r dt −dM Mdf f.(b) Derive (15.2) from the fact that g dt +E[df]f = r dt −dM Mdf f.
Derive the fundamental PDEs in Section 15.1 from the fact that the expected rate of return of an asset must equal its required rate of return, as discussed at the end of Section
Assume (15.25) holds with strict inequality. Repeating the argument at the end of Section 15.5 shows that, for any date t, Et Tt MuCu du = Et Tt Mˆ uCˆ u du + XtEt Tt
Considerthe continuous-time portfolio choice problem with exponentially decaying habit described in Section
This exercise verifies that, as asserted in Section 15.3, condition (15.9) is sufficient for MW to be a martingale. Let M be an SDF process such that MR is a martingale. Define B∗ by (15.8). Let W
Assume the market is complete, and let M denote the unique SDF process.Assume MR is a martingale. Consider T < ∞, and define the probability Q in terms of ξT = MTRT by (15.5). Define B∗ by
Suppose MdRd is a martingale and define the risk-neutral probability corresponding to Md. Assume MdXRf is also a martingale. Show that dX X = (r d − r f)dt +σx dB∗ , where B∗ is a Brownian
Adopt the notation of Exercise
Assume two dividend processes Di are independent geometric Brownian motions:dDi Di= μi dt + σi dBi for constantsμi and σi and independent Brownian motionsBi. DefineCt = D1t+D2t. Assume Mt def=
Assume aggregate consumption C and its expected growth rate μ satisfy dC C = μdt +σ dB1 dμ = κ(θ − μ)dt +γρ dB1 + 1− ρ2dB2 for constants σ, κ, θ, ρ, and γ and independent
Assume the investor has constant relative risk aversion ρ. Define optimal consumption C and terminal wealth WT from the first-order conditions (14.7), and define Wt from (14.5).(a) Show that Wt =
Adopt the assumptions of Section
Assume ertMt is a martingale.(a) Using Girsanov’s theorem, show that dD D = (μ− σ λ)dt + σ dB∗ , where B∗ is a Brownian motion under the risk-neutral probability associated with M.(b)
Adopt the assumptions of Part (a) of Exercise
Assume there is a representative investor with constant relative risk aversion ρ. Assume aggregate consumption C satisfies dC C = α(X)dt +θ (X)dB for functions α and θ, where X is the Markov
Let M be an SDF process and Y a labor income process. Assume ET 0Mt|Yt|dt< ∞for each finite T. The intertemporal budget constraint is dW = rW dt + φ(μ− rι)dt +Y dt − Cdt + φσ dB.
This exercise demonstrates the equivalence between the intertemporal and static budget constraints in the presence of labor income when the investor can borrow against the income, as asserted in
Consider an investor with power utility and a finite horizon. Assume the capital market line is constant and the investor is constrained to always have nonnegative wealth. Let M = Mp. Calculate the
Consider an investor with power utility and an infinite horizon. Assume the capital market line is constant, so we can write J(w) instead of J(x,w)for the value function.(a) Defineξ = δ −(1−
Consider an investor with log utility and an infinite horizon. Assume the capital market line is constant, so we can write J(w) instead of J(x,w) for the value function.(a) Show that J(w) = logwδ +
Consider an investor with initial wealth W0 > 0 who seeks to maximize E[logWT]. Assume ET 0|rt|dt< ∞ and ET 0κ2 t dt< ∞, where κ denotes the maximum Sharpe ratio. Assume portfolio processes
For each investor h = 1,...,H, let πh denote the optimal portfolio presented in (14.24). Using the notation of Section 14.6, set τh = 1/αh for each investor h. Then, (14.24) implies Whπh =
Assume the continuous-time CAPM holds:(μi −r)dt = ρdSi SidWm Wmfor each asset i, where Wm denotes the value of the market portfolio, ρ = αWm, and α denotes the aggregate absolute risk
Suppose W, C, and π satisfy the intertemporal budget constraint (13.38).Define W†t = Wt + Rt t0 Cs Rs ds.Note: This means consumption is reinvested in the money market account rather than in the
Suppose W > 0, C, and π satisfy the intertemporal budget constraint(13.38). Define the consumption-reinvested wealth process W† by (13.43).(a) Show that W† satisfies the intertemporal budget
For a local martingale Y satisfying dY/Y = θdB for some stochastic process θ, Novikov’s condition is that Eexp1 2T 0θθ dt < ∞.Under this condition, Y is a martingale on [0,T]. Consider
Let r d denote the instantaneous risk-free rate in the domestic currency, and let Rd denote the domestic currency price of the domestic money market account:Rd t = expt 0r ds ds.As in Section 8.6,
Consider an asset paying dividends D over an infinite horizon. Assume D is a geometric Brownian motion:dD D = μdt + σ dB for constants μ and σ and a Brownian motion B. Assume the instantaneous
For constants δ > 0 and ρ > 0, assume Mt def= e−δtCt C0−ρis an SDF process, where C denotes aggregate consumption. Assume that dC C = α dt +θdB (13.56)for stochastic processes α and
Let dMi = θi dBi for i = 1,2 and Brownian motions B1 and B2. Supposeθ1 and θ2 satisfy condition (12.5), so M1 and M2 are finite-variance martingales.Consider discrete dates s = t0 < t1 < ··· <
Suppose dMi = θ dBi for i = 1,2, where Bi is a Brownian motion and θi satisfies (12.5), so Mi is a finite-variance martingale.(a) Show that the conditional variance formula (12.28) is equivalent to
The process can be applied for more than two Brownian motions.
Let B1 and B2 be independent Brownian motions and dZ def=dZ1 dZ2=σ11 σ12σ21 σ22dB1 dB2 def= AdB for stochastic processes σij, where A is the matrix of the σij.(a) Calculatea, b, and c
Let ρ = ± 1 be the correlation process of two Brownian motions B1 and B2. Set Bˆ 1 = B1. Define Bˆ 2 by Bˆ 20 = 0 and dBˆ 2 = 1√1 −ρ2 (dB2 −ρ dB1).Show that Bˆ 1 and Bˆ 2 are
Let B1 and B2 be independent Brownian motions and let ρ ∈ [−1,1]. Set Bˆ 1 = B1. Define Bˆ 2 by Bˆ 20 = 0 and dBˆ 2 = ρ dB1 + √1− ρ2 dB2.(a) Use Levy’s theorem to show that Bˆ 2 is
Let B be a Brownian motion. Define Yt = B2 t −t.(a) Use the fact that a Brownian motion has independent zero-mean increments with variance equal to the length of the time interval to show that Y is
Suppose dS/S = μdt + σ dB for constants μ and σ and a Brownian motion B. Let r be a constant. Consider a wealth process W as defined in Section 12.2:dW W = (1 −π )r dt + πdS S , where π is a
LetX be an Ornstein-Uhlenbeck process with a long-run mean of zero; that is, dX = −κX dt + σ dB for constants κ and σ. Set Y = X2. Show that dY = ˆκ(θˆ − Y)dt + ˆσ√Y dB for constants
Assume Xt = θ − e−κt(θ −X0) +σt 0e−κ(t−s)dBs for a Brownian motion B and constants θ and κ. Show that dX = κ(θ − X)dt +σ dB.Note: The process X is called an Ornstein-Uhlenbeck
Assume S is a geometric Brownian motion:dS S = μdt +σ dB for constants μ and σ and a Brownian motion B.(a) Show that vartSt+1 St= e 2μ*eσ 2−1+.Hint: Compare Exercise 1.7.(b) Use the result
Assume X1 and X2 are strictly positive Itô processes. Use Itô’s formula to derive the following:(a) Define Yt = X1tX2t. Show that dY Y = dX1 X1+dX2 X2+dX1 X1dX2 X2.(b) Define Yt = X1t/X2t.
Assume X is an Itô process. Use Itô’s formula to derive the following:(a) Define Yt = eXt . Show that dY Y = dX +1 2(dX)2 .(b) Assume X is strictly positive. Define Yt = logXt. Show that dY = dX
Simulate the path of a Brownian motion over a year (using your favorite programming language or Excel) by simulating N standard normal random variables zi and calculating Bti = Bti−1 +zi√t for i
Take T = 2.Suppose consumption C0 is known at date 0 (before any coins are tossed).Assume the power certainty equivalent and the CES aggregator.(a) Assume two coins are tossed at date 0 determining
Consider consumption processes (ii) and (iii) in Section
Let C denote aggregate consumption, and assume consumption growth Ct+1/Ct is IID. Assume Mt+1 Mt def= δCt+1 Ct−ρ+αCt+1 Ct−γis an SDF process for some δ, ρ, α, and γ . For α > 0,
Calculate the expected market return and the risk-free return in the rare disasters model when(a) bt+1 is uniformly distributed on [0,b∗] for some constant b∗ < 1.(b) bt+1 = b∗/2 with
Calculate the unconditional standard deviation of Rft in the catching up with the Joneses model.
In the setting of Exercise 8.1, let P denote the physical probability and assume EPt+1 +Dt+1 Pt= Rf .Suppose there is an infinite horizon. Show that there is no probability Q on the space of
In the model of Exercise 8.1, calculate the unique risk-neutral probability for any given horizon T < ∞, and show that the risk-neutral probability of any path depends on νt and the parameters Rf
Consider an investor with an infinite horizon in a market with a constant risk-free return and a single risky asset with returns Rt = 1νeμ+σ εt for a sequence of independent standard normals εt
Consider the infinite-horizon model with IID returns and no labor income.Denote the investor’s utility function by u(c). Let Jˆ be a function that solves the Bellman equation. Assume (9.39) holds.
Consider the infinite-horizon model with IID returns and no labor income.Denote the investor’s utility function by u(c).(a) Case B: Assume there is a constant K such that −K ≤ u(c) ≤ K for
Suppose there is a single asset that is risk free with return Rf > 1. Consider an investor with an infinite horizon, utility function u(c) =c, and discount factor δ = 1/Rf . Suppose she is
Consider the finite-horizon model with consumption at each date, IID returns, and no labor income. Suppose one of the assets is risk free with return Rf . Let R denote the vector of risky asset
Consider the finite-horizon model with consumption at each date, state variables Xt, log utility, and no labor income. Assume maxπ Et [log(πRt+1)] is finite for each t with probability 1. The
Consider the infinite-horizon model with IID returns and no labor income.Assume max π E[logπRt+1] < ∞.(a) Calculate the unique constant γ such that J(w) = logw 1− δ + γsolves the Bellman
Consider any T < ∞, and suppose Ct is a marketed date–t payoff, for t =0,...,T. Show that there exists a wealth process W and portfolio process π such that C, W, and π satisfy Wt+1 = (Wt
Suppose the return vectors R1,R2,... are independent and identically distributed. Let w be a positive constant. Assume maxπ E[log(πRt)] > −∞ and let π∗ be a solution to maxπ
Suppose there is a risk-free asset with constant return Rf each period.Suppose there is a single risky asset with dividends given by Dt+1 =λhDt with probability 1/2 ,λDt with probability 1/2
In the two-period economy illustrated in Figs. 2.1 and 2.2 consider an asset paying a dividend at time 2 given by D2 =⎧⎪⎨⎪⎩0, for ω = 3, 5, for ω ∈ {1, 2, 4}, 10, for ω ∈ {5, 6}.(a)
Assume ε˜i ≥ −γ with probability 1, for some constant γ . Via the following steps, show that|δi| ≤αw0πi exp(αγ w0πi)var(ε˜i)Rf.(a) Show thatδi = E[exp(−αw˜
Suppose there is a risk-free asset in zero net supply and the risky asset returns have a statistical factor structure R˜i = ai +bi F˜ + ˜εi , wherethe ε˜i have zero means and are independent
Use the results on affine sharing rules in Section 4.4 to establish (7.8) and(7.9) in Section 7.2.
Showthat if uh0 and uh1 are concave for each h, thenthe social planner’s utility functions u0 and u1 are concave.
Assume in (7.16) that logR˜ and log(˜c1/c0) are joint normally distributed.Specifically, let logR˜ = ˜y and log(˜c1/c0) = ˜z with E[˜y] = μy, var(y˜) = σ2 y , E[˜z] = μ, var(z˜) = σ2,
Assume there is a representative investor with utility function u. The first-order condition E[u(R˜ m)(R˜ 1 − R˜ 2)] = 0 must hold for all returns R˜ 1 and R˜ 2. Assume there is a risk-free
Assume there is a representative investor with constant relative risk aversion ρ. Assume there is a risk-free asset and the market is complete. Use the fact that R˜ p and Rf span the mean-variance
Assume there is a risk-free asset, and let m˜ be an SDF.(a) Show that each return R˜ satisfies E[R˜] −Rf = var∗(R˜)Rf− cov(m˜ R˜,R˜), where var∗ denotes variance under the risk-neutral
This is because of two offsetting factors:Both the risk premium of the market and the volatility of the market are higher in the data than the model would predict, given reasonable values of δ and
Note that (7.36) implies risk aversion must be larger if consumption volatility is smaller or the maximum Sharpe ratio is larger. Also, using the approximation log(1+ x) ≈ x, the lower bound on ρ
Assume there is a risk-free asset and a representative investor with power utility, so (7.15) is an SDF. Let z˜ = log(c˜1/c0) and assume z˜ is normally distributed with mean μ and variance σ2.
Assume there is a representative investor with quadratic utility u(w) =−(ζ −w)2. Assume E[ ˜wm] = ζ . Show that λ in the CAPM (6.11) equals var(w˜ m)E[τ (w˜ m)],where τ (w) denotes the
Suppose there is no risk-free asset and the minimum-variance return is different from the constant-mimicking return, that is, bm = bc. From Section 5.5, we know that there is an SDF that is an affine
So, it must be that Rz = 0 in (6.36).Calculate Rz to demonstrate this.
Suppose there is no risk-free asset and the minimum-variance return is different from the constant-mimicking return, that is, bm = bc. From Section 6.2, we know there is a factor model with the
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