Questions and Answers of Equity Asset Valuation

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
Assume there are H investors with CARA utility and the same absolute risk aversion α. Assume there is a risk-free asset. Assume there are two risky assets with payoffs x˜i that are joint normally
Suppose two assets satisfy a statistical factor model with a single factor:R˜ 1 = E[R˜ 1] + ˜f + ˜ε1 , R˜ 2 = E[R˜ 2] − ˜f + ˜ε2, where E[˜f] = E[ ˜ε1] = E[˜ε2] = 0, var(˜f) = 1,
Assume the asset returns R˜i for i = 1,...,n satisfy R˜i = E[R˜i] +Cov(F˜,R˜i)−1 F (F˜ − E[F˜])+ ˜εi , where each ε˜i is mean independent of the factors F˜, that is, E[˜εi |F˜] =
Show that the CAPM holds with R ≤ Rz ≤ Rb.
Suppose investors can borrow and lend at different rates. Let Rb denote the return on borrowing and R the return on lending. Suppose B/C > Rb >R, where B and C are defined in (5.6). Suppose each
Suppose there is a risk-free asset and suppose Jensen’s alpha in (6.22) is positive. Consider an investor with initial wealth w0 who holds the benchmark portfolio and therefore has terminal wealth
Assume there is a risk-free asset, and assume that a factor model holds in which each factor ˜f1,...,˜fk is an excess return.(a) Show that each return R˜ on the mean-variance frontier equals Rf +k
Assume returns are normally distributed, investors have CARA utility, and there is no labor income. Derive the CAPM from the portfolio formula (2.22), that is, fromφh = 1αh−1(μ− Rfι), where
Assume there exists a return R˜ ∗ that is on the mean-variance frontier and is an affine function of a vector F˜; that is, R˜ ∗ = a + bF˜. Assume either (i) there is a risk-free asset and
Consider the problem of choosing a portfolio π of risky assets, a proportionφb ≥ 0 of initial wealthto borrow, and a proportionφ ≥ 0 of initial wealthto lend to maximize the expected return
Assume there is a risk-free asset.(a) Using the formula (3.45) for m˜ p, compute λ such that R˜ p = λπtangR˜ +(1− λ)Rf .(b) Show that λ in Part (a) is negative when Rf < B/C and positive
If all returns are joint normally distributed, then R˜ p, e˜p, and ε˜ are joint normally distributed in the orthogonal decomposition R˜ = R˜ p + b˜ep + ˜ε of any return R˜ (because R˜ p is
Establish the properties claimed for the risk-free return proxies:(a) Show that var(R˜) ≥ var(R˜ p + bme˜p)for every return R˜.(b) Show that cov(R˜ p,R˜ p +bze˜p) = 0.(c) Prove (5.23),
Write any return R˜ as R˜ p +(R˜ −R˜ p) and use the fact that 1− ˜ep is orthogonal to excess returns—because e˜p represents the expectation operator on the space of excess returns—to
Show that E[R˜ 2] ≥ E[R˜ 2 p] for every return R˜ (thus, R˜ p is the minimum second-moment return). The returns having a given second moment a are the returns satisfying E[R˜ 2] =a, which is
Suppose that the risk-free return is equal to the expected return of the GMV portfolio (Rf = B/C). Show that there is no tangency portfolio. Hint: Show there are no δ and λ satisfyingδ−1(μ−
Assume there is a risk-free asset. Consider an investor with quadratic utility−(w˜ −ζ )2/2 and no labor income.(a) Explain why the result of Exercise 2.5 implies that the investor will choose a
Calculate the GMV portfolio and locate it on Figure 5.1.
Suppose there are two risky assets with means μ1 = 1.08, μ2 = 1.16, standard deviations σ1 = 0.25, σ2 = 0.35, and correlation ρ =
Suppose the payoff of the market portfolio w˜ m has k possible values.Denote these possible values by a1 < ··· < ak. For convenience, suppose ai − ai−1 is the same number for each i.
Consider a model with date–0 endowmentsyh0 and date–0 consumption ch0.Suppose all investors have log utility, a common discount factor δ, and no date–1 labor income. Show that, in a
Assume the investors have time-additive utility and the date–1 allocation solves the social planner’s problem (4.1). Using the first-order condition (3.9), show that the equilibrium allocation is
Consider an economy with date–0 consumption as in Section
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
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
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,
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
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
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
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
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
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
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 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.
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
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
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

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