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equity asset valuation

Asset Pricing And Portfolio Choice Theory 2nd Edition Kerry E. Back - Solutions

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. Then, the price of the asset is Pt = Et∞t Mu Mt Du du = DtEt∞t MuDu MtDt du = Dtf(Xt)for some
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 Mue−(a−b)(u−t)du.(Section 15.5 considers t = 0.) Assume power utility: u(c−x) = 1 1−ρ (c−x)1−ρ.Assume the
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 be a positive self-financing wealth process. Define W∗ = W/R.(a) Use Itô’s formula, (13.10),
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 (15.8). Let x be a random variable that depends only on the path of the vector process B∗ up to time T
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 motion under the risk-neutral probability. Note: This is called uncovered interest parity under the
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= e−δt C0 Ct is an SDF process. (This is the MRS for a log-utility investor.) Define Xt = D1t/Ct.(a)
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 Brownian motions B1 and B2.Then, the vector process (C,μ) is Markovian. Assume Mt def= e−δtCt
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 = M−1/ρt f(t,Xt)for some function f .(b) Derive a PDE for f .(c) Explain why the optimal portfolio is
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) Calculating under the risk-neutral probability, show that the asset price is Pt def= Dt r + σ λ −μ
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 process (13.50).(a) Explain why the market price-dividend ratio is a function of Xt.(b) Denote the
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. (14.34)(a) Suppose that (C,W,φ)satisfies the intertemporal budget constraint(14.34), C ≥ 0, and the
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 Section
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 optimal portfolio as follows:(a) Using (14.12), show that, for s > t, Et M1−1/ρs= M1−1/ρt
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− ρ)rρ − (1− ρ)κ2 2ρ2 .Assume (14.26) holds, so ξ > 0. Show that J(w) = ξ −ρ 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 + K solves the HJB equation (14.25), where K = logδδ +r −δ +κ 2/2δ2 .Show that c = δw and π =
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 are constrained to satisfy ET 0πttπt dt< ∞.Recall that this constraint implies ET 0πσ
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 = τh−1(μ− rι) −j=1τhηhj−1σ νj .(a) Deduce thatμ− rι = αWπ +j=1ηjσ νj ,
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 aversion. Define σi = eiei to be the volatility of asset i, as described in Section 13.1, so we have
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 portfolio generating the wealth process as in (13.43).(a) Show that W† satisfies the intertemporal
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 constraint (13.44).(b) Show that W†t − Wt = W†t t0 Cs W†s ds for each t.Hint: Define Y = W/W†
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 Y = MW, where M is an SDF process and W is a self-financing wealth process.(a) Show that dY/Y =
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, let X denote the price of a unit of a foreign currency in units of the domestic currency. Let r f
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 risk-free rate r is constant, and assume there is an SDF process M such thatdD DdM M= −σ λdt
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 θ.(a) Apply Itô’s formula to calculate dM/M.(b) Explain why the result of Part (a) implies that the
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 < ··· < tN = u for some s < u. Show that covs(M1u − M1s,M2u − M2s) = Es⎡⎣N j=1(M1tj − M1tj−1
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 M2 it −t 0(dMis)2 (12.32)being a martingale.(b) Show that the conditional covariance formula
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 with a > 0 and c > 0 such that LL = AA, where L =a 0 b c.(b) Define Bˆ = (Bˆ 1 Bˆ 2) by Bˆi0 =
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 independent Brownian motions. Note: Obviously this reverses the process of the previous exercise. It gives
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 a Brownian motion.(b) Show that ρ is the correlation process of the two Brownian motions Bˆ 1 and
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 a martingale.(b) Apply Itô’s formula to calculate dY and verify condition (12.5) to show that Y
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 constant.(a) By observing that W is a geometric Brownian motion, derive an explicit formula for
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 κˆ, θˆ and σˆ . Note: The squared Ornstein-Uhlenbeck process Y is a special case of the
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 process. Assumingκ > 0, θ is called the long-run or unconditional mean, and κ is the rate of mean
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 of the previous part, the formula (12.22), and the approximation ex ≈ 1+ x to derive approximate
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. Show that dY Y = dX1 X1− dX2 X2−dX1 X1dX2 X2+dX2 X22.
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 X − 1 2dX X2.(c) Assume X is strictly positive. Define Yt = X−λt for a constant λ. Show that
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 = 1,...,N, wheret = 1/N and B0 = 0. (To simulate a standard normal random variable in a cell of an
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 C1 and C2. Calculate the utility U0 of the person before the coins are tossed.(b) Assume a coin is
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, condition (11.62a) of the Constantinides-Duffie model is satisfied. Take δ = 0.99 and ρ = 10 as in
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 probability 1 for some constant b∗ < 1.Explain why the ratio E[Rmt]/Rft is larger in the rare disasters
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 infinite paths that is (a) equivalent to P, and (b) satisfies E∗tPt+1 + Dt+1 Pt= Rf for each t,
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 , k, λh, and λ.
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 and a constant ν—as implied by (10.31) whenthe investor is a representative investor and
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. For arbitrary decisions(Ct,πt), assume E[u(Ct)] andE[Jˆ(Wt)] are finite for each t. Suppose
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 eachc. Show that the transversality condition (9.36) holds.(b) Case N: Assume u(c) ≤ 0 for each c
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 constrained to consume 0 ≤ Ct ≤ Wt.(a) Show that the value function for this problem is J(w) = w.(b) Show
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 returns, let μdenote the expected value of R, and let denote the covariance matrix of R. Assume is
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 value function at date T is JT(x,w) = logw.Defineγt = 1− δT+1−t 1−δ .(a) Show that Jt(x,w) =
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 equation.(b) Show that the transversality condition limT→∞ δTE[J(W∗T)] = 0 holds.(c) Show that
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 −Ct)πtRt+1 (8.26)for t = 0,...,T −1, and CT = WT. Hint: Add up the wealth processes and take a
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 E[log(πRt)].Let W∗ be the wealth process defined by the intertemporal budget constraint(8.1) with πt = π∗
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 ,where λh > λ are constants, and D0 > 0. Suppose the price of the risky asset satisfies Pt = kDt for a
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) What is the expectation at time 0 of D2? What is the expectation at time 1 of D2?Verify that the Law
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˜ m)ε˜i]RfE[exp(−αw˜ m)].(b) Show that δi = E[exp(−αw0πiε˜i)ε˜i]RfE[exp(−αw0πiε˜i)].Hint: Use
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 of each other and of F˜. Assume there is no labor income and there is a representative investor with
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, and corr(y˜, z˜) = γ .(a) Show that μ = −logδ +ργσσy + ρμ− 1 2 ρ2 σ2 − 1 2 σ2 y
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 asset. Consider any return R˜. By orthogonal projection, we have R˜ −Rf = α + β(R˜ m −Rf) +
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 frontier to show that each mean-variance efficient return is of the form a−bR˜ −ρm for b > 0.
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 probability corresponding to m˜ .(b) Assume there is a representative investor with constant
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 ρ.(d) Use δ = 0.99 and ρ = 10 and the Mehra-Prescott data on the mean and standard deviation of
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 in (7.36) is approximately κ/σ, and, using the approximation ex ≈ 1+ x, the upper bound on κ is
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. Let κ denote the maximum Sharpe ratio of all portfolios.(a) Use the Hansen-Jagannathan bound (3.35)
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 coefficient of risk tolerance of the representative investor at wealth level w. (Thus, the risk
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 function of the minimum-variance return:m˜ = γ + β(R˜ p +bme˜p) (6.37)for some γ and β. From
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 constant-mimicking return as the factor:E[R˜] = Rz +ψ cov(R˜,R˜ p + bce˜p) (6.36)for every return
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 distributed with mean vector μand nonsingular covariance matrix . Assume HU investors are unaware
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, cov(˜f ,ε˜1) = cov(˜f ,ε˜2) = 0, and cov(ε˜1,ε˜2) = 0. Assume var(ε˜1) = var(ε˜2) = σ2.
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˜] = 0 (note it is not being assumed that cov(ε˜i, ε˜j) = 0). Assume markets are complete and the
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 investor chooses a mean-variance efficient portfolio, as described in Exercise
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 w0R˜b. Assume E[u(w0R˜ b)] > 0.Consider the return R˜ 1 = R˜ +(1− β)(R˜ b −Rf) = R˜ b
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 j=1βj˜fj for some β1,...,βk. In other words, show that the risk-free return and the factors
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 αh denotes the absolute risk aversion of investor h. Show that the price of risk is αw0 var(R˜
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 R˜ ∗ = Rf , or (ii) there is no risk-free asset and R˜ ∗ is different from the GMV return. Show
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 πμ + φR − φbRb subject to the constraints(1/2)ππ ≤ k and ιπ +φ −φb = 1. Assume
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 when Rf > B/C. Note: This shows that R˜ p is on the inefficient part of the frontier, because the
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 a return and e˜p and ε˜ are excess returns). Assuming all returns are joint normally
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), showing that R˜ p +bce˜p represents the constant bc times the expectation operator on the space of
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 x˜ def= 1 E[R˜ p](1 − ˜ep)is an SDF. When there is a risk-free asset, x˜, being
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 equivalent to var(R˜) +E[R˜]2 = a ;thus, they plot on the circle x2 +y2 = a in (standard deviation,
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− Rfι) = λπmu +(1− λ)πgmv .Recall that we are assuming μ is not a scalar multiple of ι.
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 portfolio on the mean-variance frontier.(b) Under what circumstances will the investor 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. Suppose there is a risk-free asset with payoff equal to 1. Suppose there are k −1 call options on the
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 competitive equilibrium, the date–0 value of the market portfolio is δH h=1 ch0.
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 Pareto optimal. Hint: Using the first-order condition(4.4) with η˜ = ˜η1, show that(∀h)
Consider an economy with date–0 consumption as in Section

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