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

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What are the potential shortfalls of using comparables to value a private company?
After Coinbase’s 2013 financing, what was the total equity ownership by venture firms?
Describe the difference between pre-money and post-money valuation. In what settings are investors most likely to focus on pre-money valuation, and when on post-money valuation?
What are the difficulties and challenges that investors face in valuing early-stage companies?
Why do so many VC firms think their follow-on decision process needs to be changed? What do you think would be the best approach? Why?
Give three reasons why a private equity firm would syndicate a deal. How do the reasons differ for venture and buyout firms?
Why would an entrepreneur backed by a top-tier firm be more likely to start another venture?
Discuss the impact of timing and managerial skill in entrepreneurial success. Why would the price of a venture started by a serial entrepreneur with timing success tend not to be particularly high?
What are the three risks that Kaplan and Strömberg describe? Why would external risks be viewed as reasons to invest more often than reasons not to?
Why would a general partner want to do most of the reference checks themselves?
Why might an investor prefer a CEO who had failed in the past?
Why only use twelve slides in a company presentation?
How does an entrepreneur find an investor?
What are the channels for an investor to seek out an investment opportunity?
What are the disadvantages and potential risks of an auction process for both the investors and the seller? How might either party mitigate them?
What are the key points to include in a pitch to a potential investor?
How might a sector or geographic focus for a fund be beneficial in attracting a proprietary deal?
Why do investors prefer proprietary deals? What would make these appeal to an entrepreneur or a seller?
In Question 14, what if the fund is structured so that instead of having to return drawn-down capital, the manager only needs to return capital that was actually invested in companies (i.e., not including management fees). How would the investment multiple and IRR change?
If a firm with a $2 billion fund charges a 2 percent management fee for eight years, and 20 percent carried interest on all profits, and earns a 3.0× return on its investments, what is the true multiple of investment and IRR for the LP? (Assume that investments are made equally at the beginning of
Why have larger private equity firms (in terms of assets under management) come under scrutiny regarding their management fee structure?
How does the use of leverage vary among the different types of private equity firms?
What is the purpose of a “concentration limit”?
Compare and contrast the traditional fund, which has a finite life, with the evergreen fund.
In the limited partnership agreement, how are the interests of the LP and GP aligned?
Why have FOFs received criticism?
What is the primary role of a FOF and what type of investor is likely to invest in one?
Which type of LP might be inclined to increase their private equity investment allocation going forward? Why? Under what conditions might they want to decrease private equity exposure?
Why would a corporation, like Salesforce, want to have a direct venture investing operation? Would that rationale change for a consumer-focused company like Kellogg?
How might a GP be successful in raising a first-time fund without a track record?
What are the four major similarities for private equity transactions?
What are the advantages of setting up a private equity firm as an LLC?
What are the major subcategories of private equity and how do their investment approaches differ?
Use the dynamics (20.25) of Y and the definition (20.30) of X to verify the formula (20.35) for the risk of a stock return in the model of perfect competition in Section 20.5.
As in Section 20.5, assume dZ Z = μz dt + σz dB∗for constants μz and σz and a Brownian motion B∗ under the risk-neutral probability. Match industry supply (20.23) to industry demand Qt = (Zt/Pt)εto compute the equilibrium output price Pt. Define Yt = cπ P1/αt . Compute constants δ, σ,
Compute constants cq and cπ such that, at the optimal , Akα1−α = cqp(1−α)/αk , pAkα1−α − w = cπ p1/αk .
Consider the optimization problem max pAkα1−α −wthat arises in the perfect competition model of Section
In the Leland (1998) model, the firm issues at date 0 debt with principal P that pays aggregate coupons C. Prior to bankruptcy, a fraction mdt of the outstanding debt is called each instant and redeemed at par. Therefore, e−mtP of principal remains outstanding at each t > 0 from the debt issued
Repeat the previous exercise for the value of receiving κSt until S hits a boundary, for a constant κ > 0. In lieu of Part (b) of the previous exercise, show that f(s) = κs/δ satisfies the ODE.
Consider the value f(St) of receiving a cash flow c per unit of time until the asset price S hits a boundary s∗. Assume the dividend yield, volatility, and risk-free rate are constant. The value is derived in Section 19.2 by reducing the valuation problem to valuing the receipt of $1 at the
Derive the value of tax payments (19.41) and the value of issuance costs(19.42) in the model of dynamic capital structure.
Explain how (19.6), (19.17), and (19.19) are used to compute the values(19.22)–(19.25) of corporate claims in the Leland model.
Calculate the value of a perpetual call that is knocked out when the underlying asset price falls to a boundary sL. What is the optimal exercise boundary for the call? How doesthe optimal boundary comparetothe optimal boundary when there is no knock-out feature and why? Answerthe same questions for
Assume the short rate is rt = ˆrt +g(t), where drˆ = −κrˆdt + σ dB∗and g(·) is chosen to fit the current yield curve.(a) Consider a forward contract maturing at T on a discount bond maturing at u > T. Let Ft denote the forward price for t ≤ T. What is the volatility of dFt/Ft?(b) What
Assume the short rate is rt = ˆrt + g(t), where drˆ = −κrˆdt + σ dB∗for constants κ and σ and g(·) is chosen to fit the current yield curve.(a) Calculate the forward rates fs(u) using the Vasicek bond pricing formula.(b) Calculate αs(u) and σs(u)such that, as s changes, dfs(u) =
Consider the Vasicek model with time-dependent parameters:drt = κ(t)θ (t) −rtdt + σ (t)dB∗t , (18.51)where B∗ is a Brownian motion under a risk-neutral probability. Define rˆt = exp−t 0κ(s)dsr0 +t 0exp−t uκ(s)dsσ (u)dB∗u (18.52a)g(t) =t 0exp−t uκ(s)dsκ(u)θ (u)du.
Assume Mt = exp(X1t +(X2t − a)2)is an SDF process, where dX1 = μdt + σ dB1t , dX2 = −κX2t dt + φ dB2t , with μ, σ, κ, and φ being constants and with B1 and B2 being independent Brownian motions under the physical probability.(a) Derive dM/M, and deduce that r is a quadratic function of
Assumedr dY= κ(θ −r)γ (φ −Y)dt +σ 0 0 η 1 0 0 √YdB∗1 dB∗2(18.50)for constants κ, θ, σ, γ , φ, and η, where the B∗i are independent Brownian motions under a risk-neutral probability.(a) Given the completely affine price of risk specification (18.12b), where X = (r
Assume there is an SDF process M with dM M = −r dt −S(Xt)λ +S(Xt)−1ΛXtdB+dεε , (18.49)where B is a vector of independent Brownian motions under the physical probability, ε is a local martingale uncorrelated with B, S(X) is a diagonal matrix the squared elements of which are affine
Assume r = X1 + X2, where the Xi are independent square-root processes under the physical probability; that is, dXi = ˆκi(θˆi −Xi)dt +σi√Xi dBi for constants κˆi, θˆi, and σi, wherethe Bi are independent Brownian motions under the physical probability. Assume there is an SDF process M
Assume the short rate is an Ornstein-Uhlenbeck process under the physical probability, that is, dr = ˆκ(θˆ − r)dt +σ dB, for constants κˆ, θˆ, and σ, where B is a Brownian motion under the physical probability. Assume there is an SDF process M with dM M = −r dt −λdB+dεε , where
Suppose r is the sum of two independent square-root processes X1 and X2.Define Zt = σ2 1 X1t +σ2 2 X2t. Note that the instantaneous variance of r = X1 +X2 isσ1√X1t dB∗1t +σ2√X2t dB∗2t2= Zt dt .Assume σ1 = σ2. Show thatdr dZ= dt +Kr Zdt +S(r,Y)dB∗for a constant vector and
Consider a two-factor Gaussian affine model. Show that the two factors can be taken to be the short rate and its drift in the sense that drt = Zt dt + σ dBˆ ∗1t , (18.48a)dZt = (a+ brt +cZt)dt + ζ1 dBˆ ∗1t + ζ2 dBˆ ∗2t (18.48b)for constants σ,a, b,c, ζ1, and ζ2 and independent
Consider a single-factor affine model.(a) Use the fact that the short rate r is an affine function of the state variable to show that dr = φ dt −κr dt + α + βr dB∗ (18.46)for constants φ, κ, α, and β.(b) Assume β > 0 in (18.46). Show that r is a translation of a square-root
In the Vasicek model, set f(t,r) = exp(−α(T −t) −β(T −t)r) for fixed T.(a) Show that f satisfies the fundamental PDE and the boundary condition f(T,r) = 1 if and only ifβ+κβ = 1 ,α− κθβ +1 2σ2β2 = 0 , and α(0) = β(0) = 0.(b) Verify that the formula (18.18) for yields with
Consider an American call option with strike K on an asset that pays a single known discrete dividend x at a known date T < u, where u is the date the option expires. Assume the asset price S drops by x when it goes ex-dividend at date T (i.e., ST = limt↑T St −x) and otherwise is an Itô
In the Heston model (17.10), define Y1 = V/γ 2 and Y2 = logS− ρV/γ .(a) Derive the constants ai, bij, and β such thatdY1 dY2=a1 a2dt +b11 b12 b21 b22Y1 Y2dt +√Y1 0 0 √βY1dB1 dB2.(b) Consider a price of risk specificationλ1t = c10 +c11Y1t √Y1t,λ2t = c20 + c21Y1t √βY1t,
Set Vt = logσt, where σt is the volatility of a non-dividend-paying asset with price S. Assume dSt St= μt dt +σt dB1t , dVt = κ(θ −Vt)dt + γρ dB1t + 1− ρ2 dB2t, where μ, κ, θ, γ , and ρ are constants and B1 and B2 are independent Brownian motions under the physical probability
Let S denote the price of a non-dividend-paying asset. Assume dSt St= μt dt + σt dB1t dσt = φ(σt)dt + γ (σt)dB2t for some functions φ(·) and γ (·), where B1 and B2 are independent Brownian motions andμ may be a stochastic process. Assume the price of risk for B2 equalsλ(σt) for some
Consider a European call option on an asset that pays a single known discrete dividend x at a known date T < u, where u is the date the option expires.Assume the asset price S drops by x when it goes ex-dividend at date T (i.e., ST = limt↑T St − x) and otherwise is an Itô process. Suppose
Suppose the price S of a non-dividend-paying asset has a constant volatility σ. Assume the volatility at date t of a discount bond maturing at T > t isφκ1−e−κ(T−t) (17.17)for constants κ > 0 and φ > 0. Assume the discount bond and stock have a constant correlation ρ. Let P denote the
Let A denote the event S1T > S2T.(a) Show that, for i = 1,2, E[MTSiT1A] = Si0 probSi(A).Conclude that the value at date 0 of an option to exchange asset 2 for asset 1 at date T is S10 probS1 (A)− S20 probS2 (A).(b) Define Y = S2/S1. Show that dlogY = −1 2σ2 dt + σ dB∗ , where B∗ is a
Adopt the assumptions of Exercise
Suppose the prices of two non-dividend-paying assets are given by dSi Si= μi dt + σi dBi , where the Bi are Brownian motions with correlation ρ. The μi, σi, and ρ can be stochastic processes. However, assume the volatility of S1/S2 is a constant σ.Assume there is an SDF process such that MS1
Suppose V∗ is a self-financing wealth process with risky asset price S∗ =S1/S2 and money market price R∗ = 1, meaning dV∗V∗ = π∗ dS∗S∗ + (1−π∗)dR∗R∗for some π∗. Define V = S2V∗. Show that V is a self-financing wealth process with risky asset prices S1 and S2 and no
Consider a forward contract on an asset that pays a single known discrete dividend x at a known date T < u, where u is the date the forward matures.Suppose there are traded discount bonds maturing at T and u. Let S denote the price of the asset. Prove the following spot-forward parity formula for t
Calculate the value of a call on a put assuming a constant risk-free rate and a constant volatility for the underlying asset price.
A compound option is an option on an option. Suppose there is a constant risk-free rate, and the underlying asset price has a constant volatility. Consider a European call option with strike K maturing at T. Consider now a European option maturing atT S∗, and it is out of the money otherwise.
This exercise verifies the assertion (16.6) regarding the dynamics of an SDF process. Suppose the information in the economy is given by independent Brownian motions B1,...,Bk. Consider a non-dividend-paying asset with price S satisfying dS S = μdt +k i=1σi dBi for stochastic processes μ and
Consider a second risky non-dividend-paying asset with price Z. Assume dZ Z = μz dt + σz dBz , where Bz is a Brownian motion under the physical probability. Let ρ denote the correlation process of Bz and B.(a) Define ξt = MtSt = Et[MTST]. Show that dξξ = r dt + σ dB+dM M .(b) Prove
Consider an asset with a constant dividend yield q. Assume the price S of the asset satisfies dS S = (μ −q)dt +σ dB, where B is a Brownian motion under the physical probability, and μ and σ are constants. Consider a European call and a European put with strike K on the asset. Assume the
Calculate the value and delta of a European derivative security that pays S2 T at date T.
Assume constant r and σ and the “no arbitrage” assumptions of Section
Use put-call parity to show that a European put and a European call with the same strike and time to maturity have the same gamma.
Calculate the theta and gamma for the Black-Scholes call option formula and verify that it satisfies the fundamental PDE.
A butterfly spread consists of a long call with strike K − , two short calls with strikes K, and a long call with strike K + , for > 0.(a) Draw the payoff diagram of a butterfly spread. What is it a bet on?(b) As explained in Section 16.1, selling straddles is a way to sell volatility, but it
A bull spread consists of a long call with strike K and a short call with strike K > K. A collar consists of a long put with strike K and a short call with strike K > K. Draw the payoff diagram (meaning the value at maturity as a function of the underlying asset price, as in Figure 16.1) of a
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 aboutthe informativeness of a signal.Assume the investor is a representative investor, there is a
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 = logξ .(a) Show that θ satisfies the equation θ = μ+1 2 (1− ρ)σ2 +1 1− ρlog ⎡⎣1 +β N
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 with at least one of these being a strict inequality.(b) Show thatg in (25.13) is strictly monotone
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 differentiable. By differentiating v(w −π(σ ))E[λ(w+ σε)˜ ] = E[λ(w+σε)˜ v(w+ σε)˜ ], assuming
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 $6,000 99.9% chance of $0 .Show that an expected utility maximizer who prefers A to B must also
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 $0 .(a) Show that an expected utility maximizer who prefers A to B must also prefer C to D.(b) Show
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 = σv/σz. Show that the strategies P0 = eμ+ 1 2 σ2 vdPt = λPt dYt dXt = (logv˜ − μ)/λ− Yt 1− t
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 signals. Each buyer knows her own signal but does not know the signal of the other buyer. Assume 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 value of the other buyer.(a) Assume the auction is conducted as a first-price auction. Show that it
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. Suppose there is no information in market orders, which are of a unit size. Assume that the future value
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 evolves as dXt = n i=1(i −Xt−)dNit .(a) Suppose the economy is in state j just before time t
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 are Brownian motions with constant correlation ρ. Assume the risk-free rate is constant. Assume an
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 this the case?(b) Show that β is a decreasing function of κσ. Intuitively, why should the
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 imply (23.43)–(23.46).
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 var(ε˜h). Suppose, for someb, that each investor observes x˜ + by˜ in addition to her private
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, and a completely unrevealing equilibrium (an equilibrium in which the price is constant rather than
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’ beliefs Ph satisfy, for all t ≥ 0, P1(Dt+1 = 0|Dt = 0) = 1/2 , P1(Dt+1 = 1|Dt = 0) = 1/2 ,
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 planner’s utility is ET t=0δt Zt C1−ρt 1−ρfor some stochastic process Z. Show that Z is a
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 mean μh and variance σ2, where σ is the same for all investors.(a) Show that the Radon-Nikodym
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 risk-neutral probability must be the risk-free rate.

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

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