Questions and Answers of Equity Asset Valuation

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
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
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 =
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
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
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
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
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
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
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
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
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
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
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
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=
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 +
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,
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−
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
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
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 ,
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
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
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
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)−
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.
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
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
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
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
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
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
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
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
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

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