Solutions Manual – Asset Pricing and Portfolio Choice Theory, 2nd Edition by Kerry Back | Complete Step‑by‑Step Solutions for All Chapters, Dynamic Asset Pricing, Stochastic Calculus, Portfolio Optimization & Equilibrium Models (PDF)

SOLUTIONS MANUAL ASSET PRICING AND PORTFOLIO CHOICE THEORY
SOLUTIONS MANUAL BY KERRY BACK 2ND EDITION

, Asset Pricing and Portfolio Choice Theory Kerry Back



Table Of Contents




Part I Single-Period Models

1 Utility Functions and Risk Aversion Coefficients.......................................................... 3

2 Portfolio Choice and Stochastic Discount Factors........................................................15

3 Equilibrium and Efficiency........................................................................................................25

4 Arbitrage and Stochastic Discount Factors..................................................................... 35

5 Mean-Variance Analysis..............................................................................................................41

6 Beta Pricing Models.................................................................................................................. 49

7 Representative Investors............................................................................................................. 59


Part II Dynamic Models

8 Dynamic Securities Markets................................................................................................... 69

9 Portfolio Choice by Dynamic Programming...................................................................77

10 Conditional Beta Pricing Models.........................................................................................89

11 Some Dynamic Equilibrium Models.................................................................................... 91

12 Brownian Motion and Stochastic Calculus....................................................................... 95

13 Continuous-Time Securities Markets and SDF Processes....................................... 111

,4 Contents


14 Continuous-Time Portfolio Choice and Beta Pricing............................................... 127


Part III Derivative Securities

15 Option Pricing.............................................................................................................................. 145

16 Forwards, Futures, and More Option Pricing................................................................ 159

17 Term Structure Models............................................................................................................ 175


Part IV Topics

18 Heterogeneous Priors................................................................................................................. 197

19 Asymmetric Information......................................................................................................... 203

20 Alternative Preferences in Single-Period Models........................................................209

21 Alternative Preferences in Dynamic Models.................................................................. 215

22 Production Models......................................................................................................................225

, Part I

Single-Period Models

1

Utility Functions and Risk Aversion Coefficients

1.1. Calculate the risk tolerance of each of the five special utility functions in Section 1.7 to verify
the formulas given in the text.



ur(w) 1
u(w) = —e—αw ⇒ ur(w) = αe—αw , urr(w) = —α2e—αw , — = .
u rr(w) α
1 1 ur(w)
u(w) = log w ⇒ ur(w) = , urr(w) = — , — = w.
w w2 urr(w) r
1 u (w) w
u(w) = w1—ρ ⇒ ur(w) = w—ρ , urr(w) = —ρw—ρ—1 , — = .
1—ρ urr(w)r ρ
1 1 u (w)
u(w) = log(w — ⇣ ) ⇒ u (w) =
r
, u (w) = —
rr
, — =w—⇣ .
✓ ◆ ✓ ◆
w — (w — ⇣ )2 urr(w)
✓ ◆
⇣
ρ w — ⇣ 1—ρ w — ⇣ —ρ w — ⇣ —ρ—1
u(w) = ⇒ u (w) =
r
, u (w) = —
rr
,
1—ρ ρ ρ ρ
ur(w) w — ⇣
— rr = .
u (w) ρ

1.2. Let ε˜ be a random variable with zero mean and variance equal to 1. Let π(σ) be the risk
premium for the gamble σε˜ at wealth w, meaning

u(w — π(σ)) = E [u(w + σε˜)] .

Assuming π is a sufficiently differentiable function, we have the Taylor series approximation
1 rr
π(σ) ≈ π(0) + πr(0)σ + π (0)σ2
2
for small σ. Obviously, π(0) = 0. Assuming differentiation and expectation can be interchanged,
differentiate both sides of (1.13) to show that πr(0) = 0 and πrr(0) is the coefficient of absolute risk
aversion.