1. Consider a two-period economy where.

Income in period 0 and in period 1 can take two possible values: 1 or 2.

The probability that income is 1 in period 0 is 1/3, i.e., Pr(y0 = 1) = 1/3.

Moreover,

Pr(y₁ = 1|y₀ = 1) = 0.8,

Pr(y₁ = 2|y₀ = 2) = 0.9.

Individuals in this economy all have the same income and the same lifetime utility function

ln(c₀)+ ln(c₁),

where the time discount factor, , is 0.9.

It follows that individuals price any asset according to

p₀ = 0.9 E₀ [y₀/y₁] (Z₁)

in which Z₁ is the gross payoff associated with the asset considered.

(a) Calculate the price of a bond which pays 1 with certainty in period 1 (i.e., Z₁ = 1) when y₀ = 1.

(b) Calculate the price of a bond which pays 1 with certainty in period 1 (i.e., Z₁ = 1), when y₀ = 2.

(c) Why is this bond more expensive when y₀ = 2 than when y₀ = 1?

(d) Calculate the expected net rate of return on this bond.

2. Suppose that in the above economy, there is a corporate bond.

A corporate bond pays 1 in period 1 unless bankruptcy happens.

If bankruptcy happens, the corporate bond pays nothing.

Suppose that bankruptcy happens with

probability =

0.20 if y1 = 1,

0.05 if y1 = 2.

(a) Build a probability tree for the payoff of this corporate bond.

(b) What is the (unconditional) probability that Z₁ = 1?

(c) Calculate the price of this bond and its expected rate of return when y₀ = 1.

(d) Explain why the expected rate of return on this bond differs from that of the above riskless bond when y₀ = 1.