# 经济代写 | 金融代写 | 代做Python | assignment代做 – ECON8037: Financial Economics

### ECON8037: Financial Economics

#### August 15, 2022

EXERCISE 1: (10 Marks)Consider a representative agent who lives for two periods. This agent is endowed with incomey 0 and consumesc 0 and purchases shares at pricepoin the first period. This agent does not receive an endowment in the second period and therefore must use the shares purchased in the first period to fund their consumption in the second period. In the second period, this agent can sell their shares at pricep 1 and they receive a risky dividedd 1 from the asset. The agent enjoys utilityu:R+R+, whereu>0 andu<0 and therefore the agents problem is given by

``````max
{c 0 ,c 1 ,}
``````
``````{u(c 0 ) +E 0 u(c 1 )}
``````

Subject to

``````c 0 +p 0 y 0 and c 1 (d 1 +p 1 )
``````

Where(0,1) is the subjective discount factor (which captures the agents prefer- ence of time), andE 0 is the expectations operator conditional on information available at time 0

Show that the solution of the agents problem delivers the two-period version of the fundamental asset pricing equation, i.e.

``````p 0 =E 0 m 1 g 1
``````

Where

``````m 1 =u
``````
``````(c 1 )
u(c 0 )
``````
``````and g 1 =d 1 +p 1
``````
``````Email: [email protected].
``````

EXERCISE 2: (5 Marks)We can generalise the fundamental asset pricing equation in the following way

``````pt=Etmt+1gt+
``````

Where

``````mt+1=
``````
``````u(ct+1)
u(ct) and gt+1=dt+1+pt+
``````

mt+1is known as the stochastic discount factor. Let:=ct+1/ctbe the growth rate of consumption. Given the following CRRA utility function

``````u(c) = lim 1
``````
``````c^1  1
1
``````

Explain how the correlation between consumption growthandgt+1effects the price of the asset and what this implies about how this agent thinks about risk.

EXERCISE 3: (5 Marks)Consider again the the fundamental asset pricing equa- tion

``````pt=Etmt+1gt+
``````

But, in this case

``````mt+1= and gt+1=dt+1+pt+
``````

Suppose we have a non-random dividend stream i.e. dt=d >0 for allt. Show that the equilibrium asset price in this case is given by

``````p= d
1
``````

Hints: Use this closed form of an infinite horizon geometric series

``````1
1 r
``````
``````= 1 +r+r^2 +...
``````

And also note

``````klimk^1 pt+k= 0
``````

EXERCISE 4: (10 Marks)Again, given the the fundamental asset pricing equation

``````pt=Etmt+1gt+
``````

Again, where

``````mt+1= and gt+1=dt+1+pt+
``````

And defining the price dividend ratiovt:=pt/dt. Consider a growing, non-random dividend processdt+1=dtwhere 0< <1. Show that ifvt=vthat the price dividend ratio can be given by

``````v=
``````
``````
1
``````

Also, if we let= 1 +and let:= 1/1, show that the price is given by the Gordon formula

``````pt=
``````
``````1 +
dt
``````

EXERCISE 5: (10 Marks)Write a Python script that

1. Plots the price given by the Gordon formula as a function ofdt. Suppose that = 0.96 and= 0 .1 (4 Marks)
2. Plots how the Gordon formula price changes forvalues 0. 1 , 0. 2 ,…, 0 .9 as a function ofdt(3 Marks)
3. Plots how the Gordon formula price changes forvalues 0. 91 , 0. 92 ,…, 0 .99 as a function ofdt(3 Marks)

EXERCISE 6: (20 Marks)Consider a consumer that has preferences over con- sumption stream that are ordered by the utility functional

##### X
``````t=
``````
``````tu(ct)
``````

WhereEtis the mathematical expectation conditioned on the consumers timetin- formation.ctis timetconsumption.uis a strictly concave one-period utility function and(0,1) is the subjective discount factor. For the purposes of this problem, let us assume (1 +r)^1 =. The consumer maximises this stream of utility by choos- ing a consumption, borrowing plan{ct,bt+1}t=0subject to the sequence of budget constraints

``````ct+bt=
``````
##### 1
``````1 +rbt+1+yt, t^0
``````

Whereytis an exogenous endowment process,r >0 is a time-invariant risk-free net interest rate,btis one period risk-free debt maturing att. The consumer also faces initial conditionsb 0 andy 0 , which can be fixed or random.

Regarding preferences, we assume the quadratic utility function

``````u(ct) =(ct)^2
``````

Whereis a bliss level of consumption. Finally, we impose theno Ponzi scheme condition.

##### X
``````t=
``````
``````tb^2 t
``````
##### <

This condition rules out an always-borrow scheme that would allow the consumer to enjoy bliss consumption forever.

1. Observe that limt 2 t bt+1= 0. Show that the debt path is given by
``````bt=
``````
##### X
``````j=
``````
``````j(yt+jct+j)
``````
``````(5 Marks)
``````
1. Show that optimal consumption is given by
``````ct= (1)
``````
##### X
``````j=
``````
``````jEt[yt+j]bt
``````
``````= r
1 +r
``````
##### X
``````j=
``````
``````jEt[yt+j]bt
``````
``````Comment on this equation. What does it imply about optimal consumption (
Marks)
``````