ECONS4280 PS2: Risk and Return 2022

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PS2: Risk and Return

Corporate Finance – ECONS4280

2022

1. Rational Expectations and Project Evaluation Suppose that some project, X, generates payoffs in the following way:

xt  = ρxt − 1 + et

where et  is some random shock with mean 0 and all shocks are independent of one another.

(a) Find E0 [xt]

(b) Find the PV of Project X for a generic value of T

Now suppose that some alternative project, Y , generates payoffs accord- ing to the following random process:

yt  = (ρ + et ) yt − 1

(c) Find E0 [yt]. (Hint: if random variables, {e1 ,e2 , ...}, are independent, then E[e1  × e2  × ...] = E[e1]E[e2]...)

(d) Find the PV of Project Y for a generic value of T (e) Suppose that ρ = 0.9,y0  = 1;

i. Simulate Projects X and Y for 100 steps.  (Hint:   Use Excel to  do this.  First generate 100 random shocks between - 1 and 1 using the

RAND() function.   Then,  starting  at y0   =  1,  construct the series using the equations noted above)

ii. Plot the two sequences on a graph using the ’line graph’ option. Comment on what you find. In particular: which project looks more risky?

iii. Calculate the variance of the payoffs of Project X and Project Y . Which is higher? Suppose you had to select a discount rate for these projects: which would be higher, rX , or rY ? Why?


2. Volatility and Portfolios: Go to the PS2 folder on courseworks and down- load the excel file,‘ps2 data’. This file contains 100 observations of the returns on three stocks.

(a) Calculate the average (simple arithmetic is fine) and variance of the re-

turns of each stock.

(b) Plot what you find: place the average return of each stock on the x-axis,

and the variance of each stock on the y-axis, using a scatter plot.  You should have three data points in total.  Go to ‘add chart element’and add a linear trend line. What do you find?

Now we’re going to construct some portfolios.  To start with, take the following two portfolios: in Portfolio A, we have equal amounts of Stock 1 and 2, and in Portfolio B, we have equal amounts of Stock 2 and 3.

(c) Find the covariance between stocks 1 and 2, and stocks 2 and 3.  (Hint: Use the Excel function COVARIANCE.P())

(d) Using the formula from class and in the notes, calculate the portfolio variance of portfolios A and B. Which is higher?

Now suppose we construct a portfolio containing all three stocks, and we have equal amounts of all three (ω1  = ω2  = ω3  = ).

(e) Find the portfolio variance by following these steps

i. Find the covariance matrix of the portfolio.

ii. Calculate the weight matrix.

iii. Multiply the elements of these matrices and take the sum.

(f) Extra Credit1: Suppose now we allow you to choose the weights, {ω1 ,ω2 ,ω3 }.

What choice of these weights minimises the portfolio variance?  Closest answer to the truth wins extra credit of 3 additional points, plus a mys- tery prize!!  (Hint:  There are many ways you could do this.  One is to setup a minimisation problem and solve it analytically. Another might be to simulate across a range of ω’s and see which combination works out best.)











2022-07-20