Espen Henriksen
August 24, 2008
Question 1. t is governed by the following law of motion
t+1 = t +
.1 if it > .5
-.1 if it  .5
where 1 = 0 and it is independent and identically distributed (iid) and drawn
from a uniform distribution on the interval (0,1).
Generate a sequence of length 200. Plot this sequence
Algorithm 1 Answer to Question 1 (pseudo-code)
1  0
for t  1 to 199 do
d  random draw from a uniform distribution over (0, 1)
if d > 0.5 then
t+1 = t + 0.1
else
t+1 = t - 0.1
end if
end for
plot {t}
200
t=1.
Algorithm 2 Answer to Question 1 (Matlab implementation)
c l e a r a l l ;
theta (1) = 0;
f o r t = 1 : 199
d = rand ;
i f d > 0.5
theta ( t+1) = theta ( t ) + 0 . 1 ;
e l s e
theta ( t+1) = theta ( t ) - 0 . 1 ;
end
end
plot ( theta )
1
Question 2. Suppose you are going to evaluate a set of 50 fund managers over
a period of four years. You will observe their performance every month (48
observations).
Before you start your study you decide to simulate a simple model where no
manager is better than the other. You know the market has an expected drift
with a large associated variance.
After some statistical analysis of historical data you decide to simulate the
following little model: Suppose all managers starts in the first month with the
same, given amount of available funds, for simplicity normalized to 1. Monthly
returns are independent and identically distributed (iid). For any given month,
with 40% probability a manager will outperform the market and earn a return
of 0.01098% (14% annually), with 20% probability she will earn the average
market return of 0.00327% (4% annually), and with 40% probability she will
loose money and earn a return of -0.00514% (-6% annually).
Algorithm 3 Answer to Question 2 (pseudo-code)
for f  1 to 50 do {Let f be a counter for the fund managers}
1,f  1
for m  1 to 47 do {Let m be a counter for the months}
d  random draw from a uniform distribution over (0, 1)
if d > 0.4 then
m+1,f = m,f  (1 + 0.01098)
else if 0.4  d < 0.6 then
m+1,f = m,f  (1 + 0.00327)
else if d  0.6 then
m+1,f = m,f  (1 - 0.00514)
end if
end for
end for
2
Question 3. In a given quarter, state of the economy can be either high ("boom")
or low ("recession") and it follows a Markov process. The Markov transition
matrix is
 =
0.6 0.4
0.4 0.6
.
Total factor productivity of the economy is given by A = ezt
where the parameter
z is governed by the following stochastic process: If the economy is in the high
state, the z increases by 0.025 relative to trend, and if the economy is in the
low state, the z decreases by 0.025 relative to trend, i.e. z is governed by the
following law of motion
zt+1 = zt +
0.025 if the state is high
-0.025 if the state is low
.
Simulate a sequence of 40 quarters.
Algorithm 4 Answer to Question 2 (pseudo-code)
 =
0.6 0.4
0.4 0.6
z1  1
s  1 {Let s be an indicator of the state of the economy}
for q  1 to 39 do {Let q be a counter for the quarters}
d  random draw from a uniform distribution over (0, 1)
if s = 1 then
if d < (s, 1) then
zq+1  zq + 0.025
s  1
else
zq+1  zq - 0.025
s  2
end if
else if s = 2 then
if d < (s, 2) then
zq+1  zq + 0.025
s  2
else
zq+1  zq - 0.025
s  1
end if
end if
end for
3