Finance 462
Solutions to
Problem Set #1
1)
Suppose
that, each year, you have a 2% chance of being involved in a car accident.
The damages from a car accident are $10,000.
a)
Assuming that you don’t purchase insurance, calculate the expected
value of your losses. Calculate the
standard deviation of your losses.
Here, we have
two possible outcomes:
$10,000 (w/probability .02)
$0 (w/probability .98)
The expected value is the weighted average of these two outcomes where
the weights are the probabilities.
E(Loss) =
(.98)($0) + (.02)($10,000) = $200
To get the
standard deviation:
·
Subtract
the expected value from each possible outcome
·
Square
each difference
·
Calculate
the expected value of these squared differences (this is the variance)
·
Take
the square root
Variance =
(.98)($0 – ($200))^2 +
(.02)($10,000 – ($200))^2
= (.98)($200)^2 +
(.02)($9800)^2
= (.98)($40,000) + (.02)($96,040,000)
= $39,200 + $1,920,800 =
$1,960,000
Std. Dev. =
SQRT($1,960,000) = $1400
b)
Now, suppose that you by insurance.
How much should an insurance policy cost (Assuming everybody has a 2%
accident rate)? What happens to the expected value and standard deviation of
your losses?
Note that, on
average, the insurance company will suffer losses of $200 for every policy
holder ($200 is the expected loss). Therefore,
the will need to charge at least $200 to make a profit.
Lets assume that they charge a markup of 20% and charge $200(1.2) = $240.
For an insured driver, the two possible outcomes are now:
$240
(w/probability .98): you don’t get into an accident, but you must
pay
your premium.
$240
(w/probability .02): you get into an accident, the insurance company
pays the $10,000 loss (there is no deductible in
this example), you pay the premium.
Note that the
expected loss here is $240 (with certainty) and the standard deviation is zero
Now, suppose that there are two types of drivers.
Safe drivers have a 2% accident rate, while unsafe drivers have a 4%
accident rate. Assume that there are
an equal number of safe and unsafe drivers, but the insurance company can’t
distinguish between the two types. How
does your answer to (b) change?
The only
difference between this example and (b) is that the presence of unsafe drivers
raises insurance premiums. The insurance company doesn’t know which are which
and, therefore, must charge everybody a price equal to a markup above their
expected cost
E(Cost) =
(.5)($200) + (.5)($400) = $300
Premium Cost =
($300)(1.2) = $360.
Note that the
unsafe driver is unambiguously better off with insurance that without (the
unsafe driver’s expected loss without insurance is $400.)
Suppose that apples and oranges are the only two
goods available in the economy, and that they are sold in the quantities and
prices indicated in the following table.

Apples 

Oranges 

Year 
Quantity 
Price 
Quantity 
Price 
1970 
30 
$1 
70 
$1 
2000 
40 
$4 
60 
$2 
2)
Suppose
the assuming that the average household spends 30% of its income on apples and
70% of its income on oranges each year, calculate the CPI for 1970 and 2000.
What is the average annual inflation rate?
The
CPI is equal to a weighted average of individual goods prices where the weights
are predefined (in this case, .30 and .70).
P(1970)
= (.3)($1) + (.7)($1) = $1
P(2000)
= (.3)($4) + (.7)($2) = $2.60
The
inflation rate from 1970 to 2000 is (2.6 – 1)/1 x100 = 160%.
Notice that the individual inflation rates are 400% for apples, 100% for
oranges.
(.3)(400)
+ (.7)(100) = .70 + 120 The CPI inflation rate is closer to the inflation rate
for oranges because oranges have the larger weight in the index.
The
average annual inflation rate is 160%/30 = 5.33%
3)
Using
1970 as the base year, calculate real and nominal GDP for 2000. What is the
implied annual inflation rate?
First,
calculate nominal GDP in 2000 (price times quantity)
Nominal
GDP(2000) = ($4)(40) + ($2)(60) = 280
Now,
calculate GDP using 1970 prices
Real
GDP(2000) = ($1)(40) + ($1)(60) = 100
The
Deflator for 2000 = Nominal GDP/Real GDP = 280/100 = 2.8.
The deflator for the base year is always one.
Therefore, the inflation rate is
(2.8
– 1)/1 x 100 = 180% (or, 180/30 = 6% per year).
4)
Over the
past 30 years, we have seen a shift in consumer patterns in the
These
weights look very similar to the quantities in (3).
In fact, that’s the whole point behind the deflator! It should pick up
changing consumer patterns. In this
example, Consumers are actually consuming more of the product that’s becoming
more expensive (this is because services – more specifically medical services
are often subsidized by insurance, the government, etc).
Therefore, the CPI in (2) is actually underestimating inflation.
5)
Consider
an economy with 100 people in the labor force.
At the beginning of every month, 5 people lose their jobs and remain
unemployed for exactly one month; one month later, they find new jobs and become
employed. In addition, on January 1
of each year, 2 people lose their job and remain unemployed for 6 months.
Finally, on July 1 of each year, 1 person loses his job and remains
unemployed for 1 year.
a)
What is
the unemployment rate in this economy in a typical month?
The way this
question is written, there are two distinct periods in the economy…..
From January
until July, there are 8 unemployed people (5 from the most recent month, 2 that
lost their jobs in January and 1 that lost his job the previous July).
From July to December there are 6 people unemployed (The 2 people that
lost their jobs in January have found jobs).
Therefore, on average there are 7 people unemployed.
Which gives us an unemployment rate of 7%?
b) What is the average duration of unemployment
First, figure
out how many total people lose their job throughout the year:
(5 people per
month)(12 months/year) = 60
2 people lose
their job in January
= 2
+ 1 person loses his job in July
= 1
Total =
63
There are three possible outcomes for someone that loses their job.
Remain unemployed for 1 month
(with probability 60/63 = .95)
Remain unemployed for 6 months
(with probability 2/63 = .03)
Remain unemployed for 12 months(with probability 1.63 = .02)
E(Duration) = (.95)(1) + (.03)(6) + (.02)(12) = 1.37 months = 5.5 weeks