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 US .  In 1970, the average household spent 30% if its income on services while in 2000, the average household spent 40% on services (hmm…..these numbers sound familiar!).  Further, services have higher rates of inflation than goods (some goods.  What does this information tell us about the accuracy of the CPI?

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