Chapter 17 - Selected Quantitative Problems & Solutions

Quantitative Problems
Question 1
On January 1st, the shares and prices for a mutual fund at 4:00 PM are:
Stock
Shares owned
Price
1
2
3
4
5
cash
1,000
5,000
2,800
9,200
3,000
n.a.
$
1.92
$ 51.18
$ 29.08
$ 67.19
$
4.51
$5,353.40
Stock 3 announces record earnings, and the price of stock 3 jumps to $32.44 in aftermarket trading. If the fund (illegally) allows investors to buy at the current NAV, how
many shares will $25,000 buy? If the fund waits until the price adjusts, how many shares
can be purchased? What is the gain to such illegal trades? Assume 5,000 shares are
outstanding.
Solution: At 4:00 PM, the NAV is calculated as:
$1,920  $255,900  $81,424  $618,148  $13,530  $5,353.40
5,000
 $195.26/share.
NAV 
$25,000 buys 128.034 shares.
Based on the new information, NAV is:
$1,920  $255,900  $90,832  $618,148  $13,530  $5,353.40
5,000
 $197.14/share.
NAV 
$25,000 buys 126.813 shares.
If sale prices are used, the investor buys 128.034 shares. $25,000 enters the fund. After
the price increase (assuming nothing else changes), the fund is worth $1,010,683.40.
Each share is worth $1,010,683.40/5128.034  $197.09. The investor’s shares are now
worth $25,234.20, or a gain of $234.20.
Question 2 (Useful)
A mutual fund charges a 5% upfront load plus reports an expense ratio of 1.34%. If an
investor plans on holding a fund for 30 years, what is the average annual fee, as a
percent, paid by the investor?
Solution: 5%/30  0.1667%
The expense ratio is an annual charge, so it remains 1.34%.
The total fees paid are 1.34%  0.1667%  1.5067%.
Question 3 (Useful)
A mutual fund offers “A” shares which have a 5% upfront load and an expense ratio of
0.76%. The fund also offers “B” shares which have a 3% backend load and an expense
ratio of 0.87%. Which shares make more sense for an investor looking over an 18 year
horizon?
Solution: For the “A” shares, the average annual fee is 5%/18  0.76%  1.0378%
For the “B” shares, the average annual fee is 3%/18  0.87%  1.0367%
So, the investor is better off with the “B” shares.
Question 4 (Useful)
A mutual fund reported year-end total assets of $1,508 million and an expense ratio of 0.90%.
What total fees is the fund charging each year?
Solution: The fees are a percent of total assets. In this case, 0.90%  $1,508 million 
$13,572,000.
Question 5
A $1 million fund is charging a back-end load of 1%, 12b-1 fees of 1%, and an expense ratio of
1.9%. Prior to deducting expenses, what must the fund value be at the end of the year for
investors to
break even?
Solution: With the backend load, the fund value must be (after expenses):
$1 million/0.99  $1,010,101.01
The expense ratio typically includes 12b-1 fees. So, a total of 1.9% will be
charged. So, before expenses, the fund value must be: $1,010,101.01/0.981 
$1,029,664.64
Chapter 18 - Selected Quantitative Problems & Solutions
Question 1 (Useful)
Research indicates that the 1,000,000 cars in your city experience unrecoverable losses of
$250,000,000 per year from theft, collisions, etc. If 30% of premiums are used to cover
expenses, what premium must be charged to car owners?
Solution: The average loss per car  $250,000,000/1,000,000 cars  $250/car
So, 70% of the premium must equal the payout of $250.
Or, 0.70 Premium  $250
Premium  $250/0.70
Premium  $357.14
Question 2
Assume that life expectancy in the United States is normally distributed with a mean of 73
years and a standard deviation of 9 years. What is the probability that you will live to be
over 100 years old?
Solution: The Z-score is calculated as follows:
Z
100  73 27

3
9
9
The age of 100 years old is 3 standard deviations to the right of the mean.
Using a standard normal probability chart, this suggests that the probability is
less than 1%.
Question 3
Your rich uncle dies, leaving you a life insurance policy worth $100,000. The insurance
company also offers you an option to receive $8,225/year for 20 years, with the first
payment due today. Which option should you use?
Solution: Since the options are either $100,000 immediately or $8,225/year, you can
calculate the rate your are “paying” as:
PV  100,000; PMT  8225; N  20; FV  0; Calculator in BEGIN mode.
Calculate I. I  6% (roughly)
With this information, the answer depends on many factors. Do you “need’
the $100,000 today? Can you personally invest the $100,000 at a higher rate
with the same level of risk? Is there any risk that the insurance company will
not pay in the future? Etc
Question 4 (Useful)
A home products manufacturer estimates that the probability of being sued for product
defects is
1% per year per product manufactured. If the firm currently manufacturers 20
products, what is the probability that the firm will experience no lawsuits in a
given year?
Solution: The probability of not being sued is 99%. If we assume that the probabilities
are independent, then the probability of no lawsuits over all 20 products is:
0.9920  0.8179, or about 82%.
Question 5 (Useful)
Kio Outfitters estimated the following losses and probabilities from past experience:
Loss
$30,000
$15,000
$10,000
$ 5,000
$ 1,000
$ 250
$
0
Probability
0.25%
0.75%
1.50%
2.50%
5.00%
15.00%
75.00%
What is the probability Kio will experience a loss of $5,000 or greater? If an insurance
company offers a loss policy with $1,500 deductible, what is the most Kio will pay?
Solution: Losses of less than $5,000 occur 95% of the time. So, 5% of the time, losses
will be $5,000 or greater.
With a $1,500 deductible, Kio’s expected losses are:
Loss
$28,500
$13,500
$ 8,500
$ 3,500
$ 1,000
$ 250
$
0
Probability
0.25%
0.75%
1.50%
2.50%
5.00%
15.00%
75.00%
The expected (mean) loss is $475, which is the fair price of insurance.