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 / 1,000 / $ 1.92
2 / 5,000 / $ 51.18
3 / 2,800 / $ 29.08
4 / 9,200 / $ 67.19
5 / 3,000 / $ 4.51
cash / n.a. / $5,353.40

Stock 3 announces record earnings, and the price of stock 3 jumps to $32.44 in after-market 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:

$25,000 buys 128.034 shares.

Based on the new information, NAV is:

$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:

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 / Probability
$30,000 / 0.25%
$15,000 / 0.75%
$10,000 / 1.50%
$ 5,000 / 2.50%
$ 1,000 / 5.00%
$ 250 / 15.00%
$ 0 / 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 / Probability
$28,500 / 0.25%
$13,500 / 0.75%
$ 8,500 / 1.50%
$ 3,500 / 2.50%
$ 1,000 / 5.00%
$ 250 / 15.00%
$ 0 / 75.00%

The expected (mean) loss is $475, which is the fair price of insurance.