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4.5 a. Jane is 20 years old today. Jane is going to put $1,000 into her savings account on her 21st birthday and again on every birthday for 20 payments (i.e., till her 40th birthday). She will earn 5%, paid annually. How much money will be in the account after she collects her interest and makes her 20th payment? b. Calculate how much money she could take out each year for the 20 years from her 41st birthday till her 60th birthday, assuming she still earns 5% and takes out the same amount each year, leaving exactly $0 in the account after removing her 20th payment. 5.3 Are you better off playing the lottery or saving the money? Assume you can buy one ticket for $5, draws are made monthly, and a winning ticket correctly matches 6 different numbers of a total of 49 possible numbers. The probabilities: In order to win, you must pick all the numbers correctly. Your number has a 1 in 49 chance of being correct. Your second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options. But the order in which you pick them does not matter, so you actually have a few more ways to win. You can pick 6 different numbers in exactly 6 x 5 x 4 x 3 x 2 x 1 = 720 orders of choice. Any one of those orders would still win the lottery. Putting this all together, your ticket has 720/10,068,347,520 = 1/13,983,816 chance of winning. This equates to a .000000071 percentage chance. If you played one ticket every month from age 18 to age 65, you would have 47 x 12 = 564 plays. Your odds of not ever winning would be calculated using a binomial distribution to be .9999599568, meaning your chances of winning would be 1 – .9999599568 = .0000400432. So, if the lottery winnings averaged $10 million over this time period, your expected return would be less than .0000400432 x $10 million = $400.43. (It\'s less than $400.43 because your 564 plays are spread out over the next 47 years, so the present value of these future plays would be significantly less than if you were able to play all 564 immediately. The $400.43 assumes you play all 564 plays today, which makes it the highest possible expected value.) REQUIRED: A. What would your $400.43 be worth if you invested it at 1% real interest for 47 years? B. If, instead, you wrote down your 6 numbers on a piece of paper, and deposited your $5 in a bank at 1% real interest, how much would you have at the end of the first year? C. If you did this every year for 47 years, how much would you have at age 65? D. If you earned 5% real interest on your deposits, how much would you have at age 65? E. Which option would make you better off at age 65? How many times better off? 9.4 You own one share in a company called Invest Co. Inc. Examining the balance sheet, you have determined that the firm has $100,000 cash, equipment worth $900,000, and 100,000 shares outstanding. Calculate the price/value of each share in the firm, and explain how your wealth is affected if: A. The firm pays out dividends of $1 per share. B. The firm buys back 10,000 shares for $10 cash each, and you choose to sell your share back to the company. C. The firm buys back 10,000 shares for $10 cash each, and you choose not to sell your share back to the company. D. The firm declares a 2-for-1 stock split. E. The firm declares a 10% stock dividend. F. The firm buys new equipment for $100,000, which will be used to earn a return equal to the firm\'s discount rate. 11.1 In the northeast United States and in eastern Canada, many people heat their houses with heating oil. Imagine you are one of these people, and you are expecting a cold winter, so you are planning your heating oil requirements for the season. The current price is $2.25 per US gallon, but you think that in six months, when you\'ll need the oil, the price could be $3.00, or it could be $1.50. A. If you need 350 gallons to survive the winter, how much difference does the potential price variance make to your heating bills? B. If your friend Tom is running a heating oil business, and selling 100,000 gallons over the winter season, how does the price variance affect Tom? C. Which one of you benefits from the price increase? Which of you benefits from price decrease? D. What are two strategies you can use to reduce the risk you face? Could you make an agreement with Tom to mitigate your risk? E. Assuming you are both risk-averse, does such an agreement make you both better off?

Assignment Questions





















4.5

a.

Jane is 20 years old today. Jane is going to put $1,000 into her savings account on her 21st birthday and again on every birthday for 20 payments (i.e., till her 40th birthday). She will earn 5%, paid annually. How much money will be in the account after she collects her interest and makes her 20th payment?


 

b.

Calculate how much money she could take out each year for the 20 years from her 41st birthday till her 60th birthday, assuming she still earns 5% and takes out the same amount each year, leaving exactly $0 in the account after removing her 20th payment.


5.3

Are you better off playing the lottery or saving the money? Assume you can buy one ticket for $5, draws are made monthly, and a winning ticket correctly matches 6 different numbers of a total of 49 possible numbers.
 
The probabilities: In order to win, you must pick all the numbers correctly. Your number has a 1 in 49 chance of being correct. Your second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options.
 
But the order in which you pick them does not matter, so you actually have a few more ways to win. You can pick 6 different numbers in exactly 6 x 5 x 4 x 3 x 2 x 1 = 720 orders of choice. Any one of those orders would still win the lottery.
 
Putting this all together, your ticket has 720/10,068,347,520 = 1/13,983,816 chance of winning. This equates to a .000000071 percentage chance.
 
If you played one ticket every month from age 18 to age 65, you would have 47 x 12 = 564 plays. Your odds of not ever winning would be calculated using a binomial distribution to be .9999599568, meaning your chances of winning would be 1 .9999599568 = .0000400432.
 
So, if the lottery winnings averaged $10 million over this time period, your expected return would be less than .0000400432 x $10 million = $400.43.
 
(It's less than $400.43 because your 564 plays are spread out over the next 47 years, so the present value of these future plays would be significantly less than if you were able to play all 564 immediately. The $400.43 assumes you play all 564 plays today, which makes it the highest possible expected value.)
 
REQUIRED:


 

A.

What would your $400.43 be worth if you invested it at 1% real interest for 47 years?

 

B.

If, instead, you wrote down your 6 numbers on a piece of paper, and deposited your $5 in a bank at 1% real interest, how much would you have at the end of the first year?

 

C.

If you did this every year for 47 years, how much would you have at age 65?

 

D.

If you earned 5% real interest on your deposits, how much would you have at age 65?

 

E.

Which option would make you better off at age 65? How many times better off?

9.4

You own one share in a company called Invest Co. Inc. Examining the balance sheet, you have determined that the firm has $100,000 cash, equipment worth $900,000, and 100,000 shares outstanding.
 
Calculate the price/value of each share in the firm, and explain how your wealth is affected if:
 
 

 

A.

The firm pays out dividends of $1 per share.
 
 

 

B.

The firm buys back 10,000 shares for $10 cash each, and you choose to sell your share back to the company.
 
 

 

C.

The firm buys back 10,000 shares for $10 cash each, and you choose not to sell your share back to the company.

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