AFM 2013 Name ______
Harry Casey and the PA Lottery Date ______Block _____
Situation
As reported in the Philadelphia Inquirer, Harry Casey was the first winner of Pennsylvania lottery in 1972. The amount he won was $1 million, to be paid in 20 annual installments of $50,000 each year, received his last check in 1991, and was broke in the spring of 1992.
Problem
Suppose your neighbor, Margaret Anderson, has just won a state lottery with a one million dollar prize. Margaret read about Harry Casey’s problem. Like Harry, she wants to be able to enjoy her winnings, but she does not want to end up broke after 20 years. Margaret knows you are taking Advance Functions and Modeling and asks you to help her figure out various strategies she might take with her winnings and saving the rest so she can accumulate a “nest egg” amount at the end of the 20-year period. Your task is to use what you are learning about exponential functions (as well as other mathematics you know) to explore various aspects of her situation.
Your overall task is to develop mathematical models using exponential functions for various aspects of the lottery situation so that you can help Margaret understand the implications of her choices to spend or save part or all her winnings. For example, how much should she save and on what schedule in order to reach her goal of a nest egg after her payments have ended? Prepare information for Margaret in a form that will show her various options and the implications in terms of money accumulated for each option.
Part I. Initial Exploration
You know that it would be wise for Margaret to save some of her lottery winnings, but you are not sure what savings plan to suggest. To help you analyze the situation better, you decide to look at some concrete possibilities first.
Suppose Harry Casey had saved only his first payment of $50,000 and deposited it in a savings account paying 5% interest compounded annually.
1. Complete Table 1 to show how much Harry would have had in his account at the end of the specific years mentioned.
Calendar Year / Years Since 1972 / Amount1972 / 0 / 50,000
1973 / 1 / 52,500
1974 / 2
1975
1976
1977
(3 points)
2. The growth in the amount in his account over the first year would have been
$52,500-$50,000=$2,500.
The yearly growth factor (or the rate of growth) for any year is defined to be the amount at the end of the year divided by the amount at the beginning of the year. For example, the yearly growth factor for the first year is .
Complete Table 2 and watch for patterns in the growth and in the rate of growth.
Years Since 1972 / Amount of Growth / Yearly Growth Factor1
2
3
4
5
(5 pts)
Part II.
3. The amount in the account, A(n), is a function of the number of years n that have elapsed since 1972. In Table 2 it was apparent that the rate of growth was constant. That is A(n+1)/A(n)=1.05 for any n in the time period we are examining. Since the ratio of successive amounts is constant, the function A is exponential. Use this information to write an expression that defines function A.
A(n)= ______(1 point)
4. This formula can help answer many questions about Harry’s account. Graph this function in an appropriate viewing window, and then answer the following questions. (2 points)
a. How much money would be in his account after 10 years had passed? ______
b. How long would it have taken Harry to have doubled his initial investment (that is, to have $100,000 in his account)? ______
5. The constants (called parameters) that appear in your function represent particular features of this situation. The $50,000 value represents the amount that was deposited to open the account. It is known as the initial amount or the principal P. The 1.05 value (actually 1 + 0.05) represents the amount by which a current balance can be multiplied to get the next year’s balance. If the interest rate is represented by r, (1+r) is the yearly multiplier, also known as the growth factor. (6 points)
a. Using the parameters P and r, define a general function A(n) where n is the number of years since the principal was deposited.
______
b. What new function would be valid if the growth factor stayed at 1.05 (the annual interest rate remained 5%) but the principal was changed to $30,000?
______
c. How long would it take to double his investment if Harry had only deposited $30,000 initially? ______
d. What new function would be valid if the annual rate was only 4%, but the principal remained at $50,000?
______
e. How long would it take to double Harry’s initial investment if the rate is 4%?
______
6. Create a new function in which the annual interest rate is 5% but the principal is some new amount, and then determine how many years it takes for the balance to double. (2 points)
a. A(n) = ______
b. The doubling time is ______years.
7. a. From the results of Questions 4, 5, and 6, what conclusion can you make regarding “doubling time”? (1 point)
b. Use the equation A(n) = P(1.05)n to explain why your conclusion must be true. (1point)
Now consider some variations that would have been options for Harry Casey.
8. Suppose Harry had invested his money in an account that offered 5% annual interest compounded annually. (3 points)
a. How much money would have been in his account at the end of 20 years when the payments stopped if he had originally deposited all $50,000? ______
b. Only $25,000? ______
c. A “mere” $10,000? ______
9. Suppose Harry had invested his money in an account that offered 4.5% annual interest compounded annually. (3 points)
a. How much money would have been in his account at the end of 20 years if his initial deposit was $50,000? ______
b. What if he deposited $25,000? ______
c. If he deposited $10,000? ______
10. Use your graphing calculator with an appropriate viewing window to compare the graphs of functions defined for the following situations.
(i) Investing $50,000 in an account that offers 5% annual interest compounded annually.
(ii) Investing $30,000 in an account that offers 8.5% annual interest compounded annually.
Notice that the two graphs intersect. (2 points)
a. If you were studying the amount in each account over the first 20 years of investment, for what years is the amount in account (i) greater? ______
b. For what years is the amount in account (ii) greater? ______
11. Over a long period of time, which parameter has the greater effect on the amount in an account that earns interest annually: the principal or the interest rate? ______(1 point)
Part III.
Many savings institutions offer compounding intervals other than annual compounding. For example, a bank that offers quarterly compounding computes interest on an account every quarter, that is, every 3 months. Thus, instead of computing interest once each year, the interest will be computed 4 times each year. If a bank advertises that it is offering 8% annual interest compounded quarterly, it does not use 8% to determine the growth factor. Instead, it will use to determine the quarterly growth factor. In this example, 8% is known as the nominal interest rate and 2% as the quarterly interest rate.
12. Suppose Harry invested all $50,000 in an account that offered a nominal interest rate of 5% compounded quarterly. (7 points)
a. What would be the quarterly interest rate? ______
Under these circumstances, how much would Harry have in his account after (remember to pay attention to your units):
b. 3 months ______
c. 6 months ______
d. 9 months ______
e. 1 year ______
f. 4 years ______
g. n years ______
13. a. How long would it take for Harry to double his initial investment? ______
(1 point)
b. How much less time is this than it took to double when the rate was 5%, but the account accrued interest only once a year? ______(1 point)
c. Compare the graphs of A1(n)=50,000(1.05)n with that of A2(n)=50,000(1+0.05/4)(4n) in an appropriate viewing window. These two functions represent two different accounts. Compare the balance in these accounts at several different times. (1 point)
d. After how many years will the value of A2 be $1,000 more than the value of A1? ______(1 point)
14. Determine the amount that Harry would have in an account after 20 years if the principal is $50,000, the nominal rate is 5%, and the interest is compounded (5 points):
a. yearly ______
b. quarterly ______
c. monthly ______
d. daily ______
e. What affect does increasing the compounding interval have on the final amount in the account after 20 years?
15. To be able to compare the effect of the different compounding intervals more easily, one uses another rate known as the effective rate.
a. Consider an account that offers a nominal interest rate of 4% compounded quarterly. If the initial investment is $10,000, determine the amount (balance) in this account after 1 year.
Balance = ______(1 point)
b. Consider a second account with an initial investment of $10,000, but which compounds annually. What would the interest rate to have to be if this account is to generate the same balance as the account in part a. after 1 year? (1 point)
Interest rate = ______
The interest rate you just determined is the effective rate of the account in part a. That is, it is the rate that an account offering interest only once a year would have had to offer to produce the same result as a competitive account offering more frequent compounding.
16. Compute the effective rates of the four situations given in Question 14. (3 points)
a. 5% nominal rate compounded yearly: ______
b. 5% nominal rate compounded quarterly: ______
c. 5% nominal rate compounded monthly: ______
Part IV. Assessment
Write a typed explanation (using 12 point font, double spacing, and 1 inch margins) answering the following questions.
1. Should Margaret, over the long term, be more concerned with the principal or the interest rate? Be sure to include some specific examples to help support your recommendation.
2. If Harry had decided to take some portion of his $50,000 and deposit it in an interest-bearing account, which of the following nominal rates would have been his wisest choice? Be sure to explain clearly why you selected an option (show your work).
3.74% compounded weekly
3.76% compounded monthly
3.77% compounded quarterly
For questions 3 and 4, as you think about developing several options for Margaret to consider, you check the local newspaper to see what investment opportunities are currently available. You discover that the following opportunities are advertised.
Big Bucks Bank: Annual rate of 4% on amounts greater than or equal to $30,000
Annual rate of 3.7% on amounts less than $30,000
Serious Savings: Nominal rate of 3.67% compounded weekly
3. What should Margaret do if she chooses to invest all $50,000?
4. What should Margaret do if she chooses to invest only $25,000 of her first check?
For questions 3 and 4, be sure to include an explanation of why you are making these recommendations (again show your work). You should be sure to discuss the effect, if any, on the length of time she plans on leaving the money in the account.
Grading Rubric
Questions in Packet ______/ 50 divided by 2 = ______/25 points
Explanation Rubric:
0 points / 2 point / 4 points / 6 pointsShould Margaret be more concerned with principal or interest rate and why? / Did not include in explanation / Included but no/incorrect details / Included with some correct details / Included with correct and full explanation
Which rate would you choose and why? / Did not include in explanation / Included but no/incorrect details / Included with some correct details / Included with correct and full explanation
What should Margaret do if she chooses to invest all $50,000 and why? / Did not include in explanation / Included but no/incorrect details / Included with some correct details / Included with correct and full explanation
What should Margaret do if she chooses to invest only $25,000 and why? / Did not include in explanation / Included but no/incorrect details / Included with some correct details / Included with correct and full explanation
Total ______/ 50 points