1. Mathematics of Finance
Chapter 5

2. Mathematics of Finance
5.1 Simple and Compound Interest
5.2 Future Value of an Annuity
5.3 Present Value of an Annuity and Amortization
ECO B. Potter

3. 5.1 Simple and Compound Interest
Interest: The fee paid to use someone else’s money.
Principal: The amount borrowed or deposited.
Simple Interest: Interest paid only on the principal Frequently charged on loans of a year or less.
Compound Interest: Interest is paid on interest as …well as on principal.
Rate of interest: Given as a percent per year, ...expressed as a decimal (6% = .06)
ECO B. Potter

4. Simple Interest SIMPLE INTEREST
The product of the principal P, rate r, and time in years t gives simple interest, I:
I = Prt
If P and r are known, then I is a function of t
For example, assume that P = 500 and r =.12 then, I = 500(.12)t or I = 60t
ECO B. Potter

5. Simple Interest – Future Value
FUTURE OR MATURITY VALUE FOR SIMPLE INTEREST
The future value (A) of P dollars invested at simple interest r for t years.
A = P(1 + rt)
Ex: Solve for A. Suppose $3,000 is placed in a savings account that pays 6% simple interest. What is the balance in the account after 7 months.
How much interest was earned?
$ $3000 = $105
ECO B. Potter

6. Simple Interest – Future Value
Ex: Solve for r. Suppose $3,000 is placed in a savings account. If the balance in the account after 7 years is $5000, what simple interest rate did we earn ?
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7. Simple Interest – Future Value and Time
With simple interest and only simple interest, the future value (A) is a linear function of time.
Ex: Suppose $1,000 is placed in a savings account. that pays 6% simple interest. Write the future value as a function of time.
ECO B. Potter

8. Simple Interest – Future Value and Time
Year
Value
1000 1
1060 2
1120 3
1180 4
1240 5
1300 6
1360 7
1420 8
1480
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9. Present Value for Simple Interest
Simple Interest – Present Value
A sum of money that can be deposited today to yield some larger amount in the future is the present value of that future amount.
Present Value for Simple Interest
The present value P of a future amount of A dollars at a simple interest rate r for t years is
ECO B. Potter

10. Simple Interest – Present Value
Tuition of $1769 will be due when the spring term begins in 4 months. What amount should a student deposit today, at 3.25%, to have enough to pay the tuition?
A = 1769, r = .0325, t = 4/12
ECO B. Potter

11. Simple Discount Notes
Interest is deducted in advance from the loan amount before giving the balance to the borrower.
The full value of the note must be paid at maturity.
Bank Discount (or discount): The money that is deducted in advance.
Proceeds: The money actually received by the borrower.
ECO B. Potter

12. Simple Discount Notes Consider a loan of $3000 at 6% for 9 months
Simple Interest Note
Bank Discount Note
Interest on the note
3000(.06)(9/12) = $135
Borrower receives
$3000
$2865
Borrower pays back
$3135
“Actual” interest rate
ECO B. Potter

13. Simple Discount Notes Discount P = A – D
If D is the discount on a loan having a maturity value A at a rate of interest r for t years, and if P represents the proceeds, then
P = A – D
ECO B. Potter

14. Now You Try
John Matthews signs a $4200 note (A) at the bank, which charges a 12.2% discount rate (r). Find the net proceeds (P) if the loan is for 10 months (t). Find the actual interest rate (to the nearest hundredth) charged by the bank.
ECO B. Potter

15. Compound Interest
Normally used for loans or investments of a year or more.
Interest is paid (or charged) on interest as well as on principal.
A = Compound amount at the end of n periods (future value)
P = principal (present value)
r = annual interest rate
m = number of compounding periods in 1 year
i = interest rate per compounding period (r/m)
t = number of years
n = total number of compounding periods (tm)
ECO B. Potter

16. Compound Interest
Compound amount (future value) when interest is compounded m times per year.
Where
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17. P = $1,000, r = 10%, t = 20, m = 1 A B C 1 year compound simple 2 11003
1210
1200 4
1331
1300 5
1464.1
1400 6
1500 19 18
2800 20
2900 21
3000
ECO B. Potter

18. P = $1,000, r = 10%, t = 20
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19. Compound Interest - continued
Ex: Suppose $3,000 is placed in a savings account. If the annual interest rate is 6%, and interest is compounded semiannually, what is the balance in the account after 7 years?
P = 3000
i = r/m = 0.06/2 = 0.03
n = tm = 72 = 14 interest periods.
ECO B. Potter

20. Now You Try
Find the compound amount for each of the following deposits.
4.6% compounded semiannually for 11 years.
6.1% compounded quarterly for 4 years.
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21. Compound Interest – Effective Rate
Nominal rate (r) – The stated interest rate to be paid on an investment. Sometimes called Annual Percentage Rate, or APR.
Effective rate (rE) – The actual increase in an investment. Higher than the nominal rate if interest is compounded. Sometimes called Annual Percentage Yield, or APY.
ECO B. Potter

22. Example $1,000 investment r = 6% m = 12 calculate rE or, 6.168% Month
compounded
simple 1 2 3 4 5 6 7 8 9 10 11 12
1060
r = 6%
m = 12
calculate rE
or, 6.168%
ECO B. Potter

23. Now You Try
The Flagstar Bank in Michigan offered a 5-year certificate of deposit (CD) at 4.38% interest compounded quarterly. On the same day on the internet, Principal Bank offered a 5-year CD at 4.37% interest compounded monthly. Find the APY (effective rate) for each CD. Which bank pays a higher APY?
ECO B. Potter

24. Compound Interest – Present Value
My daughter would like to have $5,000 six years from now to put down on a new car. What amount deposited today at 6% compounded monthly will amount to $5,000 in 6 years?
A = 5,000, i = .06/12 = .005, n = 6  12 = 72, P = ?
(compound interest)
(Remember, i = r m and n = mt)
ECO B. Potter

25. Compound Interest – Present Value
$3, deposited today at 6% compounded monthly will amount to $5,000 in six years.
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26. Now You Try
Bill Poole wants to have $20,000 available in 5 years for a down payment on a house. He has inherited $15,000. How much of the inheritance should he invest now to accumulate the $20,000 if he can get an interest rate of 8% compounded quarterly?
ECO B. Potter

27. Compound Interest – Using logs
How long will it take for $600 to amount to $900 at an annual rate of 8% compounded quarterly ?
i = 0.08/4 = Let n be the number of interest periods it takes for a principal of P = $600 to amount to A = $900 then,
ECO B. Potter

28. Compound Interest – Using logs
To solve for n we first take the natural logs (ln) of both sides:
Solve for n:
n = quarterly interest periods, or /4 = years
ECO B. Potter

29. Now You Try
Suppose a conservation campaign coupled with higher rates causes the demand for electricity to increase at only 2% per year, as it has recently. Find the number of years before the utilities will need to double their generating capacity.
ECO B. Potter

30. Simple & Compound Interest – Summary
Simple Interest
Compound Interest
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31. 5.2 Future Value of an Annuity
Annuity – A sequence of equal payments made at equal periods of time.
Ordinary Annuity – Payments are made at the end of the time period, and the frequency of the payments is the same as the frequency of the compounding.
Sinking Fund – Periodic payments are made to meet some future obligation (college savings plan).
Amortization – Periodic payments are made to dispose of a present obligation (car loan). Section 5.4
ECO B. Potter

32. 5.2 Future Value of an Annuity
Payment period – The time between payments.
Term – The time from the beginning of the first payment period to the end of the last payment period.
Future value of the annuity – The sum of the compound amounts of all the payments, compounded to the end of the term (The final sum on deposit.).
ECO B. Potter

33. Future Value of an Annuity; Sinking Fund
If you deposit $100 at the end of each year for 4 years in a savings account that pays 10 % per year how much will you have at the end of 4 years? 1 2 3 4
Year
100
100
100
100
$100
100(1.1) = 110
100(1.1)2 = 121
100(1.1)3 =
$464.10
ECO B. Potter

34. Future Value of an Annuity; Sinking Fund
Where:
S is the future value;
R is the payment;
i is the interest rate per period;
n is the number of periods
ECO B. Potter

35. Future Value of an Annuity; Sinking Fund
Previous example: If you deposit $100 at the end of each year for 4 years in a saving account that pays 10 % per year how much will you have at the end of 4 years?
ECO B. Potter

36. Example
To meet college expenses for their newborn child, Bob and Kathy decide to invest $600 every six months in an ordinary annuity that pays 8% annual interest, compounded semi-annually. What is the value of the annuity in 18 years?
i = .08/2 = .04
n = 2(18) = 36
R = 600
ECO B. Potter

37. Example 2
A company wants to have $1,500,000 to replace old equipment in 10 years. How much must be set aside annually in an ordinary annuity at 7% annual interest?
i = .07
n = 10
S = 1,500,000
Solve for R
ECO B. Potter

38. Example 2 i = .07 n = 10 S = 1,500,000 Solve for R
The company needs to invest $108, each year to accumulate $1.5m in 10 years
ECO B. Potter

39. Retirement – Bill & Mary
Bill and Mary both graduated from UCF with similar degrees, and obtained similar jobs. Bill immediately began putting $200 per month into an ordinary annuity paying 8% interest compounded monthly for his retirement. Mary, on the other hand, decided to put off retirement planning for the current time.
Ten years later, Bill stopped paying into his annuity and decided to let the money sit and continue earning interest. In the meantime, Mary decided to start a retirement account, and began investing $200 per month into an annuity paying 8% interest compounded monthly.
Another 20 years goes by. How much money do Bill and Mary have in their retirement accounts?
ECO B. Potter

40. Retirement - Bill Ordinary annuity for 10 years
Compound interest for 20 more years
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41. Retirement - Mary Ordinary annuity for 20 years
After graduating 30 years ago, Bill invested $24,000 and now has $180,267 in his retirement account. Mary contributed $48,000 and now has $117,804 in her retirement account.
What if they decide to postpone retirement for 10 more years?
ECO B. Potter

42. Retirement - Mary
Bill – continues earning compound interest for 10 years
Mary – continues contributing $200 per month into her ordinary annuity for 10 more years (30 years total)
Bill’s total contribution = $24,000
Mary’s total contribution = $72,000
ECO B. Potter

43. Now You Try
A 45-year-old man puts $1000 in a retirement account at the end of each quarter until he reaches the age of 60 and makes no further deposits. If the account pays 8% interest compounded quarterly, how much will be in the account when the man retires at age 65?
$1000 each quarter from age 45 to 60 (15 years)
Account pays 8% interest compounded quarterly.
Value at age 65?
ECO B. Potter

44. Future Value of an Annuity Due
Annuities in which payments are made at the beginning of each time period.
The future value of an annuity due of n payments of R dollars at the beginning of consecutive interest periods, with interest compounded at the rate of i per period is
ECO B. Potter

45. Now You Try
Bill is paid on the first day of the month and $80 is automatically deducted from his pay and deposited in a savings account. If the account pays 2.5% interest compounded monthly, how much will be in the account after 3 years and 9 months?
ECO B. Potter

46. 5.3 Present Value of an Annuity; Amortization
You’ve just inherited a large amount of money from your dear Aunt Sally! The only stipulation is that you can’t touch the money for 20 years, and it must be deposited into a savings account that pays 6 percent annual interest compounded annually. You have 2 payment options to choose from:
20 annual deposits of $35,000 (20  $35,000 = $700,000);
A lump-sum deposit of $400,000
Present value of an annuity – The amount that would have to be deposited in one lump sum today (at the same compound interest rate) in order to produce the same balance at the end of t years.
ECO B. Potter

47. Present Value of an Annuity
Future value of 20 annual deposits (n) of $35,000 (R) at 6% annual interest (i) compounded annually:
What lump sum deposit at i = .06 and n = 20, would amount to $1,287,495.70?
Present value of an annuity (P)
ECO B. Potter

48. Present Value of an Annuity
The present value P of an annuity of n payments of R dollars each at the end of consecutive interest periods with interest compounded at a rate of interest i per period is
Therefore, a lump sum deposit of $401,450 would equal the future value of the annuity described.
ECO B. Potter

49. Present Value of an Annuity
Determine the present value (P) of an annuity that pays $100 per month for 10 years at 5% interest compounded monthly.
R = $100, i = .05/12, n = 120
ECO B. Potter

50. Present Value of an Annuity; Amortization
Car Payments
A car costs $32,000. After a down payment of $2000, the balance ($30,000) will be paid off in 60 equal payments with interest of 6% per year on the unpaid balance. What is the monthly payment?
A single lump sum payment of $30,000 today would pay off the loan. So, $30,000 is the present value of an annuity of 60 monthly payments with i = .06  12 = Thus,
P = 30,000, n = 60, i = Solve for R...
ECO B. Potter

51. Present Value of an Annuity; Amortization
P = 30,000, n = 60, i = Solve for R
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52. Amortization
The process of gradually reducing a debt through installment payments of principal and interest, versus paying off the debt all at once.
The periodic payment needed to amortize a loan may be found, as in the car payment example, by solving the present value equation for R.
ECO B. Potter

53. Amortization AMORTIZATION PAYMENTS
A loan of P dollars at interest rate i per period may be amortized in n equal payments of R dollars made at the end of each period, where
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54. Amortization
Bob & Kathy bought a house for $250,000. After a down payment of $25,000 they financed the remaining $225,000 for 30 years at 6% annual interest.
Calculate their monthly mortgage payments.
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55. Amortization
Bob & Kathy bought a house for $250,000. After a down payment of $25,000 they financed the remaining $225,000 for 30 years at 6% annual interest.
How much of their first mortgage payment went towards paying interest, and how much went toward the principal?
$1,125 went toward interest
$ $1125 = $224 toward the principal.
ECO B. Potter

56. Principal at End of Period
Amortization Table
Payment Number
Amount of payment
Interest for Period
Portion to Principal
Principal at End of Period $ 1 $ $
$224.00 2
225.12 3
226.25 4
227.38 5
228.51
180
802.02
546.98
360
6.65
0.00
ECO B. Potter

57. Now You Try
Kareem Adams buys a house for $285,000. He pays $60,000 down and takes out a mortgage at 6.9% on the balance. Find his monthly payment and the total amount of interest he will pay if the length of the mortgage is:
15 years
20 years
ECO B. Potter

58. 
Chapter 5
End
ECO B. Potter