FINITE - UNIT 3 - FINANCE
Simple Interest:
Interest earned I = P r t
(Revised 4/16)
Future value A = P + Prt → A = P (1 + r t)
Compound Continuously: Future value A = P er t (Will need to use logarithms when given A and asked to find r or t.)
Note: Simple interest and interest compounded continuously cannot be calculated in the TVM solver as it requires the number
of compoundings to be a finite number larger than 0.
mt
r

Compound Interest A  P1  
 m
Where I = interest earned (or owed) for the given time period
A = future value or accumulated amount
P = principal or present value or initial amount
r = annual interest rate (written as a decimal)
t = the number of years (If it is a partial year, “t” will be a fraction.)
m = number of periods per year
n = m∙t , which is the total number of compounding periods
Can use TVM Solver (on the calculator) for compound interest:
APPS → Finance → 1. TVM Solver
N = total number of compounding periods or payments
I % = annual interest rate (written as a percent)
PV = present value (starting amount)
PMT = payment amount (made at regular intervals)
FV = future value (ending amount)
P/Y = payments per year (annually = 1, semi-annually = 2, quarterly = 4, monthly = 12
C/Y = compounding periods per year
PMT: END BEGIN (If the payments are made at the end or at the beginning of the compounding period)
NOTES:
 Money is directional in the TVM Solver.
Money moving away from you is negative (amount you invested, payments you make, etc.).
Money coming back to you is positive .
 Payments made each month
Assume “end” if it doesn’t specify. Ordinary Annuity → “end”
Sinking Fund → “end”
Annuity Due → “begin” (Instructions will usually indicate this.)
Effective Rate (Also called APY, “annual percent yield”)
This corresponds to a stated rate of interest r (as a decimal) compounded m times per year.
If compounded m times per year → APY: rE = (1 +
r m
m
) − 1 (Note: the − 1 takes away the original principal)
Can use the calculator: APPS → Finance → C:EFF (I%, C/Y)
(I% is written as a percent)
If compounded continuously → APY: rE = e r − 1 (Note: the − 1 takes away the original principal)
Cannot use the calculator.
Annuity → When a sequence of payments are made at equal periods of time.
Ordinary annuity → Frequency of payments is the same as the frequency of compounding periods.
The payments are usually made at the “end” of each time period.
Sinking fund → Similar to ordinary annuity. A fund set up to receive periodic payments. The payments plus
the interest are designed to produce a given sum at some time in the future.
Annuity Due → Annuity in which payments are made at the beginning of each time period.
Present Value of Annuity → The amount that would have to be deposited in one lump sum today (at the
same compound interest rate) in order to produce exactly the same ending balance of an
ordinary annuity for the same time period.
Future Value → The account balance at the end of the allocated time. For loans this value is 0.
Amortization → When a borrower agrees to make regular payments on the principal and interest until the
loan is paid off. An amortization table lists the monthly payments, how much goes toward interest,
how much goes toward the principal, and the remaining balance of the loan.
Example of Amortization Table
Borrow $10,000 at 5% interest compounded monthly to be repaid in 3 years. To amortize the loan, it was
determined that the borrower would have to pay $299.71 each month for the next 3 years to repay the loan
and its interest. Start an Amortization Table showing the first three monthly payments.
Interest: To determine how much interest was owed for each month, multiply the previous principal balance
by the interest rate (.05) and then divide by 12 to get the interest owed for that particular month.
Portion Applied to Principal: Subtract the interest amount for that month from the monthly payment.
Principal Balance: Subtract the “Portion Applied to Principal” from the previous “Principal Balance”.
Payment
Number
Amount of
Payment
Interest
for Period
Portion Applied
to Principal
Principal Balance
at End of Period
0
-
-
-
1
$ 299.71
(.05  10,000)÷12
= $ 41.67
299.71 – 41.67
= $ 258.04
10,000.00 – 258.04
= $ 9741.96
2
$ 299.71
(.05  9741.96)÷12
= $ 40.59
299.71 – 40.59
= $ 259.12
9741.96 – 259.12
= $ 9482.84
3
$ 299.71
(.05  9482.84)÷12
= $ 39.51
299.71 – 39.51
= $ 260.20
9482.84– 260.20
= $ 9222.64
$ 10,000.00
EXAMPLES USING TVM SOLVER
1. Finding the future value
You have $1000 at 2.1% compounded monthly
for 2 years. You also deposit $100 at the
beginning of each month for 2 years.
N = 12 ∙ 2 = 24
I% = 2.1
PV = −1000
PMT = −100
FV = 0 (what we are solving for)
P/Y = 12 (monthly)
C/Y = 12
PMT: End Begin
2. Finding the present value
You want to have $30,000 in 18 years,
compounded quarterly at 2.4%. How much
is needed to start with?
N = 4 ∙ 18 = 72
I% = 2.4
PV = 0 (what we are solving for)
PMT = 0 (no payments for this example)
FV = 30,000
P/Y = 4 (quarterly)
C/Y = 4
PMT: End Begin (doesn’t matter since no
payments were made)
Move cursor back up to FV.
Press ALPHA ENTER for “Solve”
Ans: FV = 3496.07
3. Finding the interest rate
Find the interest rate for an $8000 deposit
to accumulate to $10,006, compounded
quarterly for 5 years.
Move cursor back up to PV.
Press ALPHA ENTER for “Solve”
Ans: PV = - 19501.44
You need to invest $19,501.44
4. Finding the interest rate
Find the interest rate needed to accumulate
$30,466 over 5 years, if quarterly payments
of $1275 are made at the end of each quarter.
N = 4 ∙ 5 = 20
I% = 0 (what we are solving for)
PV = −8000
PMT = 0
FV = 10,006
P/Y = 4 (quarterly)
C/Y = 4
PMT: End Begin
N = 4 ∙ 5 = 20
I% = 0 (what we are solving for)
PV = 0
PMT = −1275
FV = 30,466
P/Y = 4 (quarterly)
C/Y = 4
PMT: End Begin
Move cursor back up to I%.
Press ALPHA ENTER for “Solve”
Ans: I% = 4.5 percent
Move cursor back up to I%.
Press ALPHA ENTER for “Solve”
Ans: I% = 7.33 percent
5. Comparing two different types of investments.
What sum deposited today at 6% compounded annually for 11 years will provide the same amount as
$1600 deposited at the end of each year for 11 years at 5% compounded annually?
First, find the future value of the 2nd option.
N = 11
I% = 5
PV = 0
PMT = -1600
FV = 0 (what we solving for)
P/Y = 1 (annually)
C/Y = 1
PMT: End Begin
Move cursor back up to FV.
Press ALPHA ENTER for “Solve”
Ans: FV =22,730.85946
Next, leave the FV in the calculator, compute the 1st option.
N = 11
I% = 6
PV = 0 (what we solving for)
PMT = 0
FV = 22,730.85946
P/Y = 1 (annually)
C/Y = 1
PMT: End Begin
Move cursor back up to PV.
Press ALPHA ENTER for “Solve”
Ans: PV = -11,974.3332
6. Often lottery winnings are divided into equal payments given annually for a set number of years. So the present
value of the winnings is worth less than the actual jackpot, depending on the rate at which money could be invested .
Find the present value if the jackpot amount is $15,000,000 to be paid annually for 20 years & the interest rate is 5%.
Note: Instead of thinking of $15,000,000 as the future value, think of the 20 equal payments 15,000,000/20 = 750,000.
The $15,000,000 that is owed you is being paid each year with a $750,000 payment.
After the 20 payments, they will owe you $0 (which is the “future value”).
N = 20
I% = 5
PV = 0 (what we are solving for)
PMT = -15,000,000/20 = -750,000
FV = 0
P/Y = 1 (annually)
C/Y = 1
PMT: End Begin
(The payment for this problem could be positive or negative. It doesn’t matter .)
Move cursor back up to PV.
Press ALPHA ENTER for “Solve”
Ans: PV =9346657.757 → $9,346,657.76