Interest: Amount of money you earn on savings. Amount of money you pay for barrowing.
Principal: Amount of money barrowed or deposited.
The amount you earn or pay in interest depends on the interest rate you are paying or earned.
Amount of time you’ve had the savings or paid the loan
Simple interest
Interest = (principal) (rate) (time)
Calculated on the principal only
Ex. You deposit 2000 at 6% for 1 year
I=PRT
I= 2000 (.06) (1)
I=120
a. A student took out a simple interest loan for 1800 for 2 years @ a rate of 8%.
1800* (.08) (2) = $288 = 2288 total /24 months = $87
b. A student took out a simple interest loan for 2400 for 2 years @ a rate of 7%.
2400* .07 * 2 = $336
336+2400 /24 months = $114
Future value
Principal + interest
Student 1 1800 + 288 = 2088
Student2 2400 +336 = 2736
In general future value A = P + I
P (1+ R T)
Calculating future value for simple interest.
A=P (1 +R*T)
Ex.
A loan for 1060 has been taken out at 6.5 for 3 months. Find the future value.
1060 *1.065 *.25 = $ 1,077.23
Banks use 360 days as a year vs. the 365 days.
Compare the value of t for 125 day short term loan using the 360 days and the 365 days.
125/360 = .347 vs 125/365 = .342
The bank benefits from the bankers rule since the loan is calculated for a greater period of time.
Compound interest:
Interest is earned on the original amount deposited and the interest already posted to your account.
Ex.
Years – beginning balance 1000 at 5% interest compounded yearly for 3 years.
1. 1000: A = 1000 (1 + .05) = 1050
2. 1050: A 1050 (1 +.05) = 1102.50
3. 1102.5 : A = 1102.5 (1 + .05) = 1157.63
Simple interest would be
A = 1000 (1+.05*3) = 1150
Compound interest will give you an extra $ 7.63
In general, calculating the amount in an account for compound interest paid once per year.
A= 1000 (1 +.05)^3 from our example
A = Present value ( 1 + rate) ^ Time
Ex. Calculate compound interest
Deposit $2000 at 6% annual interest for 3 years
A = 2000 (1+ .06)^3
A = $2382.03
Ex. 1000 @ 4% for 5 years
A = 1000 (1 +.04)^5
A = $1216.65
Compounding periods:
Depending on the financial institution and account, interest may be compounded more than once a
year.
Compounded annually = 1 a year
Semiannually = 2 a year
Quarterly = 4 a year (every 3 months)
In general, when interest is paid N times per year, we say there are n compounding periods.
Calculating the amount in an account for compounded interest paid n times a year.
A = P ( 1+ r/n)^nt  total of compounding periods
Calculate compounded interest:
Deposit $7500 @ 6% annually for 5 years interest is compounded monthly.
A = 7500 ( 1+ .06/12) ^12(5)
A = 10,116.38
Deposit 4200 at 4% for 10 years compounded quarterly
A = 4200 (1+ .04/4)^4(10)
A = 6253.23
Effective annual yield/Effective Rate
The simple interest rate that produces the same amount at the end of the year as when compounded.
Present value $4000 Rate 8% time 1 year compounded monthly
A = P (1 +r/n) ^nt
A = 4000 (1 + .08/12) ^ 12*1
A = 4332.00
A = P (1+rt)
4332=4000 (1 +R)
4332/4000 = 4000/4000 (1+R)
1.083 = 1 + R
.083 = R
8.3% = R
4332 is the future value if compounded monthly
A = 4000 (1+ .083(1))
The money invested at 8% (nominal rate) compounded monthly yields the same amount as earning 8.3%
simple interest (effective rate).
Present value $6000, rate 10%, time 1 year
A = P (1 +r/n) ^nt
A = 6000 (1 +.10/12) 12(1)
A = 6,628.28
6,628.28 = 6000 ( 1 + r t)
1.1047 – 1 = r r = 10.47%
Formula for calculating effective annual yield
Y = (1 + r/n)^n -1
Annuities – A series of equal payments made at equal time periods.
IRA, 403b,401K are examples of annuities.
Values of an annuity = sum of the deposits + all interest paid
But because the money is a deposited at different times, the interest earned is calculated a little
differently.
Example 1
A person deposits $ 1,000 at the end of the year for 3 years. The interest is compounded annually at the
interest rate of 8%
12/31 year 1
12/31 year 2
1000
1000 (1+.08) +1000 = 2,080
year 3
2,080 (1+ .08) + 1,000 = 3246.4
Ex.2
A person deposits $2,000 at the end of the year for 3 years. The interest rate is 10% compounded
annually.
Year 1 2,000 = 2,000
Year 2 2000 (1 + .10) +2000 = 4,200
Year 3 4,200 ( 1 + .10) +2000 = 6620
Ex.
Determine the value of the annuity saving for retirement.
Deposit 1000 at the end of each of the next 30 years. The interest rate will be 10% compounded
annually.
A= P [(1 +r) ^t -1] / r
When interest is compounded n times per year
A = P [(1 + r/n) nt -1] / (r/n)
Determine the value of the annuity: at age 25, to save for retirement, you decide to deposit $200 at the
end of each month into the IRA that pays %7.5 compounded monthly (n = 12) Find the value at age 65 (t
= 65-25 = 40)
A = 200[(1+ .075/12) 12*40 -1] / (.075/12)
A = $604,764.43
At age 30, to save for retirement, you decide to deposit $100 at the end of each month into an IRA
paying 9.5% compounded monthly.