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Today in Precalculus
• Turn in graded worksheet• Notes: Mathematics of Finance
(need a calculator)– Interest rates– Annual Percentage Yield
• Homework
Compound Interest
• Compound interest:
1kt
rA P
k
A: final amountP: principalr: interest ratek: number of payments per yeart: number of years
Example 1
P=$18,000
k = 2
t = 6
r= 0.075(2 6)
.07518,000 1
2A
=$27,998.18
Example 1b
P=$18,000
k = 4
t = ?
r= 0.075
A=$27,00027,000=18,000(1.01875)4t
1.5=1.018754t
log1.5 = 4tlog1.01875
4t=21.827
t = 5.457 years
4.075
27,000 18,000 14
t
log1.54
log1.01875t
Example 1cA=$31,500
P=$18,000
k = 12
t = 15
r= ?
(180)
1.75 112
r
(12 15)
31,500 18,000 112
r
180 1.75 112
r
180 1.75 112
r
18012 1.75 1 r r = .0374 So an interest rate of 3.74% is required.
ContinuouslyCompound Interest
• Continuously Compounded interest: A=Pert
A: final amountP: principalr: interest ratet: number of years
Example 2
P=20,000
t = 8
r = 0.053
A=20,000e(8•.053)
= $30,561.23
Example 2b
A = $100,000P = $20,000r = 0.053t = ?
100,000 = 20,000e.053t
5 = e0.053t
ln5 = 0.053t
t = 30.367 years
ln 5
0.053t
Example 2c
A = $40,000
P = $20,000
r = ?
t = 10
40,000 = 20,000e10r
2 = e10r
ln2=10rr =.0693An interest rate of 6.93% is needed
Annual Percentage Yield (APY)
• A common basis for comparing investments with different interest rates and methods of compounding.
• The percentage rate that, if compounded annually, would yield the same return as the given interest rate with the given compounding.
Example 1Bob invests with the local bank at 2.12% interest compounded monthly. What is the equivalent APY?
So the APY is 2.14%.
This means his investment compounded monthly at 2.12% earns the same interest as if it earned 2.14%, compounded once a year.
(12).0212
1 112
x
=.0214
Example 2
Which investment is more attractive, one that pays 8.5% compounded quarterly or another that pays 8.45% compounded monthly?
x=.0877
4.085
1 14
x
12.0845
1 112
x
x=.0879
So the investment with the 8.45% rate compounded monthly is the better investment
Example 3The interest charge by your credit card company is 18.75% APY. What is the monthly interest charge on a balance of $5,000?
(12)
1.1875 112
r
(12)
.1875 1 112
r
12 1.1875 112
r
12 1.1875 112
r
1212 1.1875 1 r
r = .1731 $5000x.1731 = $865.43
Homework
• Pg 341: 3-9 odd, 21-39odd, 41-46all