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Lecture Two
Section 2.1 – Simple and Compound Interest
Simple Interest simply means money is not compounded only once between the beginning and the end
of the note (or investment). With simple interest, the interest is calculated only at the end of the time
period {rather than periodically as is done with compounding.}
PrI t
(1 ) 1
AA P rt Prt
A: Amount is the balance of the account (including the interest).
P: Principal is the initial amount of principal that is borrowed or invested.
I: Interest is the fee that is applied to the note or investment.
r: Rate is the interest rate
t: Time is time in years.
Example
Find the total amount due on a loan of $500 at 12% simple interest at the end of 30 months.
Solution
(1 )A P rt
30
12500 1 .12
$650
500 * (1 + .12(30/12))
Example
To buy furniture for a new apartment, you borrowed $5,000 at rate of 8% simple interest for 11 months.
How much should you pay?
Solution
PrI t
115000(.08)12
5000*.08*11/ 12
$366.67
2
Example
T-Bills are one of the instrument of the U.S Treasury Department uses to finance the public debit. If you
buy a 180-day T-bill with a maturity value of $10,000 for $9,828.74, what annual simple interest rate will
you earn?
Solution
180365
.5t
(1 )A P rt
10000 9828.74(1 .5 )r
10000
9828.741 .5r
10000
9828.741 .5r
100009828.74
1
.5r
(10000 / 9828.74 – 1) / .5
.03485 3.485%r or
Example
Find the maturity value for a loan of $2500 to be repaid in 8 months with interest of 9.2%.
Solution
Given: 8 212 3
2,500 0.092P r t
(1 )A P rt
22500 1 .0923
$2,653.33
3
Compounded Interest
1mt
r
mA P
A: Amount in the account (also called future value)
P: Amount invested ($)
r: Annual simple interest rate
t: Time in years
m: Number of times a year the interest is compounded
Daily: m = 365
Monthly: m = 12
Quarterly: m = 4
Semi-annually: m = 2
Annually: m = 1
Compounded Continuously
rtA Pe
Example
If $1,000 is invested at 6% compounded over an 8-year period.
Solution
a) Annually
1(8)
$1,593.06
11
.851000A
1000(1+.06/1)^(1*8)
b) Semiannually
2(8)
$1,604.06
12
.711000A
c) Quarterly
4(8)
$1,610.06
14
.321000A
d) Monthly
12(8)
$1,614.06
112
.141000A
4
Example
What amount will an account have after 1.5 years, if $8,000 is invested at annual rate of 9%
a) Compounded Weekly
52(1.5)
$9,15.09
152
5.238000A
8000(1+.09/52) ^ (52*1.5)
b) Compounded Continuously
.09(1.5) $9,156.298000eA 8000 e ^ (.09*1.5) (e: 2nd ln)
Example
How much should new parents invest now at 8% to have $80,000 toward their child’s college education
in 17 years if compounded?
a) Semiannually: m = 2
Given: r = 8% = 0.08 A = 80,000 t = 17
1
tmr
mA P
)17(2
208.01000,80
P
P
)17(2
208.01
000,80
2(17)
0.082
80,000
1
$21,084.17P
80000 / (1+ .08/2) ^ (2*17)
b) Continuously : rtA Pe
)17)(08(.000,80 eP
(.08)(17)
80,000 $20,532.86
eP
80000 / e ^ (.08*17)
5
Annual Percentage Yield (APY)
Annual 1 11mt m
r r
m mA P Pt
Amount @ simple interest = Amount @ Compound interest
(1 ) 1m
rm
P APY P Divide P both side
1 1m
rm
APY
1 1m
rm
APY
Continuously: 1reeAPY r
Effective Rate
The effective rate corresponding to a started rate of interest r compounded m times per year is
1 1m
erm
r
APY is also referred to as effective rate or true interest rate.
Example
How much should you invest now at 10% to have $8,000 toward the purchase of a car in 5 years if
compounded?
Solution
Given: r = 10% = 0.1 A = 8000 t = 5
a) Quarterly: m = 4
1
tmr
mA P
4(5)
4
0.118000 P
8000
200.1
14
$4882.17P
8000 / (1 0.1 / 4) 20
b) Continuously : rtA Pe
(0.1)(5)8000 Pe
8000 $4852.250.5
Pe
8000 / ( (.1*5))e
6
Example
A $10,000 investment in a particular growth mutual fund over a recent 10-year period would have grown
to $126,000. What annual nominal rate would produce the same growth if
Solution
a) Annually: m = 1
1(10)126000 10000 1
1
r
1012600010000
1 r
1012.6 1 r
101 12.6r
101 12.6r
1/10(12.6) 1 .28836 28.84% r or
(12.6) (1 / 10) 1
b) Continuously : rtA Pe
10126000 10000 re
1012.6 re
1012.6ln ln re
ln12.6 10 lnr e
ln12.6 10r
ln12.610
.25337 25 3 7% . 3r or
ln(12.6) / 10
Example
A bank pays interest of 4.9% compounded monthly. Find the effective rate.
Solution
Given: 0.049, 12r m
1 1m
re m
r
12
.0491 112
1 .049 / 12 ^12 1
0.0501156
Or 5.01%er
7
Example
You need to borrow money. Bank A charges 8% compounded semiannually. Another bank B charges
7.9% compounded monthly. At which bank will you pay the lesser amount of interest?
Solution
Bank A: 2
.081 1 1 1 .08162
8.16%m
rre mre
Bank B: 12
.0791 1 1 1 .0819 . 912
8 1 %m
rr ree m
Bank A has the lower effective rate, although it has a higher stated rate.
8
Exercises Section 2.1 – Simple and Compound Interest
1. If you want to earn an annual rate of 10% on your investments, how much should you pay for a note
that will be worth $5,000 in 6 month?
2. How much should you deposit initially in an account paying 10% compounded semiannually in
order to have $1,000,000 in 30 years? b) Compounded monthly? c) Compounded daily?
3. You have $7,000 toward the purchase of a $10,000 automobile. How long will it take the $7000 to
grow to the $10,000 if it is invested at 9% compounded quarterly? (Round up to the next highest
quarter if not exact.)
4. How long, to the nearest tenth of a year, will it take $1000 to grow to $3600 at 8% annual interest
compounded quarterly?
5. Jennifer invested $4,000 in her savings account for 4 years. When she withdrew it, she had
$4,350.52. Interest was compounded continuously. What was the interest rate on the account? Round
to the nearest tenth of a percent.
6. An actuary for a pension fund need to have $14.6 million grows to $22 million in 6 years. What
interest rate compounded annually does he need for this investment to growth as specified? Round
your answer to the nearest hundredth of a percent.
7. Which is the better investment: 9% compounded monthly or 9.1% compounded quarterly?
8. Sun Kang borrowed $5200 from his friend to pay for remodeling work on his house. He repaid the
loan 10 months later with simple interest at 7%. His friend then invested the proceeds in a 5-year
CD paying 6.3% compounded quarterly. How much will his friend have at the end of the 5 years?
9. The consumption of electricity has increased historically at 6% per year. If it continues to increase at
this rate indefinitely, find the number of years before the electric utilities will need to double their
generating capacity. Round up to the next highest year.
10. In the New Testament, Jesus commends a widow who contributed 2 mites (roughly ¼ cent) to the
temple treasury. Suppose the temple invested those mites at 4% compounded quarterly. How much
would the money be worth 2000 years later?
11. If $1,000 is invested in an account that earns 9.75% compounded annually for 6 years, find the
interest earned during each year and the amount in the account at the end of each year. Organize
your results in a table.
12. If $2,000 is invested in an account that earns 8.25% compounded annually for 5 years, find the
interest earned during each year and the amount in the account at the end of each year. Organize
your results in a table.
9
13. If an investment company pays 6% compounded semiannually, how much you should deposit now
to have $10,000
a) 5 years from now?
b) 10 years from now?
14. If an investment company pays 8% compounded quarterly, how much you should deposit now to
have $6,000
a) 3 years from now?
b) 6 years from now?
15. What is the annual percentage yield (APY) for money invested at
a) 4.5% compounded monthly?
b) 5.8% compounded quarterly?
16. What is the annual percentage yield (APY) for money invested at
a) 6.2% compounded semiannually?
b) 7.1% compounded monthly?
17. A newborn child receives a $20,000 gift toward a college education from her grandparents. How
much will the $20,000 be worth in 17 years if it is invested at 7% compounded quarterly?
18. A person with $14,000 is trying to decide whether to purchase a car now, or to invest the money at
6.5% compounded semiannually and then buy more expensive car. How much will be available for
the purchase of a car at the end of 3 years?
19. You borrowed $7200 from a bank to buy a car. You repaid the bank after 9 months at an annual
interest rate of 6.2%. Find the total amount you repaid. How much of this amount is interest?
20. An account for a corporation forgot to pay the firm’s income tax of $321,812.85 on time. The
government changed a penalty based on an annual interest rate of 13.4% for the 29 days the money
was late. Find the total amount (tax and penalty) that was paid. (Use 365 days a year.)
21. A bond with a face value of $10,000 in 10 years can be purchased now for $5,988.02. What is the
simple interest rate?
22. A stock that sold for $22 at the beginning of the year was selling for $24 at the end of the year. If the
stock paid a dividend of $0.50 per share, what is the simple interest rate on an investment in this
stock?
23. The Frank Russell Company is an investment fund that tracks the average performance of various
groups of stocks. On average, a $10,000 investment in midcap growth funds over a recent 10-year
period would have grown to $63,000. What annual nominal rate would produce the same growth if
1
Solution Section 2.1 – Simple and Compound Interest
Exercise
If you want to earn an annual rate of 10% on your investments, how much should you pay for a note that
will be worth $5,000 in 6 month?
Solution
(1 )A P rt
6
125000 1 .1( )P
.1
25000 1P
5000 $4761.90
.112
P 5000 / (1 + .1 / 2)
Exercise
a) How much should you deposit initially in an account paying 10% compounded semiannually in order
to have $1,000,000 in 30 years?
b) Compounded monthly?
c) Compounded daily?
Solution
1
1 1mt
mt mtA
r
m
r rr Ao
m mA P P
a) 1000000
$53,535.52601 0 10 2
P( . / )
60 1000000(1 0.10 / 2) $53,535 52 .or
b) 360
1000000(1 0.10 / 12) $50,409.83P
c) 10950
1000000(1 0.10 / 365) $49,807.53P
2
Exercise
You have $7,000 toward the purchase of a $10,000 automobile. How long will it take the $7000 to grow
to the $10,000 if it is invested at 9% compounded quarterly? (Round up to the next highest quarter if not
exact.)
Solution
10,000 7,000(1 0.09 / 4)n
10 / 7 1.0225n
ln(10 / 7) ln1.0225n Log Power Property: lnax= xlna
ln(10 / 7) ln1.0225n
ln(10 / 7)16.03
ln1.022517 quartersorn
Exercise
How long, to the nearest tenth of a year, will it take $1000 to grow to $3600 at 8% annual interest
compounded quarterly?
Solution
1nt
P rAn
4
0.083600 1000 14
t
43.6 1.02t
4ln 3.6 ln 1.02t
ln 3.6 4 ln 1.02t
ln 3.64 ln1.02
t
16.2t yr
3
Exercise
Jennifer invested $4,000 in her savings account for 4 years. When she withdrew it, she had $4,350.52.
Interest was compounded continuously. What was the interest rate on the account? Round to the nearest
tenth of a percent.
Solution
rtA Pe
44350.52 4000 re
44350.524000
re
44350.52ln ln4000
re
Inverse Property: ln
xe x
4350.52ln 44000
r
14
4350.52ln .0214000
r
2.1%r
Exercise
An actuary for a pension fund need to have $14.6 million grow to $22 million in 6 years. What interest
rate compounded annually does he need for this investment to growth as specified. Round your answer to
the nearest hundredth of a percent.
Solution
(1)(6)
122 14.6 1 r
622 14.6(1 )r
622 (1 )14.6
r
1/6
22 114.6
r
1/6
22 114.6
r
.0707
7.07%r
4
Exercise
Which is the better investment: 9% compounded monthly or 9.1% compounded quarterly?
Solution
For 9%:
120.09
1 1 9.38%12
APY er
For 9.1%:
40.091
1 1 9.42%4
re
9.1% is better
Exercise
Sun Kang borrowed $5200 from his friend to pay for remodeling work on his house. He repaid the loan
10 months later with simple interest at 7%. His friend then invested the proceeds in a 5-year CD paying
6.3% compounded quarterly. How much will his friend have at the end of the 5 years?
Solution
For 7%: 1
1A P rt
11012
5200 1 0.07 $5503.33A
For 6.3%: 2
200.0635503.33 1 $7,522 5
4. 0A
Exercise
The consumption of electricity has increased historically at 6% per year. If it continues to increase at this
rate indefinitely, find the number of years before the electric utilities will need to double their generating
capacity. {Round up to the next highest year}
Solution
0.061
2 1n
P P
2 1.06n ln both sides
ln ln2 1.06n
ln l2 06n1.n
2 1.06
1lnln
2 yearsn
5
Exercise
In the New Testament, Jesus commends a widow who contributed 2 mites (roughly ¼ cent) to the temple
treasury. Suppose the temple invested those mites at 4% compounded quarterly. How much would the
money be worth 2000 years later?
Solution
Given: $1
1001 .25 $0.00254 c
P c c
0.04 4 2000r m t
1mt
rA P
m
4(2000)
0.040.0025 14
A 0.0025 1 0.04/4 ^(4*2000)
31$9.3 10x
Exercise
If $1,000 is invested in an account that earns 9.75% compounded annually for 6 years, find the interest
earned during each year and the amount in the account at the end of each year. Organize your results in a
table.
Solution
Given: 1,000 .0975 1 6P r m t
1mt
rA P
m
1st
year: 1
1(1).09751 1000 1 $1,097.50
1t A
$1,097.50 $1,000 $97.50Interest
2nd
year: 2
22 1000 1 .0975 $1, 204.51t A
$1,204.51 $1,097.50 $107.01Interest
3rd
year: 3
33 1000 1 .0975 $1,321.95t A
4th
year: 4
44 1000 1 .0975 $1, 450.84t A
5th
year: 5
55 1000 1 .0975 $1,592.29t A
6th
year: 6
66 1000 1 .0975 $1,747.54t A
Period Amount Interest
0 $1,000.00
1 $1,097.50 $97.50
2 $1,204.51 $107.01
3 $1,321.95 $117.44
4 $1,450.84 $128.89
5 $1,582.29 $141.46
6 $1,747.54 $155.25
6
Exercise
If $2,000 is invested in an account that earns 8.25% compounded annually for 5 years, find the interest
earned during each year and the amount in the account at the end of each year. Organize your results in a
table.
Solution
Given: 2,000 8.25% .08.25 1 5P r m t
1mt
rA P
m
1st
year: 1
1(1).08251 1000 1 $2,165.00
1t A
$2,165.00 $2,000 $165.00Interest
2nd
year: 2
22 2000 1 .0825 $2,343.61t A
$2,343.61 $2,165.00 $178.61Interest
3rd
year: 3
33 2000 1 .0825 $2,536.96t A
4th
year: 4
44 2000 1 .0825 $2,746.26t A
5th
year: 5
55 2000 1 .0825 $2,972.83t A
Period Amount Interest
0 $2,000.00
1 $2,165.00 $165.00
2 $2,343.61 $178.61
3 $2,536.96 $193.35
4 $2,746.26 $209.30
5 $2,972.83 $226.57
Exercise
If an investment company pays 6% compounded semiannually, how much you should deposit now to
have $10,000
a) 5 years from now?
b) 10 years from now?
Solution
Given: 10,000 .06 2A r m
a) 5 1mt
rt A Pm
2(5)
.0610,000 12
P
1010,000 1.03P
10
10,000
1.03$7, 440.94 P
b) 10t
7
2(10)10,000 1.03P
20
10,000
1.03$5,536.76 P
Exercise
If an investment company pays 8% compounded quarterly, how much you should deposit now to have
$6,000
a) 3 years from now?
b) 6 years from now?
Solution
Given: 6,000 8% .05 4A r m
a) 3 1mt
rt A Pm
4(3)
.086,000 14
P
126,000 1.02P
12
6,000
1.02 $4,730.96P 6000 / 1.02 ^12
b) 6t
4(6)
.086,000 14
P
246,000 1.02P
24
6,000
1.02 $3,730.33P
8
Exercise
What is the annual percentage yield (APY) for money invested at:
a) 4.5% compounded monthly?
b) 5.8% compounded quarterly?
Solution
a) Given: 0.045 12r m
12
.0451 1 0.0459412
APY 1 .045 / 12 ^ 12 1
: 4.594%APY
b) Given: 0.058 4r m
4
.0581 1 0.059274
APY 1 .058 / 4 ^ 4 1
: 5.927%APY
Exercise
48.
What is the annual percentage yield (APY) for money invested at
a) 6.2% compounded semiannually?
b) 7.1% compounded monthly?
Solution
a) Given: 6.2% 0.062 2r m
2
.0621 1 0.062962
APY 1 .062 / 2 ^ 2 1
: 6.296%APY
b) Given: 7.1% 0.071 12r m
12
.0711 1 0.0733612
APY 1 .071 / 12 ^12 1
: 7.336%APY
9
Exercise
A newborn child receives a $20,000 gift toward a college education from her grandparents. How much
will the $20,000 be worth in 17 years if it is invested at 7% compounded quarterly?
Solution
Given: 20,000 7% .07 4 17P r m t
1mt
rA Pm
4(17)
.0720,000 14
20000 1 .07/4 ^ 4*17
$65,068.44
Exercise
A person with $14,000 is trying to decide whether to purchase a car now, or to invest the money at 6.5%
compounded semiannually and then buy more expensive car. How much will be available for the
purchase of a car at the end of 3 years?
Solution
Given: 14,000 6.5% .065 2 3P r m t
1mt
rA Pm
2(3)
.06514,000 12
14000 1 .065 / 2 ^ 2 *3
$16,961.66
Exercise
You borrowed $7200 from a bank to buy a car. You repaid the bank after 9 months at an annual interest
rate of 6.2%. Find the total amount you repaid. How much of this amount is interest?
Solution
Given: 97, 200 6.2% .06212
P r t
1A P rt
97200 1 .06212
7200 1 .07 9 /12
$7,534.80
10
Exercise
An account for a corporation forgot to pay the firm’s income tax of $321,812.85 on time. The government
changed a penalty based on an annual interest rate of 13.4% for the 29 days the money was late. Find the
total amount (tax and penalty) that was paid. (Use 365 days a year.)
Solution
Given: 29321,812.85 13.4% .13.4365
P r t
1A P rt
29321,812.85 1 .134365
7200 1 .07 9 /12
$325, 239.05
Exercise
A bond with a face value of $10,000 in 10 years can be purchased now for $5,988.02. What is the simple
interest rate?
Solution
The interest earned is: $10,000 $5,988.02 = $4011.98
Given: 5988.02 4011.98 10P I t
PrI t
4011.98 5988.02 10 r
4011.98 0.067
5988.02 10 r 4011.98 / 5988.02 *10
The interest rate was about 6.7%
Exercise
A stock that sold for $22 at the beginning of the year was selling for $24 at the end of the year. If the
stock paid a dividend of $0.50 per share, what is the simple interest rate on an investment in this stock?
Solution
The interest earned is: $24 $22 $0.50 $2.50
Given: 22 2.50 1P I t
PrI t
2.50 22 1 r
2.5 0.1136422
r
The interest rate was about 11.36%
11
Exercise
The Frank Russell Company is an investment fund that tracks the average performance of various groups
of stocks. On average, a $10,000 investment in midcap growth funds over a recent 10-year period would
have grown to $63,000. What annual nominal rate would produce the same growth if
a) Annually
b) Continuously
Solution
a) Annually: m = 1
)10(1
11000,10000,63
r
1011000063000 r
1013.6 r
r13.610
10 6.3 1 0.20208 20.208%r or
b) Continuously : rtA Pe
)10(000,10000,63 re
re103.6
106.3 rln lne ln xe x
6.3 10ln r
6.3 0.18405 18.4 10
05%lnr orlne
