MATH 109
Simple Interest
If you deposit money for a fixed term with simple interest, then you simply receive a
fixed percentage of your deposit at the end of the term. The interest does not
compound unless you roll over the term.
Savings accounts generally pay simple interest each month on the average daily
balance. With a fixed interest rate r (in decimal), the amount of simple interest paid on
a deposit of $ P over time t (where t is a fraction of the year) is given by
I = Pr t
The new balance is P + P r t = P ! (1 + r t) .
Example 1. On Dec. 31, 2010, you deposited $1200 into a savings account that earned
2% simple interest each month. Find the monthly interest and balance for the first six
months of 2011.
Term
Opening
Days
r = Interest rate = 0.02
I = Pr t
Balance
$1200.00
Jan. 1 – Jan. 31
31
1200!. 02 ! 31 / 365 = $2.04
$1202.04
Feb. 1 – Feb. 28
28
1202.04!.02 ! 28 / 365 = $1.84
$1203.88
Mar. 1 – Mar. 31
31
1203.88!.02 ! 31 / 365 = $2.04
$1205.92
Apr. 1 – Apr. 30
30
1205.92! .02 ! 30 / 365 = $1.98
$1207.90
May 1 – May 31
31
1207. 90!.02 ! 31 / 365 = $2.05
$1209.95
June 1 – June 30
30
1209. 95!. 02 ! 30 / 365 = $1.99
$1211.94
Suppose instead that simple interest is paid at the end of every quarter (i.e., after
every three months). Find the interest and balance for the first two quarters of 2011.
Term
Opening
Days
Interest rate = 2%
I = Pr t
Balance
$1200.00
Jan. 1 – Mar. 31
90
1200!. 02 ! 90 / 365 = $5.92
$1205.92
Apr. 1 – June 30
91
1205.92! .02 ! 91 / 365 = $6.01
$1211.93
Why is it better to have interest paid each month rather than each quarter?
Average Daily Balance
If you make deposits or withdrawals into your savings account during the month, then
your balance changes. By adding up the balances of each day and then dividing by the
number of days in the month, we obtain the average daily balance A for the account.
Then simple interest is paid on this averge balance:
" # days %
'
Int = A r t = A ! r ! $#
365 &
!
Example 2. On Jan.1, 2012, you have $3000 in a savings account that pays 2.1% simple
interest each month. On Feb. 10, an additional deposit of $600 is credited.
(a) Find the interest earned and balance at the end of January.
(b) Find the average daily balance for February.
(c) Find the interest earned and balance at the end of February.
Solution. (a) For Jan 1 – Jan 31, the interest earned is 3000 " .021" 31 / 365 = $5.35, which
gives a balance of $3005.35 at the end of January.
(b) For Feb 1 to Feb 9 (9 days) the balance is $3005.35. For Feb 10 to Feb 29 (20 days),
the balance is $3605.35. So the average daily!balance for February is
3005.35 " 9 + 3605.35 " 20
= $3419.14 .
29
(c) The interest for Feb. is then 3419.14 " .021" 29 / 365 = $5.70, and the balance at the
end of Feb. is 3605.35 + 5.70 = $3611.05.
!
!
Municipal Bonds
When cities or states need to raise money for certain construction projects, they often
sell bonds that pay simple interest each year for a long number of years. After say 20
years, investors can redeem the bonds and the interest is usually tax-exempt.
Hopefully, the city can afford to pay off the bonds at that time from all the revenue
generated by the new project.
For such an investment of $ P with yearly simple interest, the value of the bond
after t years, called the future value FV , is given by
FV = P ! (1 + r)t
Example 3. You buy a municipal bond for $10,000 that pays 4.8% simple interest per
year. You may redeem it after 25 years. (a) What is the future value of the bond? (b)
Suppose you wish the future value to be $50,000 to be used as a (tax-deductible) gift to
your alma mater. What should your initial investment be?
Solution. (a) After 25 years, the (tax-free) value of the bond is
FV = P ! (1 + r)t = 10,000 ! (1 + 0. 048)25 = 10,000 ! 1. 04825 = $32,287.32.
25
(b) Solve for P in the equation P ! 1.048
! You should invest P =
= 50,000 .
50000
= $15,485.96 in order to achieve $50,000 in 25 years.
1.04825
Example 4. In 1991, a college graduate started a new job at $27,708 per year. Due to the
booming economy, her yearly raises have averaged 4.8062% since that time.
Approximate her salary now after 21 years of such raises.
Solution. The precise salary cannot be determined without knowing the exact
percentage raise each year. But assuming a 4.8062% raise each year, then after 21 years
the salary is 27, 708 ! (1 + 0.048062 )21 = 27, 708 ! 1.04806221 ≈ $74,256. (It pays to get
that degree!)
Exercises
1. (a) In July 2011, your savings account interest rate goes up to 2.4%. Using the data
from Example 1, find the monthly interest and balance for the last six months of 2011.
(b) Assuming interest is only paid quarterly, find the interest and balance for the last
two quarters of 2011.
2. On Apr.1, 2011, you have $2000 in a savings account that pays 1.88% simple interest
each month. On Apr. 13, an additional deposit of $500 is credited.
(a) Find the average daily balance for April.
(b) Find the interest earned and balance at the end of April.
(c) On May 11, a $400 is credited. Find the interest earned and balance at the end of
May.
3. You purchase a $5000 tax-free municipal bond that can be redeemed after 15 years.
The bond pays 5.2% yearly simple interest. (a) What is its future value? (b) If you
wish to have $15,000 in 15 years, what should your initial investment be?
Solutions
1. (a) Term
End of June
Days
Interest rate = 2.4%
I = Pr t
Balance
$1211.94
July 1 – July 31
31
1211.94!.024 ! 31 / 365 = $2.47
$1214.41
Aug. 1 – Aug. 31
31
1214.41!. 024 ! 31 / 365 = $2.48
$1216.89
Sep. 1 – Sep. 30
30
1216. 89!. 024 ! 30 / 365 = $2.40
$1219.29
Oct. 1 – Oct. 31
31
1219. 29!.024 ! 31 / 365 = $2.49
$1221.78
Nov. 1 – Nov. 30
30
1221.78!.024 ! 30 / 365 = $2.41
$1224.19
Dec. 1 – Dec. 31
31
1224.19!.024 ! 31 / 365 = $2.50
$1226.69
Now using $1211.93 as the June balance for the quarterly simple interest, we have
1. (b) Term
End of June
Days
Interest rate = 2.4%
I = Pr t
Balance
$1211.93
July 1 – Sep. 30
92
1211.93!. 024 ! 92 / 365 = $7.33
$1219.26
Oct. 1 – Dec. 31
92
1219. 26!.024 ! 92 / 365 = $7.38
$1226.64
2. (a) For Apr 1 to Apr 12 (12 days) the balance is $2000. For Apr 13 to Apr 30 (18
days), the balance is $2500. So the average daily balance for April is
2000 "12 + 2500 "18
= $2300 .
30
(b) The interest for Apr. is then 2300 " .0188 " 30 / 365 = $3.55, and the balance at the
end of April is 2500 + 3.55 = $2503.55.
!
(c) For May 1 to May 10 (10 days) the balance is $2503.55. For May 11 to May 31 (21
!
days), the balance is $2903.55.
So the average daily balance for May is
2503.55 "10 + 2903.55 " 21
= $2774.52 .
31
The interest for May is then 2774.52 " .0188 " 31 / 365 = $4.33, and the balance at the end
of May is 2903.55 + 4.33 = $2907.88.
!
3. (a) After 15 years, the value of the bond is
!
15
FV = P ! (1 + r)t = 5000 ! (1 + 0. 052 )15 = 5000 ! 1. 052
15
= $10,695.62.
(b) Solve for P in the equation P ! 1.052 = 12,500 .
12500
! You should invest P =
= $5843.52 in order to achieve $12,500 in 15 years.
1.05215