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Assuming a constant rate of inflation from 1987 to 2004, what is the inflation rate? • 1. Substitute in compound interest formula. • 2. Divide both sides by.
Typology: Lecture notes
1 / 43
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Definition:Definition:
I = PrtI =
Prt
I = interest earnedI = interest earned
P = principal ( amount invested)P = principal ( amount invested)
r = interest rate (as a decimal)r = interest rate (as a decimal)
t = timet = time
^
Find the interest on a boatFind the interest on a boatloan of $5,000 at 16% for 8loan of $5,000 at 16% for 8months.months. ^
Solution:Solution
: Use I =
Use I = Prt
Prt
^
^
(8 months =(8 months =
8/12 of one year =8/12 of one year =0.6667 years)0.6667 years)
^
Find the total amount due on a loan ofFind the total amount due on a loan of$600 at 16% interest at the end of 15$600 at 16% interest at the end of 15months.months.
solution: A =P(1+rt)solution:
A =P(1+rt)A^ A
= 600(1+0.16(1.25))= 600(1+0.16(1.25)) A
=
$720.
A
=
$720.
^
What is the annual interestWhat is the annual interestrate earned by a 33-rate earned by a 33
-day Tday T-
bill with a maturity value ofbill with a maturity value of$1,000 that sells for$1,000 that sells for$996.16?$996.16? ^
Solution: Use the equationSolution: Use the equationA =P(1+rt)A =P(1+rt) ^
1,000 = 996.161,000 = 996. ^
1000 = 996.16(1+r(0.09166))1000 = 996.16(1+r(0.09166))
Solve for r:Solve for r:
33
1
360 r ⎛^
⎞
⎛^
⎞
+^
⎜^
⎟
⎜^
⎟
⎝^
⎠
⎝^
⎠
1000 = 996.16(1+r(0.09166))1000=996.16+996.16(0.09166)r
1000-996. 996.16(0.09166)
4.2%
r
r
→
→
=^
→
=^
=
amount accumulates at a faster rate than simpleinterest. The basic idea is that after the firstinterest period, the amount of interest is addedto the principal amount and then the interest iscomputed on this higher principal. The latestcomputed interest is then added to the increasedprincipal and then interest is calculated again.This process is completed over a certain numberof compounding periods.
The result is a much
faster growth of money than simple interestwould yield.
-^
-^
amount after one month
-^
(^
) (^
)
2
2
2
3
From the previous example, we arrive at a generalization: Theamount to which 1.00 will grow after n months compoundedmonthly at 6% annual interest is :
where A is the future amount, P is the principal, r is the interestrate as a decimal, m is the number of compounding periods in oneyear and t is the total number of years. To simplify the formula, l
-^
=
where
1
(1.05)
12
n
n
⎛^
⎞
+^
=
⎜^
⎟
⎝^
⎠
1
mt r
A
P
m
⎛^
⎞
=
⎜^
⎟
⎝^
⎠^
(^
)
1
n
A
P
i
=
compounded quarterly at 6.75% interest for 10years.
-^
Solution: Use
-^
Helpful hint: Be sure to do the arithmetic using the rules for orderof operations. See arrows in formula above
10(4)
Changing the number ofChanging the number of
compounding periods per yearcompounding periods per yearTo what amount will $1500 grow if compounded
10 ( 365 )
1500
1
365
A
⎛^
⎞
=
⎜^
⎟
⎝^
⎠
Effect of increasing the numberEffect of increasing the number
of compounding periodsof compounding periods
periods per year is increased whilethe principal, annual rate of interestand total number of years remain thesame, the future amount of moneywill increase slightly.
remains the samefor the next 10years, what will thehouse be worth inthe year 2014?
68, 000(
0.0253)
133,
A
=
=
How long will it take for$5,000 to grow to $15,000 ifthe money is invested at8.5% compounded quarterly?
-^
( Note: you will most unlikely see thisamount during your lifetime)
(^
(^4) )
4
4
15, 000
5, 000 1
4
3, 000
(1.02125)
ln(3, 000)
ln (1.02125)
ln(3, 000)
4 ln(1.02125)
ln(3, 000)
4 ln(1.02125)
t
t
t
t
⎛^ t
⎞
=^
+^
→
⎜^
⎟
⎝^
⎠
=^
→
=^
→
=^
→
=
=
that is invested at :
Hint: Use the correct order of operations as indicated by thenumbers
1
1 m r
APY
m
⎛^
⎞
=^
−
⎜^
⎟
⎝^
⎠ 12
Computing the Annual nominalComputing the Annual nominal
rate given the effective raterate given the effective rate
-^
-^
1
1 m r
APY
m
⎛^
⎞
=^
+^
−
⎜^
⎟
⎝^
⎠
(^
)
(^1212)
12 12
12
1
1
12
1
12
1
12
1
12
12
1
r r
r r
r
⎛^ r
⎞
=^
+^
−
⎜^
⎟
⎝^
⎠
⎛^
⎞
=^
⎜^
⎟
⎝^
⎠
⎛^
⎞
=^
⎜^
⎟
⎝^
⎠
−^
= −
=
=
