8-2
Compound Interest
Chapter
Chapter 88
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8-3
Learning Objectives
After completing this chapter, you will be able to:
> Calculate maturity value, future value, and present value in
>
>
>
>
>
>
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compound interest applications, by both the algebraic method and
the pre-programmed financial calculator method
Calculate the maturity value of compound interest Guaranteed
Investment Certificates (GICs)
Calculate the price of "strip" bonds
Calculate the redemption value of a compound interest Canada
Savings Bond
Adapt the concepts and equations of compound interest to cases of
compound growth
Calculate the payment on any date that is equivalent to one or more
payments on other dates
Calculate the economic value of a payment stream
8-4
Compound Interest (Future Value)
Compounding involves the calculation
of interest over the life of
the loan or investment
Compound interest the interest on the principal plus
the interest of prior periods
Future value (compound amount)
Present Value -
- is the final amount of the loan or
the value of a loan
investment at the end of the last period
or investment today
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8-5
Interest earned
in the first year
Interest earned in
the second year
$110
$100
A
m
o
u
n
t
$1000
$1100
$1000
Beginning
principal
for
the second
year
($1100)
Beginning
principal
for
the third
year
($1210)
Compounding
period
0
1
Time (years)
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2
8-6
Table 8.1
Compounding Frequencies and
Periods
Compounding
frequency
Annual
Semiannual
Quarterly
Monthly
Daily
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Number of
compoundings per year
1
2
4
12
365
Compounding
period
1 year
6 months
3 months
1 month
1 day
8-7
Determining values for i and n
Number of periods (n)
Rate for each period (i)
# of years * # of times the
interest is compounded per year
Annual interest rate / # times
interest is compounded per year
If you compounded $100 for 3 years at
6% annually, semiannually, or
quarterly… What are values for n and i?
Periods (n)
Annually:
3*1=3
Semiannually: 3 * 2 = 6
Quarterly: 3 * 4 = 12
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Rate (i)
Annually:
6% / 1 = 6%
Semiannually: 6% / 2 = 3%
Quarterly: 6% / 4 = 1.5%
8-8
Formula for Future Value
FV = PV(1+i) n
Calculator Steps:
i+1=
yx n =
* PV =
MH Ryerson
8-9
Steve Smith deposited $1,000 in a savings
account for 4 years at an semiannual
compounded rate of 8%. What is Steve’s
interest and compounded amount?
Compounded amount
= Future Value
PV = $1000
n=4x2=8
.08/2
i = .08/2 = .04
+ 1 1200
=
FV= 1000(1 +
1400
.04)8
1000
x
y 8
800
600
= $1,000 *1.3686 = $1,368.60 * 1000
400
Interest:
= 200
I = $1,368.60 - $1,000 = $368.60
0
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Investment
2000
2001
2002
2003
8-10
The Components of the Future Value of $100
300
250
Interest on
Interest
S=P(1+i)n
Future Value. S
200
Interest on
Original Principal
150
S=P(1+rt)
100
Original
Principal
50
0
1
2
3
4
5
6
Time (years)
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7
8
9
10
11
8-11
Simple Versus Compound Interest
Simple
Al Jones deposited $1,000 in a
savings account for 5 years at
an annual simple interest rate
of 10%. What is Al’s simple
interest and maturity value?
I = Prt
I = $1,000 * .10 * 5
= $500
MV = P + I
MV = $1,000 + $500
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= $1,500
Compounded
Al Jones deposited $1,000 in a
savings account for 5 years at
an annual compounded rate
of 10%. What is Al’s interest
and compounded amount?
FV = PV(1 +i)n
n=5*1=5
i = .10/5 = .02
FV = 1000(1.02)5 =
$1,000 * 1.6105 = $1,610.50
I = FV - PV
I = $1,610.50 - $1,000 = $610.50
8-12
Future Values of $100 at Various
Rates of Interest Compounded Annually
1800
12%
Future Value, S or FV
1600
1400
1200
10%
1000
800
8%
600
6%
400
200
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Years to Maturity, n
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8-13
Nominal Rates of Interest Compared
Beginning
balance
Nominal rate
of interest
$1,000 + 6%
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Compounding
period
End
balance
Annual
$1,060.00
Semiannual
$1,060.90
Quarterly
$1,061.40
Daily
$1,061.80
8-14
Future Values of $100 at the Same Nominal Rate
but Different Compounding Frequencies
Fu ture Valu e, S o r FV
250 0
200 0
12% Compounded
monthly
150 0
100 0
12% Compounded
annually
500
0
5
10
15
Time (y ea rs)
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20
25
8-15
Compounding Daily Interest
Calculate the future value of $2,000 compounded daily for
7 years at 6%
n = 7 * 365 = 2555
i = .06 / 365 = 0.000164
FV = 2000(1+.06/365)2555
=$2,000 * 1.5219 = $3,043.80
Calculator
steps:
.06 / 365
+1=
yx =
* 2000 =
MH Ryerson
8-16
Using Financial Calculators
n
represents the number of compounding periods
i
represents the periodic interest rate, j/m
PV
represents the principal or present value
FV
represents the maturity value or future value
PMT
represents the periodic annuity payment
(not used until chapter 10)
comp
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used to tell calculator to compute (CPT)
8-17
Using your financial calculator to solve the
previous example:
Calculate the future value of $2,000 compounded daily for
7 years at 6%
Using your Financial calculator:
FV = $2,000 * 1.5219 = $3,043.80
MH Ryerson
7 *365 =
n
6/365 =
2000
i
PV
0
PMT
comp
FV
8-18
Heather invested $6000 at 7% compounded
quarterly. After 2 years, the rate changed to 8.2%
compounded monthly. What amount will Heather
have 3 1/2 years after the initial investment?
2 years
0
$6000
i= .07/4 n=8
S1 = P2
3.5 years
i= .082/12 n=1.5*12=18
S2
S1 = 6000(1+.07/4)8
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S2 = 6893.29(1+.082/12)18
= 6000(1.1489)
= 6893.29(1.1304)
= 6893.29
= 7792.25
8-19
Shannon borrowed $6000 at 9% compounded
quarterly. On the first and second anniversaries of
the loan, she made payments of $2500. What is the
balance outstanding immediately following the
second payment?
1 year
0
$6000
i= .09/4 n=4
S1 = 6000(1+.09/4)4
= 6000(1.0931)
= 6558.50
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2 years
S1
- 2500
i= .09/4 n=4
P2 = 4058.50
S2 = 4058.50(1+.09/4)4
= 4058.50(1.0931)
= 4436.28
Outstanding balance: $1936.28
S2
8-20
Present Value of $1 at 8% for Four Periods
Present value goes from the future value to the present value
$1.20
$1.10
$1.00
$0.90
$0.80
$0.70
$0.60
$0.50
$0.40
$0.30
$0.20
$0.10
$0.00
Present
value
$.7350
0
$.7938
1
$.8573
2
$.9259
3
Number of periods
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$1.0000
4
Future
Value
8-21
Formula for Present Value
PV = FV(1+i)- n
Calculator
Steps:
i + 1=
yx n +/- =
* FV =
MH Ryerson
8-22
Calculating Present Value
Steve Smith needs $1,368.80 in 4 years. His bank offers 8%
interest compounded semiannually. How much money must Steve
put in the bank today (PV) to reach his goal in 4 years?
n=4*2=8
i = .08 = .04
PV = 1368.80(1+.04)-8
= $1,368.80 * .7307
Invest
Today
= $1,000.18
Calculator steps:
1400
1200
1000
800
400
200
0
1.04 yx 8 +/- = * 1368.80 =
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Investment
600
2000
2001
2002
2003
8-23
Figure 8.5
The Present Value Of $1000
(Discounted at 10% Compoun ded Annually)
$1,20 0
$1,00 0
Present value
$800
$600
$400
$200
$0
25 yea rs
earlier
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20 years 15 yea rs
earlier
earlier
10 yea rs
earlier
5 years
earlier
Payment
date
8-24
Two payments of $4000 each must be made 1
and 4 years from now. If money can earn 8%
compounded monthly, what single payment 3
years from now would be equivalent to the two
scheduled payments?
0
1 year
2 years
3 years
$4000
4 years
$4000
i=.08/12 n=2*12=24
FV1 =
4000(1+.08/12)24
= 4000(1.1729)
= 4691.55
i=.08/12 n=1*12=12
PV2 = 4000(1+.08/12) -12
= 4000(0.9234)
= 3693.45
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Final ans:
4691.55 + 3693.45 =
$8385.00
8-25
What amount must you invest now at
6% compounded daily to accumulate to
$5000 after 1 year?
Using your Financial calculator:
365
n
6/365=
5000
i
FV
S = FV = $5000
0
PMT
i= .06/365
comp
j= 6%
m = 365
PV
n = 1 * 365 = 365
Ans.: $4708.85
MH Ryerson
8-26
What regular payment will an investor
receive from a $10 000, 5 year, monthly
payment GIC earning a nominal rate of 7%
compounded monthly?
Interest rate per payment interval is:
i= j/m = .07/12 = 0.05833
the monthly payment will be:
PV * i= $10 000 * 0.005833
= $58.33
MH Ryerson
8-27
Guaranteed Investment Certificates (GICs)
> purchased from banks, credit unions, trust companies,
and caisses populaires (in Quebec).
> you are in effect lending money to the financial institution.
> the financial institution uses the funds raised from
selling GICs to make loans— most commonly,
mortgage loans.
> mortgage rates are typically 1.5% to 2% higher than
the interest rate paid to GIC investors.
> “Guaranteed” refers to the unconditional guarantee of
principal and interest by the parent financial institution.
> maturities in the range of 1 to 5 years.
> most GICs are not redeemable before maturity.
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8-28
Structure of Interest Rates
Fixed rate:
The interest
rate does not
change over
the term of the
GIC.
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Step-up rate:
The interest rate
is increased
every 6 months
or every year
according to a
pre-determined
schedule.
Variable rate:
The interest
rate is
adjusted every
year or every
6 months to
reflect
prevailing
market rates.
There may be
a minimum
“floor” below
which rates
cannot drop
8-29
Payment of interest
Compound interest version:
Interest is periodically converted
to principal and paid at maturity.
Regular interest version:
Interest is paid to the
investor every year or every
6 months.
MH Ryerson
8-30
Canada Savings Bonds (CSBs)
? you purchase from financial institutions, but $
goes to the federal government to help
finance its debt.
? usual term is 10 or 12 years
? variable interest rates
? interest rates is changed on each anniversary, with
minimum rates for subsequent 2 years
? may be redeemed at any time
? both regular and compound interest versions
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8-31
Valuation Principle:
> The fair market value of an investment is the
sum of the present values of the expected
cash flows.
> The discount rate used should be the
prevailing market determined rate of return
required on this type of investment.
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8-32
Strip bonds
? its owner will receive a single payment (called the face
value of the bond) on the bond’s maturity date.
? The maturity date could be as much as 30 years in the
future. No interest will be received in the interim.
Suppose a $1000 face value strip bond matures 18 years from now.
The owner of this bond will receive a payment of $1000 in 18 years.
What is the appropriate price to pay for the bond today if the
prevailing rate of return is 8.75% compounded semi-annually?
FV = $1000
MH Ryerson
PV = 1000(1+.0875/2)-36
i= .0875/2
= 1000(0.2141)
n = 18 * 2 = 36
= $214.06
8-33
THE END
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