Compound Interest
Howard Godfrey, Ph.D., CPA
Copyright © 2011, Dr. Howard Godfrey
Edited August 3, 2011
Review of compound interest
concepts and procedures.
1.Future value of an investment
made today.
2.Present value (today) of
an amount in the future.
3.Future value of an annuity
(series of payments).
4.Present value of an annuity
(series of payments).
Future value of an
investment made today.
I will invest $1,000 today
(Jan. 1, Year 1) in a savings account.
My savings account will earn
interest at the rate of 10% per year,
compounded annually.
How much money will be in the
account in 3 years? (Dec. 31, Year 3)?
Compound Interest-Future Value of $1
Invest $1,000 on Jan. 1, Yr 1 (for 3 yrs)
Interest at 10% compounded annually
Given: present value (PV). Compute FV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
$1,000
PV
Int.
100
$1,100
Int.
Int.
FV
Compound Interest-Future Value of $1
Invest $1,000 on Jan. 1, Yr 1 (for 3 yrs)
Interest at 10% compounded annually
Given: present value (PV). Compute FV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
$1,000
PV
Int.
100
$1,100
$1,100
Int.
110
$1,210
$1,210
Int.
121
FV
n
$1,331
Factor (See table) (1+i)
1.331
Conclusion.
If you invest $1,000 today
in a savings account
earning 10% interest per
year, your account will
have a balance of $1,331
in three years.
Present value of a payment
to be made in the future.
I bought computer today.
Price is $1,000, payable in 3 years.
I do not pay interest on this debt.
Assume I normally pay 10% when
borrowing money.
What is present value of the $1,000
to be paid in 3 years?
Present Value (today) of $1 in Future
I need $1,000 on Dec. 31, Yr 3 (in 3 yrs).
Interest is 10% compounded annually
How much should I invest today (PV)?
Given: Future value (FV). Compute PV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
÷ 1.10 or 110%
$909.09
÷ 1.10 or 110%
÷
1.10 or 110%
PV factor. Reciprocal of FV.
Present Value of $1 in the Future
I need $1,000 on Dec. 31, Yr 3 (in 3 yrs)
Interest is 10% compounded annually
How much should I invest today (PV)?
Given: Future value (FV). Compute PV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
÷ 1.10 or 110%
$909.09
÷ 1.10 or 110%
$826.45
÷ 1.10 or 110%
$751.31
0.7513 PV factor. Reciprocal of FV.
Conclusion.
The seller of the computer should
be willing to accept $751.31 in full
payment of the computer today, if
the seller uses the same interest
rate.
If the seller invests $751.31 today
in a savings account earning 10%
interest, the balance in 3 years
will be $1,000.
Future value of an annuity
(periodic payments)
I will save $1,000 each year and
deposit that amount in a savings
account on the last day of each of the
next 3 years.
My savings account will earn 10% per
year.
How much money will be in my
account at the end of 3 years?
FV of "Ordinary" Annuity of $1
Invest $1,000 at end of each year (3 yrs)
Interest at 10% compounded annually
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
Dep. $1,000
Int.
$1,000
$1,000
Int.
100
Dep. 1,000
$2,100
Balance in 3 years.
FV of Annuity of $1.
10%
$2,100
Int.
210
Dep. 1,000
FV $3,310
3.310
Conclusion
If I invest $1,000 in a savings
account at the end of each year
(total deposits of $3,000) the
account balance will be $3,310 in
three years.
[Note this also applies for other
business transactions involving
periodic payments.]
Present value of an annuity
(series of payments).
I bought computer today. My price is
$3,000, payable in $1,000 at the end of
year 1, $1,000 at the end of year 2 and
$1,000 at the end of year 3.
I do not pay interest on this debt.
Assume I normally pay 10% on
borrowed money.
What is present value of the payments?
Present Value (today) of annuity of $1
I will pay $1,000 each year on Dec. 31.
Interest is 10% compounded annually
What is the present value of those payments?
Given: Future value (FV). Compute PV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
÷ 1.10 or 110%
$909.09
$1,000.00
÷
1.10 or 110%
$1,000.00
÷
1.10 or 110%
PV tables
Present Value (today) of annuity of $1
I will pay $1,000 each year on Dec. 31.
Interest is 10% compounded annually
What is the present value of those payments?
Given: Future value (FV). Compute PV.
Jan 1
Dec 31
Dec 31
Dec 31
Yr. 1
Yr. 1
Yr. 2
Yr. 3
$1,000
÷ 1.10 or 110%
$909.09
$1,000.00
$1,909.09
÷ 1.10 or 110%
$1,735.54
$1,000.00
$2,735.54
÷ 1.10 or 110%
$2,486.85
PV tables
Conclusion
The actual purchase price of the
computer is $2,486.85, if a discount
rate of 10% is used (with annual
compounding).
A little over $500 is for interest for
deferred payment.
We use these approaches when
computing present values (PV) of
lease payments, PV of bonds, PV of
potential capital budgeting
investments, etc.
NTD Company issues bonds. [1 of 3]
NTD Company issues bonds with a face
value of $100,000 on Jan. 1, 2011.
These bonds pay interest twice per year
at the annual rate of 9%.
Interest is paid on 6-30 and 12-31.
Bonds mature in 2 years [12-31-2012].
Bonds were sold at a price to yield 10%
per year.
Please compute the price of the bonds
and complete the amortization table on
the next slide.
NTD Co. Bond IIlustration
PV
PV
PV
PV
PV
Price
2011
June
Dec.
$4,500
2011
June
Dec
$4,500
$4,500
$4,500
$100,000
Bonds – NTD
PV of 4 interest payments
Interest Payments
PV Factor [See Table]
=PV(0.05,4,-4500,,0)
PV of annuity of $4,500
$4,500.00
3.546
$15,956.78
PV of $100,000
(4 periods, 5% per period)
Principal Payment
Factor
=1/(1+0.05)^4
PV of $100,000 payment
Price of Bond
$100,000.00
0.822702475
$82,270.25
$98,227.02
NTD Bond Amortization Schedule [2 of 3]
On Jan. 1, 2011, issued $100,000, 9% , 2 year bonds.
Interest is paid on 6-30 and 12- 31. Mature 12-31-2012.
Bonds were sold at a price to yield 10% per year.
Yr
2011
2011
2012
2012
Book
Interest Interest
Value
Paymnt
98,227
4,500
Exp.
Unamort
Amort. Disc/Prem
Book
Value
NTD Bond Amortization Schedule [3 of 3]
On Jan. 1, 2011, issued $100,000, 9% , 2 year bonds.
Interest is paid on 6-30 and 12- 31. Mature 12-31-12.
Bonds were sold at a price to yield 10% per year.
Book
Yr
Interest Interest
Value Paymnt
Exp.
Unamort
Amort. Disc/Prem
Book
Value
2011
98,227
4,500 4,911
411
1,362
98,638
2011
98,638
4,500 4,932
432
930
99,070
2012
99,070
4,500 4,954
454
476
99,524
2012
99,524
4,500 4,976
476
0
100,000
Market rate if price exceeds par value.
What is the market rate of interest for a
bond issue which sells for more than its
par value?
a. Less than rate stated on the bond.
b. Equal to rate stated on the bonds
c. Higher than rate stated on the bond.
d. Rate is independent of rate stated
on the bonds.
(Source: CPA)
Market rate if price exceeds par value.
What is the market rate of interest for a
bond issue which sells for more than its
par value?
a. Less than rate stated on the bond.
Reason. Bond has rate of interest that is
better than the market demands. Market
pays extra for this bond. Amount paid for
the bond is based on present value of
cash flows.
Present Value of $1
Table 1. Present Value of $1 to be
received after n number of periods.
F. V.
$1
$1
$1
Period/rate
1%
2%
10%
1
0.99010 0.98039 0.90909
2
0.98030 0.96117 0.82645
3
0.97059 0.94232 0.75131
4
0.96098 0.92385 0.68301
5
0.95147 0.90573 0.62092
Future Value of $1
Table 2. Future Value of $1 (after n periods)
to be invested today.
F. V.
$1
$1
$1
$1
Period/rate
1%
2%
5%
10%
1
1.01000 1.02000 1.05000
1.10000
2
1.02010
1.04040
1.10250
1.21000
3
1.03030
1.06121
1.15763
1.33100
4
1.04060
1.08243
1.21551
1.46410
5
1.05101
1.10408
1.27628
1.61051
P. V. of Annuity of $1 for n Periods
Table 3. P. V. of $1 to be received at
end of each period for n periods.
F. V.
$1
$1
$1
Period/Rate
1%
2%
10%
1
0.99010 0.98039 0.90909
2
1.97040 1.94156 1.73554
3
2.94099 2.88388 2.48685
4
3.90197 3.80773 3.16987
5
4.85343 4.71346 3.79079
Future Value of Annuity of $1
F.V. (after n periods) of $1 to be received or paid
at the end of each period for n periods.
F. V.
$1
$1
$1
$1
Period/Rate
1%
2%
5%
10%
1
1.00000
1.00000
1.00000
1.00000
2
2.01000
2.02000
2.05000
2.10000
3
3.03010
3.06040
3.15250
3.31000
4
4.06040
4.12161
4.31013
4.64100
5
5.10101
5.20404
5.52563
6.10510
NPV Example
• Original investment (cash outflow):
$6,075
• Useful life: four years
• Annual income generated
from investment
(cash inflow): $2,000
• Minimum desired
rate of return: 10%
Net PV Example
Years Amount
0
($6,075)
1
2,000
2
2,000
3
2,000
4
2,000
Net present value
PV
Factor
1
0.9091
0.8264
0.7513
0.683
Present
Value
($6,075)
1,818
1,653
1,503
1,366
$265
NPV Example
PV
Present
Years Amount Factor Value
0
($6,075)
1 ($6,075)
1-4
2,000
3.17
6,340
Net present value
$265
The
End