Compound Interest, Future Value,
and Present Value
• When money is borrowed, the amount
borrowed is known as the loan principal.
– For the borrower, interest is the cost of using
the principal.
• Investing money is the same as
making a loan.
– The interest received is the return
on the investment.
Compound Interest, Future Value,
and Present Value
• Calculating the amount of interest depends
on the interest rate and the interest period.
• Types of interest:
– Simple interest - the interest rate multiplied by
an unchanging principal amount
– Compound interest - the interest rate multiplied
by a changing principal amount
• The unpaid interest is added to the principal balance
and becomes part of the new principal balance for
the next interest period.
Future Value
• Future value - the amount accumulated over
time, including principal and interest
– For example, if a person lets $10,000 sit in a
bank account that pays 10% interest per year
for 3 years, the future value of the $10,000 is
$13,310 and is determined as follows:
Year 1:
Year 2:
Year 3:
$10,000 x 1.10 = $11,000
$11,000 x 1.10 = $12,100
$12,100 x 1.10 = $13,310
Future Value
• The general formula for computing the
future value (FV) of S dollars in n years at
interest rate i is:
FV  S(1i)
n
n refers to the number of periods the funds are invested. The
interest rate must be stated consistently with the time period.
Future Value
• The calculations for future values can be
very tedious. Most people use future value
tables to determine future values.
– In the table, each number is the solution to the
expression (1 + i)n.
– The value of i is given in the column heading.
– The value of n is given in the row label for the
number of periods.
Future Value
To how much will $25,000 grow if left in
the bank for 20 years at 6% interest?
The answer is determined as follows:
$25,000 x 3.2071* = $80,177.50
*3.2071 is the future value factor for 20 periods at 6%
interest.
Present Value
• Present value - the value today of a future
cash inflow or outflow
• Present value calculations are the reverse of
future value calculations.
– In future value calculations, you determine how
much money you will have at a date in the
future given a certain interest rate.
– In present value calculations, you determine
how much must be invested today given a
certain interest rate to get to how much money
you want in the future.
Present Value
• For example, if $1.00 is to be received in
one year and the interest rate is 6%, you
will have to invest $0.9434 ($1.00 / 1.06).
– Thus, $0.9434 is the present value of $1.00 to
be received in one year at 6% interest.
Present Value
• The general formula for the present value
(PV) of a future value (FV) to be received or
paid in n periods at an interest rate of i per
period is:
FV
PV 
n
(1  i )
Present Value
• Just as with future values, tables can be
helpful in determining the present value of
amounts.
– In the table, each number is the solution to the
expression 1/(1 + i)n.
– The value of i is given in the column heading.
– The value of n is given in the row label for the
number of periods.
Present Value
• Interest rates are sometimes called discount
rates in calculations involving present
values.
• Present values are also called discounted
values, and the process of finding the
present value is discounting.
– Present values can be thought of
as decreasing the value of a future
cash inflow or outflow because
the cash is to be received or paid
in the future, not today.
Present Value
A city wants to issue $100,000 of noninterest-bearing bonds to be repaid in a
lump sum in 5 years. How much should
investors be willing to pay for the bonds if
they require a 10% return on their
investment?
$100,000 x .6209* = $62,090
*.6209 is the present value of $1 factor for 5 years at 10%
interest.
Present Value
• Remember to pay attention to the number
of periods. Interest is often compounded
semiannually instead of annually.
– If interest is compounded semiannually, the
number of periods is twice the number of years,
and the interest rate is one-half of the annual
interest rate.
– In the previous example, if interest were
compounded semiannually, the number of
periods is 10 instead of 5, and the interest rate
is 5% instead of 10%.
Present Value of an Ordinary Annuity
• Annuity - a series of equal cash flows to
take place during successive periods of
equal length
• The present value of an annuity is the sum
of the present values of each cash receipt or
payment.
– If a note has a series of payments, its present
value can be determined by finding the present
value of each payment and adding those present
values together.
Present Value of an Ordinary Annuity
• Again, tables can be helpful in determining
the present value of an ordinary annuity.
• The factors in a present value of an annuity
table are merely the cumulative sum of the
present value of $1 factors in the present
value of $1 table for the number of annuity
periods.
– The present value of an ordinary annuity tables
are especially helpful if the cash payments or
receipts extend into the future over many
periods.
Present Value of an Ordinary Annuity
A city wants to issue $1,000,000 of noninterest-bearing bonds to be repaid
$100,000 per year for 10 years. How much
should investors be willing to pay for the
bonds if they require a 10% return on their
investment?
$100,000 x 6.1446* = $614,460
*6.1446 is the present value of an annuity of $1 for 10
periods at 10% interest.
Present Value of an
Ordinary Annuity
• Notice that the higher the interest rate, the
lower the present value factor.
This occurs because at higher
interest rates, less must be
invested to obtain the same
stream of future annuity
payments or a certain amount in
the future.
Valuing Bonds
• Because bonds create cash flows in future periods,
they are recorded at the present value of those
future payments, discounted at the market interest
rate in effect when the liability is created.
• Bond - formal certificate of indebtedness that is
typically accompanied by:
– A promise to pay interest in cash at a specified annual
rate plus
– A promise to pay the principal at a specific maturity
date
Valuing Bonds
• When valuing bonds, the present value tables are
used to determine the amount of proceeds that
will be received.
– The present value of $1 table is used to determine the
present value of the face amount of the bonds.
– The present value of an annuity of $1 is used to
determine the present value of the series of interest
payments.
– The amounts are added together to determine the
amount of proceeds and any premium or discount.
Valuing Bonds
• Discount on bonds - occurs when the market
interest rate is greater than the coupon rate.
• Premium on bonds - occurs when the market
interest rate is less than the coupon rate.
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Valuing Bonds
A company issues $20,000,000 of 5-year
bonds with a coupon rate of 7%. Interest is
to be paid semiannually on June 30 and
December 31 of each year. At the time of
the issuance, the market rate is 10%. What
is the amount of the proceeds and any
premium or discount on the bonds?
Valuing Bonds
• To determine the proceeds:
$20,000,000 x .6139* = $12,278,000
$700,000‡ x 7.7217* = 5,405,190
$17,683,190
===============================
‡($700,000
= ($20,000,000 x 7%) / 2)
*PV factors are for 10 periods at 5%
The company will receive $17,683,190 upon
issuance.
The bonds are issued at a discount of $2,316,810.
Bonds Issued at a Discount
• When bonds are issued at at discount, the
amount of proceeds received from the
issuance is less than the actual liability.
• The difference must be recorded in a
separate account on the books.
Cash
Discount on bonds payable
Bonds payable
17,683,190
2,316,810
20,000,000
Bonds Issued at a Discount
• The discount on bonds payable is a contra
account; it is deducted from bonds payable.
• Balance sheet presentation:
Bonds payable, 7%
Deduct: Discount on bonds payable
Net liability
$ 20,000,000
2,316,810
$ 17,683,190
============================
Bonds Issued at a Discount
• For bonds issued at a discount, the discount can be
thought of as a second interest amount payable to
the investors at the maturity date.
– Rather than recognizing the extra interest
expense all at once upon maturity, the issuer
should spread the extra interest over the life of
the bonds.
– This is accomplished by discount amortization.
• The amortization of a discount increases the
interest expense of the issuer at each cash interest
payment date, but it has no effect on cash paid.
Bonds Issued at a Discount
• Discount amortization can be calculated
using two methods.
– Straight-line amortization
• The amortization of the discount is an equal amount each
period, but the effective interest rate is different each
period.
– Effective-interest amortization
• The effective interest rate is the same each period, but the
amortization of the discount is a different amount each
period.
Bonds Issued at a Discount
• Amortization using the effective-interest
method:
– For each period, interest expense is equal to the
carrying value of the debt multiplied by the
market rate of interest in effect when the bond
was issued.
– The cash interest payment is the coupon rate
times the face amount of the bonds.
– The difference between the interest expense and
the cash interest payment is the amount of
discount amortization for the period.
Bonds Issued at a Discount
• Journal entries:
To record the issuance of the bonds:
Cash
Discount on bonds payable
Bonds payable
xxxxxx
xxxx
xxxxxx
To record the payment of interest and discount
amortization:
Interest expense (Carrying value x Market rate)
Discount on bonds payable
Cash (Face value x Coupon rate)
xxx
xx
xxx
Bonds Issued at a Premium
• Accounting for bonds issued at a premium
is just the opposite of accounting for bonds
issued at a discount.
– The cash proceeds exceed the face amount.
– The amount of the contra account Premium on
Bonds Payable is added to the face amount to
determine the net liability reported in the
balance sheet.
– The amortization of bond premium decreases
the interest expense to the issuer.
Bonds Issued at a Premium
A company issues $20,000,000 of 5-year
bonds with a coupon rate of 7%. Interest is
to be paid semiannually on June 30 and
December 31 of each year. At the time of
the issuance, the market rate is 6%. What is
the amount of the proceeds and any
premium or discount on the bonds?
Bonds Issued at a Premium
To determine the proceeds:
$20,000,000 x .7441* = $14,882,000
$700,000‡ x 8.5302* =
5,971,140
$20,853,140
===========================
‡($700,000
= ($20,000,000 x 7%) / 2)
*PV factors are for 10 periods at 3%
The company will receive $20,853,140 upon
issuance.
The bonds are issued at a premium of $853,140.
Early Extinguishment
• When a company redeems its own bonds
before the maturity date, the transaction is
called an early extinguishment.
– Early extinguishment usually results in a gain
or loss to the company redeeming the bonds.
– The gain or loss is the difference between the
cash paid and the net carrying amount (face
amount less unamortized discount or plus
unamortized premium) of the bonds.
Early Extinguishment
Allen Company purchased all of its bonds
on the open market at 98. The bonds have a
face amount of $100,000 and a $12,000
unamortized discount. Determine any gain
or loss on the early extinguishment, and
prepare the journal entries to record the
transaction.
Early Extinguishment
Carrying amount:
Face value
$100,000
Deduct: Unamortized discount
12,000
$88,000
Cash required ($100,000 x 98%)
98,000
Loss on early extinguishment
$10,000
==================
Bonds payable
Loss on early extinguishment
Cash
Discount on bonds payable
100,000
10,000
98,000
12,000
Accounting for Leases
• Lease - a contract whereby an owner
(lessor) grants the use of property to a
second party (lessee) for rental payments
– Some leases are recorded simply as if one party
is renting property from another.
• Other leases are recorded as liabilities
and assets when the lease contract is
• signed.
Operating and Capital Leases
• Capital lease - a lease that transfers
substantially all the risks and benefits of
ownership to the lessee
– They are the same as installment sales which
provide for payments over time along with
interest.
– The leased item must be
recorded as if it were sold
by the lessor and purchased
by the lessee.
Operating and Capital Leases
• Operating lease - a lease that should be
accounted for by the lessee as ordinary
rental expenses; any lease other than a
capital lease
– Examples include rental of an apartment or
rental of a car on a daily basis.
Operating and Capital Leases
• Differences in accounting for operating and
capital leases:
– Operating - treat as rental expense
Rent expense
Cash
xxx
xxx
– Capital - treat as if the lessee borrowed the
money and purchased the leased asset
Leased property
Capital lease liability
xxxx
xxxx
Differences in Income Statements
• The major difference in the income statements for
a capital lease and an operating lease is the timing
of the expenses.
– A capital lease tends to bunch heavier charges
in the early years. These charges are the
amortization of the lease plus the interest factor.
– An operating lease records the payments
directly as expenses, generally in a straight-line
manner.
– For comparable leases, the total expenses are
the same.
Criteria for Capital Leases
• Before GAAP established criteria for leases
to be classified as capital leases, many
companies were keeping “off balance sheet
financing” by treating noncancellable leases
as monthly rentals.
– These leases created assets and
liabilities that the companies
were not recognizing.
Criteria for Capital Leases
• Under GAAP, a capital lease exists if one or more
of the following conditions are met:
– Title to the leased property is transferred to the lessee
by the end of the lease term.
– An inexpensive purchase option is available to the
lessee at the end of the lease.
– The lease term equals or exceeds 75% of the estimated
economic life of the property.
– At the start of the lease, the present value of minimum
lease payments is at least 90% of the property’s fair
value.