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: $10,000 x 1.10 = $11,000

Year 2: $11,000 x 1.10 = $12,100

Year 3: $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:

niSFV )1(

n refers to the number of periods the funds are invested. Theinterest 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:

ni

FVPV

)1(

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 ValueA city wants to issue $100,000 of non-interest-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 AnnuityA city wants to issue $1,000,000 of non-interest-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 anOrdinary 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 BondsA 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 17,683,190

Discount on bonds payable 2,316,810

Bonds payable 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% $ 20,000,000

Deduct: Discount on bonds payable 2,316,810

Net liability $ 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 xxxxxx

Discount on bonds payable xxxx

Bonds payable xxxxxx

To record the payment of interest and discount amortization:Interest expense (Carrying value x Market rate) xxx

Discount on bonds payable xx

Cash (Face value x Coupon rate) 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 PremiumTo 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 ExtinguishmentCarrying 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 100,000

Loss on early extinguishment 10,000

Cash 98,000

Discount on bonds payable 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 xxx

Cash xxx

– Capital - treat as if the lessee borrowed the money and purchased the leased assetLeased property xxxx

Capital lease liability 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.