Practical financial manaegment lasher 7th ed chapter 06

Time Value of MoneyPresent Value – The amount that must be deposited today to have a future sum at a certain interest rate Terminology – The discounted value of a future sum is its prese

Chapter 6 - Time Value of Money

Time Value of Money

A sum of money in hand today is worth more than the

same sum promised with certainty in the future

Think in terms of money in the bank

The value today of a sum promised in a year is the amount

you'd have to put in the bank today to have that sum in a year

Example: Future Value (FV) = $1,000

k = 5%

Then Present Value (PV) = $952.38 because $952.38 x 05 = $47.62

and $952.38 + $47.62 = $1,000.00

Time Value of Money

Present Value

– The amount that must be deposited today to have a future sum at a certain interest rate Terminology

– The discounted value of a future sum is its present value

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Outline of Approach

– Amounts

Present value Future value

– Annuities

Present value Future value

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Four different types of problem

Outline of Approach

Develop an equation for each

Time lines - Graphic portrayals

Place information on the time line

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The Future Value of an Amount

How much will a sum deposited at interest rate k grow into over some period of time

If the time period is one year:

The Future Value of an Amount

(1 + k)n depends only on k and n

Define Future Value Factor for k,n as:

FVFk,n = (1 + k)n

Substitute for:

FVn = PV[FVFk,n]

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The Future Value of an Amount

Problem-Solving Techniques

– All time value equations contain four

variables

In this case PV, FVn, k, and n

Every problem will give you three and ask for the fourth.

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Concept Connection Example 6-1

Future Value of an Amount

How much will $850 be worth in three years at 5% interest? Write Equation 6.4 and substitute the amounts given.

FVn = PV [FVFk,n ] FV3 = $850 [FVF5,3]

Concept Connection Example 6-1 Future Value of an Amount

Look up FVF5,3 in the three-year row under the 5% column of Table 6-1, getting 1.1576

Concept Connection Example 6-1

Future Value of an Amount

Financial Calculators

Work directly with equations

How to use a typical financial calculator – Five time value keys

Use either four or five keys

– Some calculators require inflows and

outflows to be of different signs

If PV is entered as positive the computed FV is negative

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Financial Calculators

Basic Calculator functions

0 5000

The Present Value of an Amount

Future and present value factors are reciprocals

– Use either equation to solve any amount problems

PVF

=

Concept Connection Example 6-3

Finding the Interest Rate

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Finding the Interest Rate

what interest rate will grow $850 into $983.96 in three years Here we have FV3, PV, and n, but not k

PV= FVn [PVFk,n ]

Use Equation 6.7

Concept Connector Example 6-3

PV= FVn [PVFk,n ]

Substitute for what’s known

$850= $983.96 [PVFk,n ] Solve for [PVFk,n ] [PVFk,n ] = $850/ $983.96

[PVFk,n ] = 8639

Find 8639 in Appendix A (Table A-2) Since n=3 search only row 3, and find the answer to the problem is (5% ) at top of column.

Concept Connection Example 6-3 Finding the Interest Rate

Annuity Problems

Annuities

– A finite series of equal payments separated

by equal time intervals

Ordinary annuitiesAnnuities due

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Figure 6-1 Future Value: Ordinary Annuity

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Figure 6-2 Future Value: Annuity Due

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The Future Value of an Annuity—

Developing a Formula

Future value of an annuity

– The sum, at its end, of all payments and all interest if each payment is deposited when received

– Figure 6-3 Time Line Portrayal of an

Ordinary Annuity

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Figure 6-4 Future Value of a Three-Year Ordinary Annuity

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For a 3-year annuity, the formula is:

i=1

obtain:

FVA = PMT ∑ 1+k −

The Future Value of an Annuity—

Concept Connection Example 6-5 The Future Value of an Annuity

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Brock Corp will receive $100K per year for 10 years and will invest each payment at 7% until the end of the last year

How much will Brock have after the last payment is received?

Concept Connection Example 6-5 The Future Value of an Annuity

FVAn = PMT[FVFAk,n]

FVFA 7,10 = 13.8164

– FVA10 = $100,000[13.8164] = $1,381,640

The Sinking Fund Problem

Companies borrow money by issuing

bonds

– No repayment of principal is made during the bond’s life

– Principal is repaid at maturity in a lump sum

A sinking fund provides cash to pay off principal

at maturitySee Concept Connection Example 6-6

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Compound Interest and

Figure 6-5

The Effect of Compound Interest

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The Effective Annual Rate

Effective annual rate (EAR)

– The annually compounded rate that pays the same interest as a lower rate

compounded more frequently

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Table 6.2

Year-end Balances at Various Compounding

Periods for $100 Initial Deposit and knom = 12%

The Effective Annual Rate

EAR can be calculated for any compounding period using the formula

m is number of compounding periods per year

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Effect of more frequent compounding is greater at higher interest rates

m nom

The APR and the EAR

The annual percentage rate (APR)

associated with credit cards is actually the nominal rate and is less than the EAR

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Compounding Periods and the

Time Value Formulas

n must be compounding periods

k must be the rate for a single

– E.g for quarterly compounding

k = knom divided by 4, and

n = years multiplied by 4

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Concept Connection Example 6-7

Compounding periods and Time Value Formulas

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A “Save Up” problem Payments plus interest accumulates to a known amount

Save ups are always FVA problems

Save up to buy a $15,000 car in 2½ years

Make equal monthly deposits in a bank account which

pays 12% compounded monthly How much must be deposited each month?

Concept Connection Example 6-7

Compounding periods and Time Value Formulas

Calculate k and n for monthly compounding,

and

n = 2.5 years x 12 months/year = 30 months.

%

112

%

1212

nomk

Concept Connection Example 6-7

Compounding periods and Time Value Formulas

Concept Connection Example 6-7

Compounding periods and Time Value Formulas

Figure 6-6 Present Value of a period Ordinary Annuity

Three-40

The Present Value of an Annuity

Developing a Formula

Present value of an annuity

– Sum of the present values of all of the annuity’s

payments

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PVFAk,n

( ) ( ) ( ) ( ) ( ) ( )

PVA = PMT 1+k PMT 1+k PMT 1+k Generalized for any number of periods:

PVA = PMT 1+k PMT 1+k PMT 1+k Factoring PMT and using summation, we o

btain:

PVA =PMT  1+k − 

The Present Value of an Annuity— Solving Problems

There are four variables

– PVA present value of the annuity – PMT payment

– k interest rate

– n number of periods

– Problems give 3 and ask for the fourth

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Concept Connection Example 6.9

PVA - Discounting a Note

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Shipson Co will receive $5,000 every six months (semiannually) for 10 years The firm needs cash now and asks its bank to discount the

contract and pay Shipson the present value of the expected annuity

This is a common banking service called discounting

If the payer has good credit, the bank will discount the contract at the current rate of interest, 14% compounded semiannually and pay

Shipson the present value of the annuity of the expected payments How much should Shipson receive?

Solution:

The loan amount is the present value of the

annuity of the payments

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Concept Connection Example 6-11 Amortized Loan – Finding PMT

$10,000 = PMT [34.0426]

PMT = $293.75

Concept Connection Example 6-12

Amortized Loan – Finding Amount Borrowed

PVA= 500 (30.1075) PVA =$15,053.75

Loan Amortization Schedules

Shows interest and principal in each loan payment

Also shows beginning and ending balances of unpaid principal for each period

To construct we need to know

– Loan amount (PVA)

– Payment (PMT)

– Periodic interest rate (k)

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Note that the Interest portion of the

payment is decreasing while the

Principal portion is increasing.

Mortgage Loans

Used to buy real estate

Often the largest financial transaction in

PVA= PMT [PVFAk,n ]

$100,000 = PMT [PVFA.5,360 ]

$100,000 = PMT [166.792 ]

PMT = $599.55

First month’s interest = $100,000 x 005 = $500

leaving $99.55 to reduce principal

First payment is 83.4%

Concept Connection 6-13

Interest Content of Early Loan Payment

Next, solve for the monthly payment

The $500 of the first payment goes to interest, leaving $99.55

to reduce principal The first payment is 83.4%

Concept Connection 6-13

Interest Content of Early Loan Payment

Mortgage Loans

Implications of mortgage payment pattern

– Early mortgage payments provide a large tax

savings, reducing the effective cost of borrowing– Halfway through a mortgage’s life, half of the loan

is not yet paid off

Long-term loans result in large total interest amounts over the life of the loan

Adjustable rate mortgage (ARM)

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The Annuity Due

Payments occur at beginning of periods The future value of an annuity due

– Each PMT earns interest one period longer – Formulas adjusted by multiplying by(1+k) – FVAdn = PMT [FVFAk,n](1+k)

– PVAdn = PMT [PVFAk,n](1+k)

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Figure 6-7 Future Value of a

Three-Period Annuity Due

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Concept Connection Example 6-17

Annuity Due

Baxter Corp started 10 years of $50,000 quarterly sinking fund deposits today at 8% compounded quarterly What will the fund be worth in 10 years?

Solution:

k = 8%/4 = 2%

n = 10 years x 4 quarters/year x 40 quarters

FVAdn = PMT [FVFAk,n](1+k) FVAd40 = $50,000[FVFA2,40](1+.02) FVAd40 = 60.4020 from Appendix A (Table A-3).

FVAd40 = $50,000[60.4020](1.02)

=$3,080,502

Recognizing Types of Annuity Problems

Annuity problems always involve a stream of equal payments with a transaction at either the end or

the beginning

– End — future value of an annuity

– Beginning — present value of an annuity

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A stream of regular payments goes on forever

– An infinite annuity

Future value of a perpetuity

– Makes no sense because there is no end point Present value of a perpetuity

– The present value of payments is a diminishing series

– Results in a very simple formula

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p

PMTPV

k

=

Example 6-18 Perpetuities – Preferred Stock

Longhorn Corp issues a security that pays $5 per quarter indefinitely Similar issues earn 8% compounded How much can Longhorn sell this security for?

Solution: Longhorn’s security pays a quarterly

perpetuity It is worth the perpetuity’s present value

calculated using the current quarterly interest rate.

k = 08 / 4 = 02 PVP = PMT / k = $5.00/.02 = $250

Continuous Compounding

Compounding periods can be any length – As the time periods become infinitesimally short, interest is compounded continuously

To determine the future value of a

continuously compounded value:

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( )kn n

FV = PV e

Example 6-20 Continuous Compounding

Multipart Problems

Time value problems are often combined due to the complexity of real situations

– A time line portrayal can be critical to

keeping things straight

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Concept Connection Example 6-21

Simple Multipart

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Exeter Inc has $75,000 in securities earning 16%

compounded quarterly The company needs $500,000

in two years

Management will deposit money monthly at 12%

compounded monthly to be sure of having the cash How much should Exeter deposit each month.

Solution: Calculate the future value of the $75,000 and subtract it from $500,000 to get the contribution

required from the deposit annuity

Then solve a save up problem (future value of an

annuity) for the payment required to get that amount.

Concept Connection Example 6-21 Simple Multipart

Concept Connection Example 6-21

Simple Multipart

Find the future value of $75,000 with Equation 6.4.

FVn = PV [PVFk,n ] FV8 = $75,000 [FVF4,8]

= $75,000 [1.3686]

= $102,645

Then the savings annuity must provide:

$500,000 - $102,645 = $397,355

Concept Connection Example 6-21

Uneven Streams and Imbedded

Annuities

Many real problems have uneven cash flows

– These are NOT annuities

For example, determine the present value of the following stream of cash flows

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Must discount each cash flow individually

Example 6-23 Present Value of an

Uneven Stream of Payments

answer is between 8% and 9%.

Calculate the interest rate at which the present value of the stream of payments shown below is

Imbedded Annuities

Sometimes uneven streams cash have annuities embedded within them

– Use the annuity formula to calculate the

present or future value of that portion of the problem

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Present Value of an Uneven Stream

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