Practical financial managment 7e LASHER chapter 6

Time Value of Money A sum of money in hand today is worth more than the same sum promised with certainty in the future.. Concept Connection Example 6-5 The Future Value of an Annuity...

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

Outline of Approach

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:

FV1 = PV(1 + k)

If leave in bank for a second year:

FV2 = PV(1 + k)(1 ─ k)FV2 = PV(1 + k)2

Generalized:

FVn = PV(1 + k)n

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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

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

Use either four or five keys

If PV is entered as positive the computed FV is negative

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Financial Calculators

Basic Calculator functions

0 5000

4,716.98 Answer What is the present value of $5,000 to be received in one year if the interest rate is 6%? Input the following values on the calculator and compute the PV:

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

Concept Connection Example 6-3 Finding the Interest Rate

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

deposited when received

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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—Solving Problems

Four variables in the future value of an annuity equation

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

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

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)

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 m

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

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.

%

1 12

%

12 12

nom k

Concept Connection Example 6-7

Compounding periods and Time Value Formulas

FVAN = PMT [FVFAk,n ] Write the future value of an annuity expression and substitute.

$15,000 = PMT [FVFA1,30 ] From Appendix A (Table A-3) FVFA1,30 = 34.7849 substituting

$15,000 = PMT [34.7849] Solve for PMT

PMT = $431.22

Concept Connection Example 6-7

Compounding periods and Time Value Formulas

Figure 6-6 Present Value of a Three-period Ordinary Annuity

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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

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

Concept Connection Example 6.9

PVA - Discounting a Note

Amortized Loans

An amortized loan’s principal is paid off over its life along with interest Constant Payments are made up of a varying mix of principal and interest 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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Table 6-4

Partial Amortization Schedule

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Develop an amortization schedule for the loan in Example 6 -12

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 a person’s life

years

Early years most of payment is interestLater on principal is reduced quickly

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Concept Connection 6-13

Interest Content of Early Loan Payment

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Calculate interest in the first payment on a 30-year,

$100,000 mortgage at 6%, compounded monthly

Solution:

n= 30 years x 12 months/year = 360 k=6%/12 months/year = 5%

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

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

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Figure 6-7 Future Value of a

Three-Period Annuity Due

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Concept Connection Example 6-17

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

PMT PV

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

continuously

To determine the future value of a continuously compounded value:

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( )kn n

Example 6-20 Continuous Compounding

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

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We’ll start with a guess of 12% and discount each amount separately at that rate.

This value is too low, so we need to select a lower interest rate Using 11% gives us $471.77

The answer is between 8% and 9%.

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

Imbedded Annuities

Sometimes uneven streams cash have annuities embedded within them

portion of the problem

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Present Value of an Uneven Stream

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