8 8 Chapter Time Value of Money Time Value of Money Slides Developed by: Terry Fegarty Seneca...

88Chapt

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Chapt

er Time Value

of MoneyTime Value

of Money

Slides Developed by:

Terry FegartySeneca College

© 2006 by Nelson, a division of Thomson Canada Limited 2

Chapter 8 – Outline (1)

• The Time Value of Money Time Value Problems Amount Problems—Future Value Other Issues Financial Calculators Spreadsheet Solutions The Present Value of an Amount Finding the Interest Rate Finding the Number of Periods



• Annuities The Future Value of an Annuity The Future Value of an Annuity—Developing a

Formula The Future Value of an Annuity—Solving Problems The Sinking Fund Problem Compound Interest and Non-Annual Compounding The Effective Annual Rate The Present Value of an Annuity—Developing a

Formula The Present Value of an Annuity—Solving Problems



Amortized Loans Loan Amortization Schedules Mortgage Loans The Annuity Due Perpetuities Continuous Compounding

• Multipart Problems Uneven Streams Imbedded Annuities


The Time Value of Money

• $100 in your hand today is worth more than $100 in one year Money earns interest

• The higher the interest, the faster your money grows

Q: How much would $1,000 promised in one year be worth today if the bank paid 5% interest?

A: $952.38. If we deposited $952.38 after one year we would have earned $47.62 ($952.38 × .05) in interest. Thus, our future value would be $952.38 + $47.62 = $1,000.

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• Present Value The amount that must be deposited today to

have a future sum at a certain interest rate The discounted value of a sum is its

present value In our example, what is the present value?

$952.38

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• Future Value The amount a present sum will grow into at a

certain interest rate over a specified period of time

In our example, • The present sum (value) is $952.38• The interest rate is 5%• The time is 1 year• The future value is $1,000

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Time Value Problems

• Time value deals with four different types of problems Amount— a single amount that grows at

interest over time• Future value • Present value

Annuity— a stream of equal payments that grow at interest over time• Future value• Present value


Time Value Problems

• 4 methods to solve time value problems Use formulas Use financial tables Use financial calculator Use financial functions in spreadsheet


Amount Problems—Future Value

• The future value (FV) of an amount How much a sum of money placed at interest

(k) will grow into in some period of time• If the time period is one year

• FV1 = PV + kPV or FV1 = PV(1+k)

• If the time period is two years• FV2 = FV1 + kFV1 or FV2 = PV(1+k)2

• If the time period is generalized to n years• FVn = PV(1+k)n


Amount Problems—Future Value

• The (1 + k)n depends on Size of k and n

• Can develop a table depicting different values of n and k and the proper value of (1 + k)n

• Can then use a more convenient formula

• FVn = PV [FVFk,n]

These values can be looked up in an interest

factor table.Q: If we deposited $438 at 6%

interest for five years, how much would we have?

A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13

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Table 8-1: The Future Value Factor for k and n FVFk,n = (1+k)n

1.3382

6%

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1.33825


Other Issues

• Problem-Solving Techniques Three of four variables are given

• We solve for the fourth

• The Opportunity Cost Rate The opportunity cost of a resource is the

benefit that would have been available from its next best use• Lost investment income is an opportunity cost


Financial Calculators

• How to use a typical financial calculator in time value Five time value keys

• Use either four or five keys

Some calculators distinguish between inflows and outflows

• If a PV is entered as positive the computed FV is negative



N

I/Y

PMT

FV

PV

Number of time periods

Interest rate (%)

Future Value

Present Value

Payment

Basic Calculator Keys



N

I/Y

PMT

PV

FV

1

6

0

5000

4,716.98 Answer

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Q: What is the present value of $5,000 received in one year if interest rates are 6%?A: Input the following values on the calculator and compute the PV:


Spreadsheet Solutions

• Time value problems can be solved on a spreadsheet such as Microsoft® Excel®

• Click on the fx (function) button• Select for the category Financial• Select the function for the unknown variable• For example, to solve for present value:

Use PV(k, n, PMT, FV) Interest rate (k) is entered as a decimal, not a

percentage


Spreadsheet Solutions

• To solve for: FV use =FV(k, n, PMT, PV) PV use =PV(k, n, PMT, FV) k use =RATE(n, PMT, PV, FV) N use =NPER(k, PMT, PV, FV) PMT use =PMT(k, n, PV, FV) Of the three cash variables (FV, PMT or PV)

• One is always zero• The other two must be of the opposite sign• Reflects inflows (+) versus outflows (-)

Select the function for the unknown

variable, place the known variables in

the proper order within the

parentheses and input 0, for the

unknown variable.


The Present Value of an Amount

• Either equation can be used to solve any amount problems

n

n

n n

Interest Factor

FV PV 1+k

Solve for PV

1PV = FV

1 k

k,nk,n

1FVF

PVF Solving for k or n involves

searching a table.


Spreadsheet Solution—PV of Amount

Click on the fx (function) button Select for the category Financial In the insert function box, select PV You will see PV(rate,nper,pmt,fv,type) Press OK Fill in rate = B4, nper = C1, pmt = 0, fv = C2 Enter OK

Select the function for the unknown

variable, place the known variables in

the proper order within the

parentheses and input 0, for the

unknown variable.

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Spreadsheet Solution—PV of Amount E

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Example 8.3: Finding the Interest Rate

N

PV

I/Y

PMT

FV

3

-850

0

983.96

5.0 Answer

Calculator

Q: What interest rate will grow $850 into $983.96 in three years?A:

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Interest factor is 850.00 / 983.96 = 0.8639

Look up in Table A-2 with n = 3

Interest = 5%

Financial Table


Example 8.3: Spreadsheet SolutionE

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Example 8.3: Spreadsheet Solution

Click on the fx (function) button Select for the category Financial In the insert function box, select RATE You will see RATE(n, PMT, PV, FV) Press OK Fill in n = E1, PMT = 0, PV = -B2, FV = E2 Enter OK

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Example 8.4: Finding the Number of Periods

PV

FV

N

I/Y

PMT

-1

2

14

0

5.29 Answer

Calculator

Interest factor is 2 / 1 = 2

Look up in Table A-2 with k = 14%

n = 5 – 6 years

Financial Table

Q: How long does it take money invested at 14% to double?

A: The future value is twice the present value. If the present value is $1, the future value is $2

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Example 8.4: Finding the Number of Periods

Click on the fx (function) button Select for the category Financial In the insert function box, select NPER You will see NPER(k, PMT, PV, FV) Enter OK Fill in the cell references for

• k (= .14)• PMT = 0• PV (= -1)• FV = 2

Enter OK NPER = 5.29

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Annuities

• Annuity A finite series of equal payments separated

by equal time intervals•Ordinary annuity

• Payments occur at the end of the time periods• Monthly lease, pension, and car payments are annuities

•Annuity due• Payments occur at the beginning of the time periods


Figures 8.1 and 8.2: Ordinary Annuity and Annuity Due

Ordinary Annuity

Annuity Due


Figure 8.3: Timeline Portrayal of an Ordinary Annuity


Future Value of an Annuity

• Future value of an annuity The sum, at its end, of all payments and all

interest if each payment is deposited when received


Figure 8.4: FV of a Three-Year Ordinary Annuity


The Future Value of an Annuity—Developing a Formula• Thus, for a 3-year annuity, the formula is

FVFAk,n

0 1 2

0 1 2 n -1

n

nn i

ni=1

FVA = PMT 1+k PMT 1+k PMT 1+k

Generalizing the Expression:

FVA = PMT 1+k PMT 1+k PMT 1+k PMT 1+k

which can be written more conveniently as:

FVA PMT 1+k

Factoring PMT outside the summation, we

n

n i

ni=1

obtain:

FVA PMT 1+k


The Future Value of an Annuity—Solving Problems

• There are four variables in the future value of an annuity equation The future value of the annuity itself The payment The interest rate The number of periods


Example 8.5: The Future Value of an Annuity

Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time.

If the invested money is expected to earn 7%, how much will Brock have after the last payment is received?

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Example 8.5: The Future Value of an Annuity

A: Use the future value of an annuity equation:

FVAn = PMT[FVFAk,n]

In Table A-3, look up the interest factor at an n of 10 and

a k of 7

Interest factor = 13.8164

Future value = $100,000[13.8164] = $1,381,640

FV

N

PMTI/Y

1,381,645 Answer

10

1000007

0PV

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Example 8.5: Spreadsheet Solution

Click on the fx (function) button Select for the category Financial In the insert function box, select FV You will see FV(k, n, PMT, PV) Enter OK Fill in the cell references for

• k (=.07)• n (= 10)• PMT (=100000)• PV (=0)

Enter OK FV = 1,381,644.80

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The Sinking Fund Problem

• Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the

bonds’ lives• Principal is repaid at maturity in a lump sum

• A sinking fund provides cash to pay off a bond’s principal at maturity

• Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem


Example 8.6: The Sinking Fund Problem

Q: The Greenville Company issued bonds totaling $15 million for 30 years. A sinking fund must be maintained after 10 years, which will retire the bonds at maturity. The estimated yield on deposited funds will be 6%.

How much should Greenville deposit each year to be able to retire the bonds?

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Example 8.6: The Sinking Fund Problem

A: The time period of the annuity is the last 20 years of the bond issue’s life. On your calculator:

PMT

N

FVI/Y

407,768.35 Answer

20

150000006

0PV

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Compound Interest and Non-Annual Compounding

• Compounding Earning interest on interest

• Compounding periods Interest is usually compounded annually,

semiannually, quarterly or monthly• Interest rates are quoted by stating the nominal

rate followed by the compounding period


The Effective Annual Rate

• Effective annual rate (EAR) The annually compounded rate that pays the

same interest as a lower rate compounded more frequently


The Effective Annual Rate—Example

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Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest?

A: If your initial deposit were $100, you would have $112.68 after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($112.68 $100) – 1] would have to be earned.



• EAR can be calculated for any compounding period using the following formula:

m

nominalEAR - 1k

1 m

Effect of more frequent compounding is greater at higher interest rates


Impact of Compounding Frequency

$1,097

$1,098

$1,099

$1,100

$1,101

$1,102

$1,103

$1,104

$1,105

$1,106

Annual Semi-Annual

Quarterly Monthly Daily

$1,000 Invested at 10% Nominal Rate for One Year



• The APR and EAR Annual percentage rate (APR)

• Is actually the nominal rate and is less than the EAR

• Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single

compounding period• For instance, with a quarterly compounding period the knominal must be divided by 4 and the n must be multiplied by 4


Example 8.7: The Effective Annual Rate

PMT

N

FV

I/Y

431.22 Answer

30

15000

1

0PV

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Q: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month?A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. On your calculator:


The Present Value of an Annuity—Developing a Formula• Present value of an annuity

Sum of all of the annuity’s payments

2 3

1 2 3

1 2 n

PMT PMT PMTPVA =

1+k 1+k 1+k

which can also be written as:

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

n

i

i=1

btain:

PVA PMT 1+k

PVFAk,n


Figure 8.6: PV of a Three-Payment Ordinary Annuity


The Present Value of an Annuity—Solving Problems

• There are four variables in the present value of an annuity equation The present value of the annuity itself The payment The interest rate The number of periods

• Problem usually presents 3 of the 4 variables


Example 8.9: The Present Value of an Annuity

Q: The Shipson Company has just sold a large machine on an installment contract. The contract calls for payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to it the present (discounted) value. The bank is willing to discount the contract at 14% compounded semiannually.

How much should Shipson receive?

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Example 8.9: The Present Value of an Annuity

A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity.

PV

N

FV

I/Y

52,970.07 Answer

20

0

7

5000PMT

This can also be calculated using the PVA Table A4.

Look up n = 20 and k = 7%.

PVA = $5,000[10.594]

= $52,970

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Amortized Loans

• An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant

payment is made periodically• Represents the present value of an annuity


Example 8.10: Amortized Loans

PMT

N

PV

I/Y

293.75

48

10000

1.5

0FV

Answer

This can also be calculated using the PVA formula of

PVA = PMT[PVFAk, n]

n = 48 and k = 1.5% $10,000 = PMT[34.0426]

= $293.75.

Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?

A: Adjust your interest rate and number of periods for monthly compounding. On your calculator:

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Example 8.11: Amortized Loans

Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?

A: Adjust your k and n for monthly compounding. On your calculator:

PV

N

FV

I/Y

15,053.75 Answer

36

0

1

500PMT

This can also be calculated using the PVA Table A4.

Look up n = 36 and k = 1%.

PVA = $500[30.1075]

= $15,053.75

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Loan Amortization Schedules

• Detail the interest and principal in each loan payment

• Show the beginning and ending balances of unpaid principal for each period

• Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k)


Example 8.11: Loan Amortization Schedule

Q: Develop an amortization schedule for the loan demonstrated in Example 8.11

Note that the Interest portion of the payment is decreasing while the Principal portion is

increasing.

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Mortgage Loans

• Mortgage loans (AKA: mortgages) Loans used to buy real estate

• Often the largest single financial transaction in a person’s life Typically an amortized loan over 30 years

• During the early years of the mortgage nearly all the payment goes toward paying interest

• This reverses toward the end of the mortgage

Halfway through a mortgage’s life half of the loan has not been paid off


Mortgage Loans

• Implications of mortgage payment pattern Long-term loans like mortgages result in large total

interest amounts over the life of the loan• At 6% interest, compounded monthly, over 25 years,

borrower pays almost the amount of the loan just in interest!

• Canadian banks compound semi-annually, thus lowering interest charges

Early mortgage payments and more frequent mortgage payments provide a large interest saving


Mortgage Loans—Example E

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PMT

N

FV

I/Y

1,015.65

360

0

0.5979

150000PV

Answer143.76%Interest / Principal

$215,634Total Interest

$150,000- Original Loan

$365,634Total payments

360X # of payments

$1,015.65Monthly payment

Q: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan.

A: Adjust the n and k for monthly compounding and input the following calculator keystrokes.


The Annuity Due

• In an annuity due payments occur at the beginning of each period

• The future value of an annuity due Because each payment is received one period

earlier, it spends one period longer in the bank earning interest

n -1

n

n k,n

FVAd = PMT + PMT 1+k PMT 1+k 1 k

which written with the interest factor becomes:

FVAd PMT FVFA 1 k


Figure 8.7: The Future Value of a Three-Period Annuity Due


Example 8.12: The Annuity Due

FV

N

PMT

I/Y

3,020,099 x 1.02 = 3,080,501 Answer

40

50000

2

0PV

NOTE: Advanced calculators allow you to switch from END (ordinary annuity) to BEGIN

(annuity due) mode.

Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time?

A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes.

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The Annuity Due

• The present value of an annuity due Formula

k,nPVAd PMT PVFA 1 k

Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one

end of the stream of payments• End of stream—future value of an annuity• Beginning of stream—present value of an annuity


Perpetuities

• A perpetuity is a stream of regular payments that 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 A diminishing series of numbers

• Each payment’s present value is smaller than the one before

p

PMTPV

k


Example 8.13: Perpetuities

p

PMT $5PV $250

k 0.02 You may also work this by inputting

a large n into your calculator (to simulate infinity), as shown below.

PV

N

PMTI/Y

250

999

52

0FVAnswer

Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Investors can earn 8% compounded quarterly on their money. How much can Longhorn sell this security for?

A: Convert the k to a quarterly k and plug the values into the equation.

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Continuous Compounding

• Compounding periods can be shorter than a day As the time periods become infinitesimally short,

interest is said to be compounded continuously

• To determine the future value of a continuously compounded value:

Where k = nominal rate, n = number of years, e = 2.71828

knnFV PV e


Example 8.15: Continuous Compounding

Q: The First National Bank of Cardston is offering continuously compounded interest on savings deposits.

If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have?

What is the equivalent annual rate (EAR) of 12% compounded continuously?

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Example 8.15: Continuous Compounding

A: To determine the future value of $5,000, plug the appropriate values into the equation

n = knFV PV e

To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return

.065 3.53.5=FV $5,000(2.71828) = $6.277.29

11

.12=FV $100 e

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=

.12= $100 2.71828 = $112.75

$112.75-$100EAR= =12.75%

$100

NOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding


Table 8.5: Time Value Formulas


Multipart Problems

• Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to

keeping things straight


Example 8.16: Multipart Problems

Q: Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank account will pay 12% compounded monthly.

How much should Exeter deposit with the bank each month ?

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Example 8.16: Multipart Problems

A: Two things are happening in this problem Exeter is saving money every month (an annuity) and The money invested in securities (an amount) is growing

independently at interest

We have three steps First, we need to find the future value of $75,000 Then, we subtract that future value from $500,000 to

determine how much extra Exeter needs to save via the annuity

Then, we solve a future value of an annuity problem for the payment

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Example 8.16: Multipart Problems E

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To find the future value of the $75,000…

PV

N

PMT

I/Y

102,645

8

0

4

75000FV Answer

To find the savings annuity value

PMT

N

PV

I/Y

14,731

24

0

1

397355FVAnswer

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


Uneven Streams

• Many real world problems have sequences of uneven cash flows These are NOT annuities

• For example, if you were asked to determine the present value of the following stream of cash flows

$100 $200 $300

Must discount each cash flow individually Not really a problem when attempting to determine either a

present or future value Becomes a problem when attempting to determine an interest rate


Example 8.18: Uneven Streams

$100 $200 $300

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A: Start with a guess of 12% and discount each amount separately at that rate.

This value is too low; select a lower interest rate. Using 11% gives us $471.77. The answer is

between 8% and 9%.

1 k,1 2 k,2 3 k,3

12,1 12,2 12,3

PV = FV PVF FV PVF FV PVF

$100 PVF $200 PVF $300 PVF

$100 .8929 $2

$462.2

00 .7972 $300 .71 8

7

1

Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500.


Imbedded Annuities

• Sometimes uneven streams of cash flows will have annuities imbedded within them We can use the annuity formula to calculate

the present or future value of that portion of the problem

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