Chapter 5Chapter 5The Time Value of MoneyThe Time Value of Money

TIME VALUE OF MONEYTIME VALUE OF MONEY DISCOUNTED CASH FLOW DISCOUNTED CASH FLOW

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: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 MONEYTIME VALUE OF MONEY DISCOUNTED CASH FLOW DISCOUNTED CASH FLOW

Time LinesTime Lines

0 1 2 3 4 5 6

k=5 % 0 1

$952.38 $1,000.00

Outline of Approach: amount - present value amount - future value

annuity - present value annuity - future value

TM 5-1 Slide 2 of 2

THE FUTURE VALUE OF AN AMOUNTTHE FUTURE VALUE OF AN AMOUNT

FV1 = PV + kPV FV1 = PV(1+k)

FV2 = FV1 + kFV1

FV2 = FV1(1+k)

Substitute for FV1

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

In General, FVn = PV(1+k) n

THE FUTURE VALUE OF AN AMOUNTTHE FUTURE VALUE OF AN AMOUNT

Define Future Value Factor for k and n =

[FVFk,n] = (1+k)n

then FVn = PV [FVFk,n]

[FVFk,n] = (1+k)n is tabulated for common combinations of k and n in Appendix A-1

The Future Value Factor for k and n FVFk,n = (1+k) n

k n 1% 2% 3% 4% 5% 6% ... 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 ... 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 ... 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 ... 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 ... 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 ... 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 ...

7 . . . . . . . . . . . . .

Example 5-1Example 5-1How much will $850 be worth if deposited for three years at

5% interest?

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

Look up FVF5,3 = 1.1576

FV3 = $850 [1.1576] = $983.96

Problem Solving Techniques Equations all contain four variables (In this case PV, FVn, k, and n) Every problem will give three and ask for the fourth.

Example 5-2Example 5-2 Ed Johnson sold land to Harriet Smith for $25,000. Terms: $15,000 down, $5,000 a year for two years. What was the "real" purchase price if the interest rate available to Ed is 6%?

Solution: Price today is $15,000 plus PV of two $5,000 payments in future FVn = PV [FVFk,n] $5,000 = PV [FVF6,1] $5,000 = PV [1.0600] PV = $4,716.98. and $5,000 = PV [FVF6,2] $5,000 = PV [1.1236] PV = $4,449.98

$15,000.00 + $4,716.98 + $4,449.98 = $24,166.96

The terms of sale imply an equivalent discount of $833.04 even though the real estate records indicate a price of $25,000

The Opportunity Cost RateThe Opportunity Cost RateExample 5-2 (continued)

6% was available to the seller nothing was actually invested at that rate

In a sense, seller lost income at that rate by giving the deferred payment terms.

Suppose Harriet Smith borrows to pay for land at 10%. Her opportunity cost rate is 10% And the deferred payment terms are worth a discount of $1,317.31 to her

Deferred terms are worth more to the recipient than to the donor!

The opportunity cost of a resource is the amount it could earn in the next best use.

More on Problem Solving TechniqueMore on Problem Solving Technique

T H E P R E S E N T V A L U E O F A N A M O U N TT H E P R E S E N T V A L U E O F A N A M O U N T

P V F k , n

P r e s e n t V a l u e F a c t o r f o r k a n d n ( A p p e n d i x A - 2 )

P V = F V n [ P V F k , n ]

T h e R e c i p r o c a l R e l a t i o n B e t w e e n F V F a n d P V F

T M 5 - 5 S l i d e 1 o f 2

F V P V ( 1 + k )

P V = F V 1( 1 + k )

( 1 + k )

nn

n n

- n

P V F V n

F V F 1P V Fk , n

k , n

If unknown is k or n, can't solve equations algebraically Solve for factor and use table

Example 5-3 What interest rate will grow $850 into $983.96 in three years?

Solution: PV = FVn[PVFk,n] $850.00 = $983.96 [PVFk,3] PVFk,3 = $850.00 / $983.96 = .8639

Find .8639 in Table A-2, along the row for three years and read 5% at top

Example 5-4How long does it take money invested at 14% to double?

Solution: FVn = PV [FVFk,n] FVF14,n = FVn / PV = 2.0000

(Search for 2.0000 in Appendix A-1, along the column for k = 14%

Table value is between 5 and 6 years)

THE FUTURE VALUE OF AN ANNUITY DEVELOPING A FORMULA

0 1 2 3

PMT PMT PMT

Each PMT earns interest at rate k from the time it appears on the time line until the end of the last period

The future value of the annuity is the sum of all the payments and all the interest

Equivalent to summing the future value of each PMT treated as an amount

TM 5-8 Slide 1 of 3

Future Value of a Three Year Ordinary Annuity Figure 5-4

0 1 2 3

PMT PMT PMTFuture Values PMT PMT(1+k) PMT(1+k)2

FVA3 = PMT + PMT(1+k) + PMT(1+k)2

TM 5-8 Slide 2 of 3

Future Value of a Three Year Ordinary Annuity

The Three Year Formula FVA = PMT(1+k)0 + PMT(1+k)1 + PMT(1+k)2

Generalizing the Expression

FVAn = PMT (1+k)n-i

The Future Value Factor for an Annuity

FVAn = PMT (1 + k)n-i

= FVFAk,n (Appendix A-3) FVAn = PMT [FVFAk,n]

TM 5-8 Slide 3 of 3

i

n

1

i

n

1

THE FUTURE VALUE OF AN ANNUITYTHE FUTURE VALUE OF AN ANNUITYSOLVING PROBLEMSSOLVING PROBLEMS

Example 5-5 The Brock Corporation will receive fees of $100,000 a yearfor ten years and will invest each payment at 7%. How much will it haveafter the last payment?

Solution: years @ k=7%

0 1 2 3 8 9 10

$10K $10K $10K $10K $10K $10K FVA10 FVAn = PMT [FVFAk,n] FVA10 = $100,000 [FVFA7,10] FVFA7,10 = 13.8164 (from Appendix A-3) FVA10 = $100,000 [13.8164] = $1,381,640

TM 5-9 Slide 1 of 2

COMPOUND INTEREST AND NON-ANNUAL COMPOUNDINGCOMPOUND INTEREST AND NON-ANNUAL COMPOUNDING

Compound InterestEarning interest on previously earned interest

The Effective Annual Rate (EAR)The Effective Annual Rate (EAR)The rate of annually compounded interest equivalent to the nominal rate

compounded more frequently Compounding Final balance

Annual $112.00 Semiannual $112.36 Quarterly $112.55 Monthly $112.68

Table 5-2 Year End Balances at Various Compounding Periods$100 Initial Deposit and knom = 12%

In general:

Compounding PeriodsCompounding PeriodsFrequency with which interest is credited for calculating future interest,

usually annually, semiannually, quarterly, or monthly.

The shorter the period, the more interest is earned on interest Annually

12%

$100 $112

Semiannually 6% 6%

$100 $106 $112.36

Quarterly 3% 3% 3% 3%

$100 $103 $106.09 $109.27 $112.55

Quote the annual (nominal) rate (knom) stating the compounding period immediatelyafterward "12% compounded quarterly"

TM 5-10 Slide 2 of 2

COMPOUNDING PERIODS AND THE TIME VALUE FORMULASCOMPOUNDING PERIODS AND THE TIME VALUE FORMULASTime periods must be compounding periods and the interest rate must be the rate for a single compounding period

Semiannually: k = knom / 2 n = years 2 Quarterly: k = knom / 4 n = years 4

Monthly: k = knom / 12 n = years 12

Example 5-7

Save up to buy a $15,000 car in 2 1/2 years. Bank pays 12% compounded monthly. How much must be deposited each month?

Solution: k = knom/12 = 12%/12 = 1% n = 2.5 yr 12 mo/yr = 30 months

FVAn = PMT [FVFAk,n] $15,000 = PMT [FVFA1,30] $15,000 = PMT [34.785] PMT = $431.22

Generalizing:

PVA = PMT(1+k)-1 + PMT(1+k)-2 + . . . + PMT(1+k)-n

PVA = PMT

PVA = PMT [PVFAk,n]

Appendix A-4

THE PRESENT VALUE OF AN ANNUITYTHE PRESENT VALUE OF AN ANNUITY

0 1 2 3

PMT PMT PMT PVsPMT/(1+k)PMT/(1+k)2

PMT/(1+k)3

PVA = PMT/(1+k) + PMT/(1+k)2 + PMT/(1+k)3

Figure 5-6 Present Value of a Three Period Ordinary Annuity

TM 5-13 Slide 1 of 2

AMORTIZED LOANSAMORTIZED LOANSPrincipal is paid off gradually during loan's life Generally structured so that a constant payment

is made periodically, usually monthly

Each payment contains one month's interest and an amount to reduce principal

Interest is charged on the month beginning loan balance

As loan's principal is reduced interest charges become smaller

Since monthly payments are constant successive payments contain larger proportions of principal repayment and smaller proportions of interest

THE PRESENT VALUE OF AN ANNUITY -THE PRESENT VALUE OF AN ANNUITY -SOLVING PROBLEMSSOLVING PROBLEMS

Example 5-9 The Shipson Company will receive payments of $5,000 everysix months (semiannually) for ten years on a sales contract which the bankwill discount at 14% compounded semiannually. How much will Shipsonreceive?

Solution: k = knom/2 = 14%/2 = 7%

n = 10 yrs 2 = 20 Half years @ k=7%

0 1 2 3 18 19 20

$5K $5K $5K $5K $5K $5K PVA PVA = PMT [PVFAk,n], PVA = $5,000 [PVFA7,20] PVA = $5,000 [10.5940] PVA = $52,970

TM 5-14

Example 5-10 How much is the monthly payment on a $10,000 loan to be repaid in monthly installments over four years at 18% (compounded monthly)?

Solution: k = knom/12 = 18%/12 = 1.5%

n = 4 yrs 12 mo/yr = 48 months PVA = PMT [PVFAk,n] $10,000 = PMT [PVFA1.5,48] $10,000 = PMT [34.0426] PMT = $293.75

Example 5-11 How much can you borrow at 12% compounded monthly over three years if you can make payments of $500 per month?

Solution: k = knom/12 = 12%/12 = 1% n = 3 yrs 12 mo/yr = 36 months

PVA = PMT [PVFAk,n] PVA = $500 [PVFA1,36] PVA = $500 [30.1075] PVA = $15,053.75

A loan is always a PVA problemAmount borrowed is always PVA

Loan payment is always PMT

LOAN AMORTIZATION SCHEDULESLOAN AMORTIZATION SCHEDULES Beginning Interest Principal Ending

Period Balance Payment @ 1% Reduction Balance 1 $15,053.75 $500.00 $150.54 $349.46 $14,704.29 2 $14,704.29 $500.00 $147.04 $352.96 $14,351.33 3 _________ $500.00 _______ _______ __________ 4 _________ $500.00 _______ _______ __________ . . . . . . . . . . . . . . . . . .

MORTGAGE LOANSMORTGAGE LOANS

Early payments are nearly all interest Later Payments are nearly all principal

ExampleA thirty year, $100,000 mortgage at 12% (compounded monthly)

has a monthly payment of $1,028.61

First month's interest is $1,000 (1% of $100,000)Only $28.61 is applied to principal The first payment is 97.2% interest

Reverses in last months

Tax Effect of Mortgage Payments Mortgage interest is tax deductible Effective first payment at 28%:

Payment $1,028.61 Tax Savings 280.00 Net $ 748.61

Payoff Timing Halfway through a mortgage's life, it isn't half paid off: Present value of the second half of the payment stream The amount one could borrow making 180 payments of $1,028.61

PVA = PMT [PVFAk,n] = $1,028.61 [PVFA1,180] = $1,028.61 [83.3217] = $85,705.53

Total Interest Paid

Total payments = $1,028.61 360 = $370,299.60Less original loan = 100,000.00Total Interest = $270,299.60Tax Savings @ 28% 75,683.89

Net Interest Cost $194,615.71

Example 5-12The Baxter Corporation began 10 years of quarterly $50,000 sinking fund deposits today at 8% compounded quarterly. What will the fund be worth in 10 years?

Solution: k = 8%/4 = 2% n = 10 yrs 4 qtrs/yr = 40 qtrs

FVAdn = PMT [FVFAk,n] (1+k)FVAd40 = $50,000 [FVFA2,40] (1.02)FVAd40 = $50,000 [60.4020] (1.02)

= $3,080,502.00

THE ANNUITY DUETHE ANNUITY DUE Payments occur at the beginning of time periods

The Future Value of an Annuity Due 0 1 2 3

PMT PMT PMT PMT

PMT(1+k) PMT(1+k) (1+k) PMT(1+k)2 (1+k)

FVAd3 = [PMT + PMT(1+k) + PMT(1+k)2](1+k) Figure 5-7 Future Value of a Three Period Annuity Due

FVAd3 = PMT(1+k) + PMT(1+k)(1+k) + PMT(1+k)2(1+k)

FVAdn = PMT [FVFAk,n] (1+k) Recognize by: starting now, today, or immediately

TM 5-18 Slide 1 of 2

The Present Value of an Annuity Due PVAd = PMT [PVFAPVAd = PMT [PVFAk,nk,n] (1+k)] (1+k)

CONTINUOUS COMPOUNDINGCONTINUOUS COMPOUNDING

FVn = PV (ekn)

Where k = nominal rate in decimal form n = years e = 2.71828...

RECOGNIZING TYPES OF ANNUITYRECOGNIZING TYPES OF ANNUITYPROBLEMSPROBLEMS

0 1 2 n -2 n -1 n

PVA FVA Transaction Transaction Here Here

A loan is always a present value of an annuity problem

The annuity is the stream of loan payments The transaction is the transfer of the amount borrowed from the

lender to the borrower

Saving up is always a future value of an annuity problem

TM 5-19 Slide 1 of 3

Example 5-15: a. Future value of $5,000 at 6 1/2% compounded continuously for 3 1/2 years b. The Equivalent Annual Rate (EAR) of 12% compounded continuously?

Solution: a. FVn = PV (ekn) FV3.5 = $5,000 (e(.065)(3.5)) = $5,000 (e.2275) = $5,000 (1.255457) FV3.5 = $6,277.29

b. Deposit $100 for one year: FVn = PV (ekn) FV1 = $100 (e(.12)(1)) = $100 (e.12) = $100 (1.1275) = $112.75 Initial deposit = $100, Interest earned = $12.75,

EAR = $12.75 / $100 = 12.75%

First find the future value of the $75,000

FVn = PV [FVFk,n] FV8 = $75,000 [FVF4,8] = $75,000 [1.3686] = $102,645

Then the savings annuity must provide

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

FVAn = PMT [FVFAk,n] $397,355 = PMT [FVFA1,24] $397,355 = PMT [26.9735] PMT = $14,731

MULTI-PART PROBLEMSMULTI-PART PROBLEMS

Example 5-16 Exeter Inc. has $75,000 earning 16% compounded quarterly.The company needs $500,000 in two years. How much should it deposit eachmonth in an account paying 12% compounded monthly?

Solution:Quarters @ 4%

0 1 2 3 4 5 6 7 8

$75,000 FV Product

Months @ 1% Launch 0 1 2 3 22 23 24 $500,000

PMT PMT PMT PMT PMT FVA

TM 5-21 Slide 1 of 2

Example 5-17Example 5-17The Smith family plans to buy a $200,000 house using a traditional thirty year mortgage.

Banks allow roughly 25% of income to be applied to mortgage payments.

The Smiths expect their income will be $48,000. and the mortgage interest rate will be 9% when they buy the house.

They now have $10,000 in a bank account which pays 6% compounded quarterly.

How much will they have to add to the account each quarter to buy the house in three years?

Solution: Required savings will be $200,000 less amount borrowed onmortgage less future value of the $10,000

Qtrs @ 1.5% 0 1 2 10 11 12

$10K FV12 Amount Months @ .75%

0 1 2 359 360

PVA $1K $1K $1K $1K $1K Qtrs @ 1.5%

0 1 2 10 11 12

PMT PMT PMT PMT PMT Mortgage

FVA12 Payments Savings Payments Time of Purchase

$200K Required

Problem is focused around the date of purchase

TM 5-22 Slide 2 of 2

Mortgage: k = 9%/12 = .75%, n = 360 PMT = ($48,000/12) x .25 = $1,000

PVA = PMT [PVFAk,n] = $1,000 [PVFA.75,360] = $1,000 [124.2819] = $124,282Future value of the $10,000 already in bank: k = 6%/4 = 1.5%, n = 12 FV12 = $10,000 [FVF1.5,12] = $10,000 [1.1956] = $11,956Savings requirement: $200,000 - $124,281.90 - $11,956.00 = $63,762.10 = the FVA of savings deposits

FVAn = PMT [FVFAk,n] $63,762.10 = PMT [FVFA1.5,12] $63,762.10 = PMT [13.0412] PMT = $4,889

UNEVEN STREAMSUNEVEN STREAMS

Require treatment as individual amountsSolving for k requires an iterative approach

IMBEDDED ANNUITIESIMBEDDED ANNUITIES Example 5-19

Calculate present value: 0 1 2 3 4 5 6 7 8

$5 $7 $3 $3 $3 $3 $6 $7 PV PV

PV PV

PV

TM 5-24 Slide 1 of 2

PVA

Solution:

Payment 1: PV = FV1[PVF12,1] = $5(.8929) = $4.46 Payment 2: PV = FV2[PVF12,2] = $7(.7972) = $5.58 Payment 7: PV = FV1[PVF12,7] = $6(.4523) = $2.71 Payment 8: PV = FV1[PVF12,8] = $7(.4039) = $2.83

Annuity: PVA = PMT [PVFA12,4] = $3(3.0373) = $9.11 and PV = FV2[PVF12,2] = PVA(.7972) =

$9.11(.7972)= $7.26 $22.84