Time Value of Money

Chapter # 2

A dollar received today is worth more than a dollar received tomorrow› This is because a dollar received today can be

invested to earn interest› The amount of interest earned depends on the

rate of return that can be earned on the investment

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!

Which would you prefer -- $10,000 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?

TIMETIME allows you the opportunity to postpone consumption and earn

INTERESTINTEREST.

Why is TIMETIME such an important element in your decision?

Compound InterestCompound InterestInterest paid (earned) on any previous

interest earned, as well as on the principal borrowed (lent).

Simple InterestSimple Interest

Interest paid (earned) on only the original amount, or principal borrowed (lent).

FormulaFormula SI = P(i)(n)

SI: Simple InterestP: Deposit today i: Interest Rate per Periodn: Number of Time Periods

Simple Interest(SI) = P(i)(n)=

$1,000(.07)(2)= $140$140

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

FVFV = P + SI = $1,000 + $140= $1,140$1,140

Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

What is the Future Value Future Value (FVFV) of the deposit?

The Present Value is simply the $1,000 you originally deposited. That is the value today!

Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

What is the Present Value Present Value (PVPV) of the previous problem?

Assume that you deposit $1,000$1,000 at a compound interest rate of

7% for 2 years2 years.

0 1 22

$1,000$1,000

FVFV22

7%

FVFV11 = PP (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070

Compound InterestYou earned $70 interest on your

$1,000 deposit over the first year.This is the same amount of interest

you would earn under simple interest.

FVFV11 = P P (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070

FVFV22 = FV1 (1+i)1 = PP (1+i)2 =

$1,000$1,000(1.07)2

= $1,144.90$1,144.90You earned an EXTRA $4.90$4.90 in Year 2 with

compound over simple interest.

Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 5 yearsyears.

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%

Calculation based on general formula: FVFVnn = P (1+i)

FVFV55 = $10,000 (1+ 0.10)= $16,105.10$16,105.10

n5

Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.

0 1 22

$1,000$1,000

7%

PV1PVPV00

PVPV = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 = FVFV22 / (1+i)2 = $873.44$873.44

0 1 22

$1,000$1,000

7%

PVPV00

PVPV = FVFV11 / (1+i)1

PVPV = FVFV22 / (1+i)2

etc.

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a Interest rate of 10%.

0 1 2 3 4 55

$10,000$10,000PVPV

10%

Calculation based on general formula: PVPV = FVFVnn / (1+i)n

PVPV = $10,000$10,000 / (1+ 0.10)5 = $6,209.21$6,209.21

The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit

today (present value).

Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period.

Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period.

An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring at regular intervals.

Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd of

Period 1EndEnd of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning of

Period 1BeginningBeginning of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0

R R R

0 1 2 n n n+1

FVAFVAnn

R = Periodic Cash Flow

Cash flows occur at the end of the period

i% . . .

FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 +

$1,000(1.07)0

= $1,145 + $1,070 + $1,000 = $3,215$3,215

$1,000 $1,000 $1,000

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

7%

$1,070

$1,145

Cash flows occur at the end of the period

The future value of an ordinary annuity can be viewed as occurring at the endend of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginningbeginning of the last cash flow period.

FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) = $3,215$3,215Period 6% 7% 8%

1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 +

R(1+i)1 = FVAFVAn n (1+i)

R R R R R

0 1 2 3 n-1n-1 n

FVADFVADnn

i% . . .

Cash flows occur at the beginning of the period

FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 +

$1,000(1.07)1

= $1,225 + $1,145 + $1,070 = $3,440$3,440

$1,000 $1,000 $1,000 $1,070

0 1 2 3 3 4

$3,440 = FVAD$3,440 = FVAD33

7%

$1,225

$1,145

Cash flows occur at the beginning of the period

FVADFVADnn = R (FVIFAi%,n)(1+i)FVIFA i ni n = (1+i)n -1/ i

FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)= $1,000 (3.215)(1.07) =

$3,440$3,440Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

PVAPVAnn = R/(1+i)1 + R/(1+i)2

+ ... + R/(1+i)n

R R R

0 1 2 n n n+1

PVAPVAnn

R = Periodic Cash Flow

i% . . .

Cash flows occur at the end of the period

PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 +

$1,000/(1.07)3

= $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32

$1,000 $1,000 $1,000

0 1 2 3 3 4

$2,624.32 = PVA$2,624.32 = PVA33

7%

$ 934.58$ 873.44 $ 816.30

Cash flows occur at the end of the period

The present value of an ordinary annuity can be

viewed as occurring at the beginningbeginning of the first cash flow

period, whereas the present value of an annuity due can be viewed as occurring at the endend

of the first cash flow period.

PVAPVAnn = R (PVIFAi%,n)

PVAPVA33 = $1,000 (PVIFA7%,3)= $1,000 (2.624) =

$2,624$2,624Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAPVAn n (1+i)

R R R R

0 1 2 n-1n-1 n

PVADPVADnn

R: Periodic Cash Flow

i% . . .

Cash flows occur at the beginning of the period

PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02

$1,000.00 $1,000 $1,000

0 1 2 33 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44

Cash flows occur at the beginning of the period

PVADPVADnn = R (PVIFAi%,n)(1+i)

PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) =

$2,808$2,808Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%? 0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

PVPV00

10%10%

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%

$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow

General Formula:FVn = PVPV00(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per Year

i: Annual Interest RateFVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of 12%.

Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)

= 1,262.481,262.48

Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73

Daily FV2 = 1,0001,000(1+[.12/365])(365)(2) = 1,271.201,271.20

The term loan amortization refers to the determination of equal periodic loan payments.

Julie Miller is borrowing $10,000 $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments

are made for 5 years.Step 1: Payment

PVPV00 = R (PVIFA i%,n)

$10,000 $10,000 = R (PVIFA 12%,5)

$10,000$10,000 = R (3.605)

RR = $10,000$10,000 / 3.605 = $2,774$2,774

End of Year

Payment Interest Principal Ending Balance

0 --- --- --- $10,000

1 $2,774 $1,200 $1,574 8,426

2 2,774 1,011 1,763 6,663

3 2,774 800 1,974 4,689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

$13,871 $3,871 $10,000

[Last Payment Slightly Higher Due to Rounding]

2. Calculate Debt Outstanding:2. Calculate Debt Outstanding: The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.

1.1. Determine Interest Expense: Determine Interest Expense: Interest expenses may reduce taxable income of the firm.