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(Time Value of Money)

Topic Coverage:

The Interest Rate

Simple Interest Rate

Compound Interest Rate

Amortizing a Loan

Compounding Interest More Than Once per Year

The Time Value of Money

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

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

Why is TIME such an important element in your decision?

TIME allows you the opportunity to postpone consumption and earn INTEREST.

Principles Used in this Chapter:

• The Time Value of Money – A Dollar Received Today Is Worth More Than a Dollar Received in The Future.

• Point out the importance of interest rates, which will serve a variety of functions, including discounting/compounding rates and representing opportunity costs.

Interest rate is viewed as compensation for bearing risk.

Types of Interest:

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

Compound Interest - Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

When interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum.

Simple Interest Formula:

Formula SI = P0 × I × n

SI : Simple Interest

P0 : Deposit today (t=0)

I : Interest Rate per Period

n : Number of Time Periods

Simple Interest Formula:

• 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?

SI = P0 × I × n = $1,000 × 0.07 × 2 = $140

Future Value (FV):

• What is the Future Value (FV) of the deposit?

FV = P0 + SI = $1,000 + $140

= $1,140

• Future 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.

SI = P0 × I × n

Present Value (PV):

• What is the Present Value (PV) of the previous problem?

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

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

Why Compound Interest?

0

5000

10000

15000

20000

1st Year 10th

Year

20th

Year

30th

Year

Future Value of a Single $1,000 Deposit

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Future Values of $100 with Compounding

0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Years

FV

of

$100

0%

5%

10%

15%

Future Value Single Deposit (Graphic)

• Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years.

0 1 2

$1,000

FV2

7%

Future Value Single Deposit (Formula)

FV1 = P0 (1+i)1

= $1,000 (1.07)

= $1,070

Compound Interest

You 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.

Future Value Single Deposit (Formula)

FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070

FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07)

= P0 (1+i)2 = $1,000(1.07)2 = $1,144.90

You earned an EXTRA $4.90 in Year 2 with compound over simple interest.

General Future Value Formula

FV1 = P0(1+i)1

FV2 = P0(1+i)2

General Future Value Formula:

FVn = P0 (1+i)n

or FVn = P0 (FVIFi,n) -- See Table I

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.166

3 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

General Future Value Formula

FV2 = $1,000 (FVIF7%,2)

= $1,000 (1.145) = $1,145 [Due to Rounding]

• Using Future Value Tables -- See Table I

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.166

3 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

General Future Value Formula (Example)

• Mr. A wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years.

0 1 2 3 4 5

$10,000 FV5

10%

General Future Value Formula (Example)

Calculation based on general formula:

FVn = P0 (1+i)n

FV5 = $10,000 (1+ 0.10)5

= $16,105.10

Calculation based on Table I:

FV5 = $10,000 (FVIF10%, 5) = $10,000 (1.611)

= $16,110 [Due to Rounding]

Double Your Money!!!

• Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

• Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

Approx. Years to Double = 72 / i %

= 72/12

= 6

[Actual Time is 6.12 Years]

We will use the “Rule-of-72”.

Present Value Single Deposit (Graphic)

• Assume that you need $1,000 in 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 2 7%

PV1 PV0

$1,000

7%

Present Value Single Deposit

PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2

= FV2 / (1+i)2 = $873.44

General Present Value Formula

PV0 = FV1 / (1+i)1

PV0 = FV2 / (1+i)2

General Present Value Formula:

PV0 = FVn / (1+i)n

or PV0 = FVn (PVIFi,n) -- See Table II

Using Present Value Tables

PV2 = $1,000 (PVIF7%,2)

= $1,000 (.873) = $873 [Due to Rounding]

• Please See Table II

Period 6% 7% 8%

1 .943 .935 .926

2 .890 .873 .857

3 .840 .816 .794

4 .792 .763 .735

5 .747 .713 .681

The Power of High Discount Rates

Periods

0 2 4 6 8 10 12 14 16 18 20 22 24

0.5

0.75

1.00

0.25

10%

5%

15%

20%

0%

General Present Value Formula (Example)

• Mr. A wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%.

0 1 2 3 4 5

$10,000 PV0

10%

General Present Value Formula (Example)

Calculation based on general formula:

PV0 = FVn / (1+i)n

PV0 = $10,000 / (1+ 0.10)5 = $6,209.21

Calculation based on Table I: PV0 = $10,000 (PVIF10%, 5) = $10,000 (.621) = $6,210.00 [Due to Rounding]

Types of Annuities:

• An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

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

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

Examples of Annuities

Student Loan Payments

Car Loan Payments

Insurance Premiums

Mortgage Payments

Retirement Savings

Ordinary Annuity:

An annuity is a series of equal dollar payments that are made at the end of equidistant points in time, such as monthly, quarterly, or annually. If payments are made at the end of each period, the annuity is referred to as ordinary annuity.

Ordinary Annuity:

0 1 2 3

$100 $100 $100

End of Period 1

End of Period 2

Today Equal Cash Flows Each 1 Period Apart

End of Period 3

Overview of an Ordinary Annuity--FVA

Cash flows occur at the end of the period

FVAn = PMT(1+i)n-1 + PMT(1+i)n-2 + ... + PMT(1+i)1 + PMT(1+i)0

PMT PMT PMT

0 1 2 n n+1

FVAn

PMT = Periodic Cash Flow

i% . . . . .

The Future Value of an Ordinary Annuity:

• FVn = FV of annuity at the end of nth period.

• PMT = annuity payment deposited or received at the end of each period

• i = interest rate per period

• n= number of periods for which annuity will last

Example of an Ordinary Annuity--FVA Cash flows occur at the end of the period

FVA3 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0

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

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

0 1 2 3 4

$3,215 = FVA3

7%

$1,070

$1,145

Valuation Using Table III:

FVAn = (FVIFAi%,n)

FVA3 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) = $3,215

• Please See Table III

Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.246

4 4.375 4.440 4.5065 5.637 5.751 5.867

Overview of an Ordinary Annuity -- PVA Cash flows occur at the end of the period

PVAn = PMT/(1+i)1 + PMT/(1+i)2 + ... + PMT/(1+i)n

PMT PMT PMT

0 1 2 n n+1

PVAn

PMT = Periodic Cash Flow

i% . . .

The Present Value of an Ordinary Annuity:

• PVn = PV of annuity at the end of nth period.

• PMT = annuity payment deposited or received at the end of each period

• i = interest rate per period

• n= number of periods

Example of an Ordinary Annuity -- PVA Cash flows occur at the end of the period

PVA3 = $1,000/(1.07)1 +$1,000/(1.07)2 + $1,000/(1.07)3

= $934.58 + $873.44 + $816.30

= $2,624.32

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

0 1 2 3 4

$2,624.32 = PVA3

7%

$934.58 $873.44 $816.30

Valuation Using Table IV:

PVAn = PMT (PVIFAi%,n)

PVA3 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624

• Please See Table IV

Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.577

4 3.465 3.387 3.3125 4.212 4.100 3.993

Annuity Due:

Annuity due is an annuity in which all the cash flows occur at the beginning of each period. For example, rent payments on apartments are typically annuities due because the payment for the month’s rent occurs at the beginning of the month.

Annuity Due:

0 1 2 3

$100 $100 $100

Beginning of Period 1

Beginning of Period 2

Today Equal Cash Flows Each 1 Period Apart

Beginning of Period 3

Overview of an Annuity Due -- FVAD Cash flows occur at the beginning of the period

FVADn = PMT(1+i)n + PMT(1+i)n-1 + ... + PMT(1+i)2 + PMT(1+i)1

= FVAn (1+i)

PMT PMT PMT PMT PMT

0 1 2 3 n-1 n

FVADn

i% . . .

Annuity Due: Future Value

Computation of future value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity.

Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period

FVAD3 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1

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

= $3,440

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

0 1 2 3 4

$3,440 = FVAD3

7%

$1,225

$1,145

Valuation Using Table III:

FVADn = R (FVIFAi%,n)(1+i)

FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440

• Please See Table III

Period 6% 7% 8%

1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.246

4 4.375 4.440 4.5065 5.637 5.751 5.867

Overview of an Annuity Due -- PVAD Cash flows occur at the beginning of the period

PVADn = PMT/(1+i)0 + PMT/(1+i)1 + ... + PMT/(1+i)n-1

= PVAn (1+i)

PMT PMT PMT PMT

0 1 2 n-1 n

PVADn

PMT: Periodic Cash Flow

i% . . .

Annuity Due: Present Value

Since with annuity due, each cash flow is received one year earlier, its present value will be discounted back for one less period.

Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period

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

= $2,808.02

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

0 1 2 3 4

$2,808.02 = PVADn

7%

$934.58 $873.44

Valuation Using Table IV:

PVADn = PMT (PVIFAi%,n)(1+i)

PVAD3 = $1,000 (PVIFA7%,3)(1.07)

= $1,000 (2.624)(1.07) = $2,808

• Please See Table IV

Period 6% 7% 8%

1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.577

4 3.465 3.387 3.3125 4.212 4.100 3.993

Steps to Solve Time Value of Money Problems

1. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a Present Value (PV) or Future

Value (FV) problem 5. Determine if solution involves a single cash flow,

annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional)

Mixed Flows Example:

Mr. A will receive the set of cash flows below. What is the Present Value at a discount rate of 10%.

0 1 2 3 4 5

$600 $600 $400 $400 $100 PV0

10%

How to Solve Mixed Flows Question?

1. Solve a “piece-at-a-time” by discounting each piece back to t=0.

2. Solve a “group-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow groups. Then discount each group back to t=0.

1. The method of “piece-at-a-time” :

0 1 2 3 4 5 10%

$545.45 $495.87 $300.53 $273.21 $ 62.09

$1677.15 = PV0 of the Mixed Flow

$600 $600 $400 $400 $100

2. The method of “group-at-a-time” :

0 1 2 3 4 5

$600 $600 $400 $400 $100

10%

$1,041.60 $ 573.57 $ 62.10

$1,677.27 = PV0 of Mixed Flow [Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57

$100 (PVIF10%,5) = $100 (0.621) = $62.10

2. The method of “group-at-a-time” : (Optional)

0 1 2 3 4

$400 $400 $400 $400

PV0 equals $1,677.30.

0 1 2

$200 $200

0 1 2 3 4 5

$100

$1,268.00

$347.20

$62.10

Plus

Plus

Frequency of Compounding:

General Formula:

FVn = PV0(1 + [i/m])mn

n: Number of Years

m: Compounding Periods per Year

i: Annual Interest Rate

FVn,m: Future Value at the end of Year n

PV0: Present Value of the Cash Flow today

Impact of Frequency:

Mr. A has $1,000 to invest for 2 Years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2)

= 1,254.40

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

= 1,262.48

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

= 1,266.77

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

= 1,269.73

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

= 1,271.20

Annual Percentage Rate (APR):

The annual percentage rate (APR) indicates the interest rate paid or earned in one year without compounding. APR is also known as the nominal or quoted (stated) interest rate.

We cannot compare two loans based on APR if they do not have the same compounding period.

To make them comparable, we calculate their equivalent rate using an annual compounding period. We do this by calculating the effective annual rate (EAR)

Effective Annual Interest Rate (EAR):

Effective Annual Interest Rate (EAR) is the actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year.

ABC Company has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)?

EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%

EAR= (1 + [ i / m ] )m - 1

Perpetuities

A perpetuity is an annuity that continues forever or has no maturity. For example, a dividend stream on a share of preferred stock. There are two basic types of perpetuities:

– Growing perpetuity in which cash flows grow at a constant rate from period to period over time.

– Level perpetuity in which the payments are constant over time.

Perpetuities:

• Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

• PV of Perpetuity Formula

where;

PV = present value of the perpetuity

PMT = periodic cash payment (constant dollar amount provided by the of perpetuity)

i = interest rate (annuity interest or discount rate)

PMTPV

i

Example - Perpetuity

• You want to create an endowment to fund a scholarship, which pays $15,000 per year, forever, how much money must be set aside today if the rate of interest is 5%?

15,000

300,0000.05

PMTPV

i

Cash flows that never end are known as perpetuities. • Suppose you plan to invest in a utility stock that will pay a $2

dividend for the life of the company. You don’t expect the dividend to ever grow, and similar stocks have an 8% required rate of return. How much should the stock be worth today?

t = 0 t = 1 t = 2

A2 = $2 A3 = $2

r = 8% r = 8%

Present Value of a Growing Perpetuity

• Suppose you plan to invest in a different utility stock that will pay a $2 dividend for the life of the company. You expect the dividend to grow (g) by 2% per year, and similar stocks have an 8% required rate of return. How much should the stock be worth today?

t = 1 t = 2 t = 3

A2 = $2

A3 = $2(1.02)

t = 0

A3 = $2(1.02)(1.02)

r = 8% g = 2%

r = 8% g = 2%

r = 8% g = 2%

r = 8% g = 2%

Amortizing a Loan:

CF2 = ? CF1 = ? CF3 = ? CF4 = ? CF360 = ?

r = 1/3% r = 1/3% r = 1/3% r = 1/3% r = 1/3%

PV0 = $688,000

t = 1 t = 2 t = 3 t = 0 t = 4 t = 360

Steps to Amortizing a Loan:

1. Calculate the payment per period.

2. Determine the interest in Period t. (Loan Balance at t-1) x (i% / m)

3. Compute principal payment in Period t.

(Payment - Interest from Step 2)

4. Determine ending balance in Period t.

(Balance - principal payment from Step 3)

5. Start again at Step 2 and repeat.

Table of Amortizing a Loan:

End of Year (t)

Beginning Balance

Payment (1)

Interest (2)

Principal (3)

Ending Balance (4)

0 (Loan Amount) - - - (Loan Amount)

1 = (4) at t-1 PMT = (4) at t-1 × i = (1) – (2) = (4) at t-1 - (3)

2 = (4) at t-1 PMT = (4) at t-1 × i = (1) – (2) = (4) at t-1 - (3)

3 = (4) at t-1 PMT = (4) at t-1 × i = (1) – (2) = (4) at t-1 - (3)

4 = (4) at t-1 PMT = (4) at t-1 × i = (1) – (2) = (4) at t-1 - (3)

5 = (4) at t-1 PMT = (4) at t-1 × i = (1) – (2) 0

Total

Amortizing a Loan Example:

Mr. A is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years.

Step 1: Payment

PV0 = PMT (PVIFA i%,n)

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

$10,000 = PMT (3.605)

PMT = $10,000 / 3.605

= $2,774

Amortizing a Loan Example:

End of Year (t)

Beginning Balance

Payment (1)

Interest (2)

Principal (3)

Ending Balance (4)

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

1 $10,000 $2,774 $1,200 $1,574 $8,426

2 $8,426 $2,774 $1,011 $1,763 $6,663

3 $6,663 $2,774 $800 $1,974 $4,689

4 $4,689 $2,774 $563 $2,211 $2,478

5 $2,478 $2,774 $297 $2,478 0

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

Usefulness of Amortization

1. Determine Interest Expense -- Interest expenses may reduce taxable income of the firm.

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

Basic Principles of Time Value of Money

• You cannot add or subtract cash flows that occur at different times without first compounding or discounting them to the same point in time. – Once at the same point in time, we can add or subtract

the resulting equivalency cash flows. – This is known as the cash flow additivity principle:

• The period of time associated with the cash flows must match the period of time associated with the discounting or compounding rate (use periodic rates for compounding and discounting).

• Pay close attention to the “when” in time for which you need an answer.

Basic Principles of Time Value of Money • Organizing TVM problems with cash flow diagrams helps us visualize

and solve complex problems. These are also known as time lines.

• Cash flow diagrams depict

– The timing of each cash flow, its amount, and its sign

– The rate at which cash flows will be compounded/be discounted/grow.

– The value for which we are attempting to solve.

CF2 = CF1 = CF3 = CF4 = CF5 =

FV5 =

r = r = r = r = r =

PV0 =

t = 1 t = 2 t = 3

t = 0

t = 4 t = 5