Chapter 4Chapter 4

NPV and the

Time Value of Money

Chapter OutlineChapter Outline

4.1 The Timeline4.2 Solving “Single Sum” problems 4.3 Valuing a Stream of Cash Flows4.4 The Net Present Value of a Stream of Cash Flows4.5 Annuities and Perpetuities5.2 Discount Rates and Loans

– Amortization Schedule

Learning ObjectivesLearning Objectives How to construct a cash flow timeline Calculate Present Value and Future Value of a

Single Sum Value a series of cash flows and compute the net

present value (NPV)– Uneven Cash Flows

Calculate Present Value and Future Value of multiple cash flows:– Annuities and Perpetuities

Calculate loan payments and construct loan amortization schedules

Types of Time Value ProblemsTypes of Time Value Problems

Single Sum: One dollar amount Uneven Cash Flows: A series of unequal cash

flows– Net Present Value (NPV)

Annuity: A series of equal cash flows for a specified number of periods at a constant interest rate

Perpetuity: An infinite series of equal payments

Basic TermsBasic Terms

Present Value (PV)– Value as of today Future Value (FV) – Value at some point in the

future Interest rate (r) – rate of return (FV) or discount

rate (PV)– Cost of capital– Required return

Number of Periods (n) - usually measured in months or years

Payment (C) – used for annuity problems

4.1 The Timeline4.1 The Timeline

A series of cash flows lasting several periods is defined as a stream of cash flows. We can represent a stream of cash flows on a timeline, a linear representation of the timing of the expected cash flows.

Constructing a Timeline Constructing a Timeline 

Date 0 represents the present. Date 1 is one year later

The first cash flow at date 0 (today) is represented as negative because it is an outflow

Distinguishing Cash Inflows Distinguishing Cash Inflows from Outflowsfrom Outflows

Example 4.1 Constructing a Example 4.1 Constructing a TimelineTimeline

Suppose you must pay tuition of $10,000 per year for the next four years. Your tuition payments must be made in equal installments of $5,000 each every 6 months. What is the timeline of your tuition payments?

Example 4.1 Constructing a Example 4.1 Constructing a Timeline Timeline

Assuming today is the start of the first semester, your first payment occurs at date 0 (today). The remaining payments occur at 6-month intervals. Using one-half year (6 months) as the period length, we can construct a timeline as follows:

4.2 Valuing Cash Flows at 4.2 Valuing Cash Flows at Different Points in TimeDifferent Points in Time

Three Important Rules Central to Financial Decision Making – Rule 1: Comparing and Combining Values– Rule 2: Compounding – Future Value– Rule 3: Discounting – Present Value

Rule 1: Comparing and Rule 1: Comparing and Combining ValuesCombining Values

A dollar today and a dollar in one year are not equivalent. Having money now is more valuable than having money in the future; if you have the money today you can earn interest on it.

Rule 2: CompoundingRule 2: Compounding

The process of moving forward along the timeline to determine a cash flow’s value in the future (its future value) is known as compounding.

Rule 2: CompoundingRule 2: Compounding

We can apply this rule repeatedly. Suppose we want to know how much the $1,000 is worth in two years’ time. If the interest rate for year 2 is also 10%, then:

Rule 2: CompoundingRule 2: Compounding

This effect of earning interest on both the original principal plus the accumulated interest, so that you are earning “interest on interest,” is known as compound interest.

Compounding the cash flow a third time, assuming the competitive market interest rate is fixed at 10%, we get:

$1000 (1.10) (1.10) (1.10) $1000 (1.10)3 $1331

Figure 4.1: The Compounding of Figure 4.1: The Compounding of Interest over TimeInterest over Time

Future Value of a Single Sum:Future Value of a Single Sum:Text Book FormulaText Book Formula

(1 ) (1 ) (1 ) (1 )

times

nnFV C r r r C r

n

Future Value of a Cash Flow

(Eq. 4.1)

Rule 3: DiscountingRule 3: Discounting

The process of finding the equivalent value today of a future cash flow is known as discounting.  

Present Value of a Single SumPresent Value of a Single Sum Text Book Formula Text Book Formula

PV C (1 r)n C

(1 r )n (Eq. 4.2)

Single Sum Formulas SimplifiedSingle Sum Formulas Simplified

No need to have “C” in both formulas. So we will replace it to simplify the equation and, hopefully, avoid using the wrong formula:– Future Value Formula:

FV = PV x (1 + r) n

– Present Value Formula:PV = FV / (1 + r) n

Future Value ExampleFuture Value Example

Suppose you purchased a house ten years ago for $125,000 and houses in your neighborhood have increased in value by an average of 10% per year. What should your house be worth today?

FV = PV(1 + r) n

FV = 125,000 (1.10) 10 = 324,218

Example 4.2 Example 4.2 Personal Finance Personal Finance Present Value of a Single SumPresent Value of a Single Sum

You are considering investing in a savings bond that will pay $15,000 in ten years. If the competitive market interest rate is fixed at 6% per year, what is the bond worth today?

PV = FV / (1 + r) n

PV = 15,000 / (1.06) 10 = 8,375.92

4.3 Valuing a Stream of Cash Flows4.3 Valuing a Stream of Cash Flows

Previous examples were easy to evaluate because there was one cash flow. What do we need to do if there are multiple cash flow?– Equal Cash Flows: Annuity or Perpetuity– Unequal/Uneven Cash Flows

4.3 Valuing a Stream of Cash Flows4.3 Valuing a Stream of Cash Flows

0 1 2 3

?? 1000 1000 20002000 3000 3000

Uneven cash flows are when there are different cash flow streams each year

Treat each cash flow as a Single Sum problem and add the PV amounts together.

4.3 Valuing a Stream of Cash Flows4.3 Valuing a Stream of Cash Flows

What is the present value of the preceding cash flow stream using a 12% discount rate?– PV = FV / (1 + r) n

Yr 1 $1,000 / (1.12)1 = $ 893Yr 2 $2,000 / (1.12)2 = 1,594Yr 3 $3,000 / (1.12)3 = 2,135 $4,622

4.4 The Net Present Value of 4.4 The Net Present Value of a Stream of Cash Flowsa Stream of Cash Flows

We can represent investment decisions on a timeline as negative cash flows.

The NPV of an investment opportunity is also the present value of the stream of cash flows of the opportunity:

If the NPV is positive, the benefits exceed the costs and we should make the investment.

NPV = PV(benefits) PV(costs) PV (benefits costs)

4.4 The Net Present Value of a 4.4 The Net Present Value of a Stream of Cash FlowsStream of Cash Flows

You have been offered the following investment opportunity: If you invest $1 million today, you will receive the cash flow stream below. If you could otherwise earn 12% per year on your money, should you undertake the investment opportunity?

Yr 1 $250,000Yr 2 $150,000Yr 3 $200,000Yr 4 $300,000Yr 5 $400,000Yr 6 $500,000

4.5 Annuities and Perpetuities 4.5 Annuities and Perpetuities

Annuity: A series of equal cash flows for a specified number of periods at a constant interest rate– Ordinary annuity: cash flows occur at the end

of each period– Annuity due: cash flows occur at the beginning

of each period (not covered in class)Perpetuity: An infinite series of equal

payments

Annuity: – If you buy a bond, you will receive equal

coupon interest payments over the life of the bond.

– If you borrow money to buy a house or a car, you will pay a stream of equal payments.

Amortization Schedule Perpetuity:

– If you buy preferred stock, you will receive equal dividend payments forever.

Assuming infinite life of corporation

Examples of Annuities and Examples of Annuities and PerpetuitiesPerpetuities

Annuity Formulas in TextAnnuity Formulas in Text

PV (annuity of C for N periods with interest rate r ) C 1

r1

1

(1 r)N

(Eq. 4.5)

(annuity) (1

11 (1 )

(1 )

1((1 ) 1)

N

NN

N

FV PV r)

Cr

r r

C rr

(Eq. 4.6)

Annuities Formulas - SimplifiedAnnuities Formulas - Simplified

This presentation utilizes a simplified version of the formulas that eliminates the 1/r component:

r

rCFV

rr

CPV

N

N

1)1(

)1(1

1

Interest Rates and Time PeriodsInterest Rates and Time Periods Keep in mind that payments (C) are not always

annual. Thus the interest rates (r) and number of periods (N) need to be adjusted if there is more than 1 payment per year. For example, here are interest rates and number of periods for a 5-year loan at 6% interest:

– Annual Payments: N = 5 r = 6.0%– Semi-Annual Payments: N = 10 r = 3.0%– Quarterly Payments: N = 20 r = 1.5%– Monthly Payments: N = 60 r = 0.5%

Present Value Annuity ExamplePresent Value Annuity Example You can afford to pay $632 per month towards a

new car. If you can get a loan that charges 1% interest per month and has a 40-month term, how much can you borrow?

Present Value Annuity ExamplePresent Value Annuity Example You can afford to pay $632 per month towards a

new car. If you can get a loan that charges 1% interest per month and has a 40-month term, how much can you borrow?

752,2001.

)01.1(1

1632

40

PV

Example 4.8 Example 4.8 Personal Finance Personal Finance Retirement Savings Plan AnnuityRetirement Savings Plan Annuity

Ellen is 35 years old, and she has decided it is time to plan seriously for her retirement. At the end of each year until she is 65, she will save $10,000 in a retirement account. If the account earns 10% per year, how much will Ellen have saved at age 65?

Example 4.8 Example 4.8 Personal Finance Personal Finance Retirement Savings Plan AnnuityRetirement Savings Plan Annuity

Ellen is 35 years old, and she has decided it is time to plan seriously for her retirement. At the end of each year until she is 65, she will save $10,000 in a retirement account. If the account earns 10% per year, how much will Ellen have saved at age 65?

millionFV 645.110.

1)10.1(000,10

30

ProblemsProblems1. With $10,000 to invest and assuming a 10%

interest rate, how much will you have at the end of 5 years?

2. Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?

3. Which of the following is better?a) Investing $500 at a 7% annual interest rate for 5

years.

b) Investing $500 today and receiving $750 in 5 years.

ProblemsProblems4. An investment will provide you with $100 at the

end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?

5. Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

6. You are saving for a new house, and you put $10,000 per year in an account paying 8%. How much will you have at the end of 3 years?

Present Value of a PerpetuityPresent Value of a Perpetuity

PV (C in perpetuity)

C

r(Eq. 4.4)

– PV is the value of the perpetuity today– C is the recurring payment – r is the required interest rate (not dividend rate)

If we know any two variables, we can solve for the third

Example 4.6 Example 4.6 Personal Finance Personal Finance Endowing a PerpetuityEndowing a Perpetuity

You want to endow an annual graduation party at your alma mater that is budgeted to cost $30,000 per year forever. If the university can earn 8% per year on its investments and the first party is in one year’s time, how much will you need to donate to endow the party?

Example 4.6 Example 4.6 Personal Finance Personal Finance Endowing a PerpetuityEndowing a Perpetuity

You want to endow an annual graduation party at your alma mater that is budgeted to cost $30,000 per year forever. If the university can earn 8% per year on its investments and the first party is in one year’s time, how much will you need to donate to endow the party?

PV C / r $30, 000/0.08 $375,000 today

Perpetuity Example Perpetuity Example Preferred StockPreferred Stock

There is a preferred stock with a par value of $25 and a dividend rate of 5% per year. If preferred stock with a similar risk profile is yielding 4%, what price should the stock be trading at?

Loan Payments and Loan Payments and Amortization SchedulesAmortization Schedules

C P

1

r1

1

(1 r)N

(Eq. 4.8)

One of the most useful applications of the annuity formula is calculating the payment on a loan. To do so, we revise the annuity formula to solve for Payment (C):

Example 4.10Example 4.10Computing a Loan PaymentComputing a Loan Payment

Your firm plans to buy a warehouse for $100,000. The bank offers you a 30-year loan with equal annual payments and an interest rate of 8% per year. The bank requires that your firm pay 20% of the purchase price as a down payment, so you can borrow only $80,000. What is the annual loan payment?

Example 4.10Example 4.10Computing a Loan PaymentComputing a Loan Payment

Eq. 4.8 calculates the payment as follows:

C P

1

r1

1

(1 r)N

80, 000

1

0.081

1

(1.08)30

$7106.19

Loan Payments and Loan Payments and Amortization SchedulesAmortization Schedules

In Example 4.10, our firm will need to pay $7,106.19 each year to fully repay the loan over 30 years.

To understand how the loan will be repaid, we must create an Amortization Schedule for the loan.

Amortization ScheduleAmortization Schedule An Amortization Schedule contains five

columns, as illustrated below:

The payment is typically monthly, quarterly, or annually, so we may need to adjust the interest rate and number of periods accordingly.

Pmt # Payment

Interest Principal Balance

$80,000

1 $7,106 $6,400 $706 $79,294

Amortization Schedule:Amortization Schedule:Equal PaymentsEqual Payments

Payment – Stays the Same

Interest Expense– Declines Over Time

Principal Repayment – Increases Over Time

Balance of Loan – Fully repaid by maturity

Amortization Schedule:Amortization Schedule:Equal PaymentsEqual Payments

Example 4.10:– Purchase Warehouse for $100,000– $20,000 Down Payment; $80,000 Loan– 30-year Term – 8% Interest Rate– Annual End of Year Payments

Amortization ScheduleAmortization Schedule

1. Beginning Balance is Loan Amount

2. Calculate Payment:

Pmt # Payment Interest Principal Balance

$80,000

1 $7,106

C P

1

r1

1

(1 r)N

80, 000

1

0.081

1

(1.08)30

$7106.19

Amortization ScheduleAmortization Schedule

3. Calculate Interest Principal x Rate x Time: ($80,000)(8%)(1 yr)

4. Principal = Payment - Interest

5. Balance = Prior Balance - Principal

Pmt # Payment Interest Principal Balance

$80,000

1 $7,106 $6,400 $706 $79,294

Amortization ScheduleAmortization ScheduleLoan

Payment Interest Principal BalanceBeginning Balance 80,000

1 7,106 6,400 706 79,294 2 7,106 6,344 763 78,531 3 7,106 6,282 824 77,707 4 7,106 6,217 890 76,818 5 7,106 6,145 961 75,857 6 7,106 6,069 1,038 74,819 7 7,106 5,986 1,121 73,699 8 7,106 5,896 1,210 72,489 9 7,106 5,799 1,307 71,181 10 7,106 5,695 1,412 69,770 11 7,106 5,582 1,525 68,245 12 7,106 5,460 1,647 66,599 13 7,106 5,328 1,778 64,820 14 7,106 5,186 1,921 62,900 15 7,106 5,032 2,074 60,825 16 7,106 4,866 2,240 58,585 17 7,106 4,687 2,419 56,166 18 7,106 4,493 2,613 53,553 19 7,106 4,284 2,822 50,731 20 7,106 4,058 3,048 47,683 21 7,106 3,815 3,292 44,392 22 7,106 3,551 3,555 40,837 23 7,106 3,267 3,839 36,998 24 7,106 2,960 4,146 32,851 25 7,106 2,628 4,478 28,373 26 7,106 2,270 4,836 23,537 27 7,106 1,883 5,223 18,314 28 7,106 1,465 5,641 12,673 29 7,106 1,014 6,092 6,580 30 7,106 526 6,580 0

213,186 133,186 80,000

Amortization Example:Amortization Example:Equal PaymentsEqual Payments

Create an amortization schedule for the first month based on the following terms:– Purchase Building for $1,000,000– $200,000 Down Payment, Balance Borrowed– 5 yr. Term – 9% Interest Rate– Monthly Payments

Calculating Payment in ExcelCalculating Payment in Excel

Excel can easily calculate the payment for a loan:

=PMT (interest rate, number of payments, loan amount)

Interest Rate 0.75% No. of Payments 60 Loan Amount 800,000

PMT (16,606.68)