Chapter 61674

Corporate Finance

FINA1003, Fall 2009

Xiaohui Gao

Faculty of Business and Economics

University of Hong Kong

Corporate Finance, Xiaohui Gao1

2

Chapter 6Discounted Cash Flow Valuation

Corporate Finance, Xiaohui Gao

Time Value of Money(summary of single cash flow)

Time value of money refers to A dollar today is better than a dollar in

the future To compare money flows, we must

convert them to the same time point. Basic formula:

where: PV = value of original investment FV = future value of investment n = number of periods r = interest rate per period

3

nrPVFV )1(


Present Value and Future Value of

a Stream of Cash Flows The present value or future value of

a stream of cash flows is the sum of the PV or FV of each cash flow using given interest rate.

For example: uneven cash flow stream with r = 10%

4 Corporate Finance, Xiaohui Gao

0 1 2 3

$100 $200 $300 $400

PV=? FV=?

Multiple Cash Flows – FV Suppose you invest $500 in a mutual fund

today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?

FV = 500(1.09)2 + 600(1.09) = 1248.05

How much will you have in 5 years if you make no further deposits?

First way: FV = 500(1.09)5 + 600(1.09)4 = 1616.26

Second way – use value at year 2:

FV = 1248.05(1.09)3 = 1616.265 Corporate Finance, Xiaohui Gao

PV of Multiple Cash Flows-cash up front v.s. an installment plan

Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (or the discount rate) is 8%, which method do you prefer?

6

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV


PV of Multiple Cash Flows To calculate the PV for a stream of cash

flows, we first calculate the PV for each cash flow, then add them together.

If we want to find the FV at time t for the same stream of cash flows, we can use the basic formula after finding FV.

7

....22

11

)1()1(0 r

C

r

CCPV

trPVFV )1(


Using Timeline – interest rate 12%

8

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1432.93 FV = 2254.74


Multiple Cash Flows Using the Calculator

Using N, I/Y, PV, and FV for each cash flow, then add up

Another way is to use the cash flow keys Press CF and enter the cash flows beginning with year

0. You have to press the “Enter” key for each cash flow Use the down arrow key to move to the next cash flow The “F” is the number of times a given cash flow

occurs in consecutive periods Use the NPV key to compute the present value by

entering the interest rate for I, pressing the down arrow and then compute

Clear the cash flow keys by pressing CF and then CLR Work


Annuities and Perpetuities

Annuity A stream of equal (level) cash payments received

at regular intervals for a specified number of periods.

Perpetuity A stream of equal (level) cash payments received

at regular intervals forever.


Annuity

Annuity - - a stream of equal (level) payments received at regular intervals for a specified number of periods.

Example of an annuity - 48 equal monthly payments on a car loan.

Example of an annuity - 20 equal annual payments from winning the Lottery.


PV of Annuity Formula

PV of Annuity Formula

C = cash payment r = interest rate t = number of years cash payment is

received13

1

11 1(1 )

1

t

t

r

r r rPV C or PV C

r


Annuity – Sweepstakes Example

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?

PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29

Using financial calculator: N = 30; I/Y = 5; PMT = 333,333.33; FV = 0; PV = -5,124,150.29


Present Value of an Annuity

What is the present value of a 10 year annuity of $1,000 a year at a discount rate of 9%?

Use the formula:

Use the financial calculator: PMT = 1,000, n = 10, i = 9% : PV = - 6417.66

Use spreadsheet

15

66.417,6$*1000$ ][ 10)09.1(09.1

09.1

PV


Spreadsheet Calculation

16

Interest Rate 9%Discount

Year Cashflow Factor PV1 1,000.00$ 0.9174 917.43$ 2 1,000.00$ 0.8417 841.68$ 3 1,000.00$ 0.7722 772.18$ 4 1,000.00$ 0.7084 708.43$ 5 1,000.00$ 0.6499 649.93$ 6 1,000.00$ 0.5963 596.27$ 7 1,000.00$ 0.5470 547.03$ 8 1,000.00$ 0.5019 501.87$ 9 1,000.00$ 0.4604 460.43$ 10 1,000.00$ 0.4224 422.41$

PV Annuity Factor 6.4177 6,417.66$ Another way: PV=PV(rate, n, PMT)


Annuity--Value of Payments

You want to buy a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

17

PV

PV

4 000

947 41

110

110 1 10 3,

$9, .

. . ( . )


Buying a House

You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?


Buying a House - Continued

Bank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment = .28(3,000) = 840 PV = 840[1 – 1/1.005360] / .005 = 140,105

Total Price

Closing costs = .04(140,105) = 5,604 Down payment = 20,000 – 5,604 = 14,396 Total Price = 140,105 + 14,396 = 154,501


Annuity-- Calculation of periodic payments

You purchase a house that costs $125,000. You put up $25,000 of your personal wealth and borrow the remaining $100,000 from a mortgage lender. You repay the loan of $100,000 in 360 monthly installments starting a month from today. What is the value of each monthly installment if the interest rate per month is 1%?


Solution This is an example of an annuity, whose PV is $100,000 You need to solve for the level cash payment C Using the formula for PV of an annuity:

or 100,000 = C*97.218. Hence C = $1028.61

This type of a loan in which the monthly payment is fixed over the life of the mortgage is called an amortizing loan.

Using your financial calculator to confirm the answer.

21

36001.1

11*

01.0

1*000,100 C


Note on Annuities Due The annuities we have discussed thus far are

called Ordinary Annuities because the first payment is received one period in the future.

If the first payment is received immediately, the annuity is called an Annuity Due.

This distinction is unnecessary from a valuation standpoint since every n period Annuity Due is equivalent to the sum of the payment amount and an n-1 period Ordinary Annuity.


Annuity Value and Annuity Due Value

Annuity Value. Your landscaping company can lease a truck for $8,000 a year (paid at year end) for 6 years. It can instead buy the truck for $40,000. The truck will be valueless after 6 years. If the interest rate your company can earn on its funds is 7%, is it cheaper to buy or to lease? PV(annuity) = 38,132.3 < 40,000

Annuity Due Value. Reconsider the previous problem. What if the lease payments are an annuity due, so that the first payment comes immediately? Is it cheaper to buy or to lease?

PV(Annuity Due) = 32,805 + 8,000 = 40,805 > 40,000


Future Value of an Annuity The future value of an annuity can be calculated by

computing the Present Value of an annuity and multiplying it by the Future Value Interest Factor.

24

r

rr

rrr

tt

t

1)1(1

)1(

11=

Factor Annuity Value Future

tt

rrrr

1

)1(

11C=Annuity of Value Future


Future Value of an Annuity

What is the future value of a 10 year annuity of $1,000 a year at an interest rate of 9%?

Using financial calculator -- PMT = 1,000, n = 10, i = 9% : FV =15,192.9297

25

1929.1509.

1)09.1(= FVAF

10

9.192,15$1929.15*$1,000= FVAF*CFV


Future Value of an Annuity Retirement planning makes

extensive use of this methodology about the future value of an annuity.

ExampleYou plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

26

FV

FV

4 000 1 10

100

110

110 1 10

2020, ( . )

$229,

. . ( . )


Another Example You need to accumulate $1,000,000 by

the time you retire 30 years later. You decide to set aside a fixed amount each year for the next 30 years, starting one year from today, to pay for your retirement. Assuming an annual interest rate of 10%, what amount should you set aside each year?

Answer: C = $6079.25 Using both the formula and your financial

calculator to confirm the answer.


Retirement Planning – PV and FV together

A couple will retire in 50 years; they plan to spend about $30,000 a year in retirement, which should last about 25 years. They believe that they can earn 8 percent interest on retirement savings.

a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year.

b. How would the answer to part (a) change if the couple also realizes that in 50 years, right before they start the retirement, they would spend $60,000 on a beach vacation house?


Perpetuity The value of a perpetuity is the amount you

would be willing to pay for the right to receive a constant payment forever.

If I invest $100 in a perpetuity that pays 10%, I should receive $10 each year forever. $100 x .10 = $10

29

Years

Cashflows

0 1 2 3 4

- $100 $10 $10 $10 $10


Perpetuity

Therefore the formula for the present value of a perpetuity is:

Or

C = cash payment r = interest rate

30

PV Cr

Rate Discount Required

Payment CashPerpetuity of ValuePresent


Valuing Perpetuities

Example Suppose you wish to endow a chair at your old

university. The aim is to provide $100,000 forever and the interest rate is 10%.

A donation of $1,000,000 will provide an annual income of .10 x $1,000,000=$100,000 forever.

Another example: preferred stock.

31

PV 100 00010 000 000,. $1, ,


Delayed Perpetuity A perpetuity might make its first payment

after t+1 years. This is called a delayed perpetuity.

How would you value this? Suppose you are now at the end of year t

Now, it is like a normal perpetuity So, what is the PV today?

Hence,

32

rCPVt

1*

tt rPVPV )1/(

1 1* *

(1 )delayed perpetuity tPV C

r r


Delayed Perpetuity - Example

Again, you wish to endow a chair at your old university. But now the first perpetuity payment will not be received until four years from today, how much money needs to be set aside today? (Note: t + 1 = 4 means t = 3.)

33

315,751$3)10.1(

000,000,1

PV


Growing Annuities and Perpetuities

The value of a growing annuity is

The value of a growing perpetuity is


111

Growing Annuity Present Value = C , is the growth rate

tgr

gr g

1Growing Perpetuity Present Value = C , is the growth rateg

r g

Table 6.2: summary of annuity and perpetuity


Perpetuity - Example

A local bank advertises the following deal: “Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever.” Is this a good deal if the interest rate available on other deposits is 8 percent?

Answer: NO!! At I = 8% At year 10, the value of a 10-yr $100 annuity is 1,448 The value of a perpetuity is 1,250 < 1,448!


Periodic Interest Rates Interest rates can be quoted for days, months,

quarters, or any convenient period other than years. These rates are often multiplied by the number of periods in one year to get the Annual Percentage Rate (APR).

Example A monthly interest rate 1.5% is referred to as

an 18% APR because 1.5%*12 = 18%.

If APR is 20% and the interest is compounded quarterly, then the interest rate per period (one quarter here) is 5%.


Annual Percentage Rate (APR)

Alternatively, APR is the interest rate that is annualized using simple interest.

If APR is 18% and interest is compounded monthly, what is the actual equivalent annually compounded rate?

When compounded, we get the following result:

(1+.18/12)12 = 1.01512 = 1.1956 or 19.56%.

Therefore this 18% APR is equivalent to an annual

rate of 19.56%. 38 Corporate Finance, Xiaohui Gao

Effective Annual Rates Effective Annual Rate (EAR) is the

interest rate that is annualized using compound interest. In other words, EAR is the interest rate expressed as if it were compounded once per year.

Example: You invest $100 in a bank that pays a monthly

rate of 1%. It will grow to $100(1+.01)12 = $112.68 by the end of the year. So, the effect annual interest rate or EAR is 12.68%.

APR is simply 12*1% = 12%.


EAR & APR When comparing interest rates, it is important to be

consistent with the periods and the compounding. Most institutions report APRs (like banks). Never compare APRs. Compare EARs. The relationship between EAR and APR is as

follows:

where n is the number of compounding periods per year with monthly compounding, n = 12 with quarterly compounding, n = 4 and so on…

40

1)1( n

n

APREAR


Decisions, Decisions

You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?

First account: EAR = (1 + .0525/365)365 – 1 = 5.39%

Second account: EAR = (1 + .053/2)2 – 1 = 5.37%

Which account should you choose and why?


Things to Remember

You ALWAYS need to make sure that the interest rate and the time period match. If you are looking at annual periods, you

need an annual rate. If you are looking at monthly periods,

you need a monthly rate.

If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly.


Example Al’s Autos has a 2.4% APR 48 monthly

payment financing option on new $20,000 Corolla sedan. What is the monthly payment?

Solution: 2.4% APR is equivalent to 0.2% monthly interest rate. Hence, r = 0.2%, n = 48, PV = $20,000. Using the formula for PV of an annuity:

20,000 = C*45.72. Hence C = $437.4028

What is the EAR of this car loan? (2.43%)

43

48002.1

11*

002.0

1*000,20 C


Future Values with Monthly Compounding

Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?

Monthly rate = .09 / 12 = .0075 Number of months = 35(12) = 420 FV = 50[1.0075420 – 1] / .0075 =

147,089.2237


Continuous Compounding

Sometimes investments or loans are figured based on continuous compounding

EAR = eq – 1 The e is a special function on the calculator

normally denoted by ex. e is the number 2.71828. The formula can be derived from taking the limit of EAR/APR formula.

Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 = .0725 or 7.25%


Example

Savings and loan associations (S&Ls) were restricted to pay at most an APR of 5.5%. What is the rate that S&Ls are effectively paying their depositors?


0.055 1 5.65406%EAR e

Loan Types Pure Discount Loans: the borrower receives

money today and repays a single lump sum at some time in the future. (Example: treasury bills)

Interest-Only Loans: the borrower pays interest each period and repay the entire principal at some point in the future. (Example: corporate bonds)

Amortized Loans: the borrower pays off the loan by making regular payment.

A fixed amount each period A fixed amount for principal reduction plus

interest for that period47 Corporate Finance, Xiaohui Gao

Amortized Loan with Fixed Principal Payment - Example

Consider a $5,000, 5 year loan at 9% interest. The loan agreement requires the firm to pay $1,000 in principal each year plus interest for that year.

Since the principal is decreasing every year, the interest amount paid every year is decreasing as well.


Amortized Loan with Fixed Principal Payment - Example


Year Beginning-of-year Balance

Year-end Interest Due on Balance

Principal Paid

Total Payment

End-of-Year Balance

1 $5,000 $450 $1,000 $1,450 $4,000

2 $4,000 $360 $1,000 $1,360 $3,000

3 $3,000 $270 $1,000 $1,270 $2,000

4 $2,000 $180 $1,000 $1,180 $1,000

5 $1,000 $90 $1,000 $1,090 $0

Consider a $5,000, 5 year loan at 9% interest. The loan agreement requires the firm to pay $1,000 in principal each year plus interest for that year.

Amortized Loan with Fixed Payment - Example

Each payment covers the interest expense plus reduces principal

Consider a 4 year loan with annual payments. The interest rate is 10% and the principal amount is $1,000.

What is the annual payment? N = 4 I/Y = 10 PV = 1,000 CPT PMT = -315.47


Amortizing “ Amortizing” means that part of the monthly

payment is used to pay interest on the loan and part is used to reduce the amount of the loan. ($1000, 10%, 4 years)

51

Year Beginning-of-year Balance

Year-end Interest Due on Balance

Year-end Payment

Amortization of Loan

End-of-Year Balance

1 $1,000.00 $100.00 $315.47 $215.47 $784.53

2 $784.53 $78.45 $315.47 $237.02 $547.51

3 $547.51 $54.75 $315.47 $260.72 $286.79

4 $286.79 $28.68 $315.47 $286.79 $0


Partial Amortization

You take out a 30-year $100,000 mortgage loan with an APR of 8 percent and monthly payment. In 12 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?

Answer: monthly payment is $733.76 and you have to pay $83,861.87 to pay off the mortgage.

$733.76*144 = $105,661.44 > $100,000


Chapter 61674

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