discounted cash flow valuation chapter 5 summer 2008

Discounted Cash Flow Valuation

Chapter 5

Summer 2008

Summer 2008 Yunling Chen 2

Objectives

Be able to compute the future value of multiple cash flows

Be able to compute the present value of multiple cash flows

Be able to compute loan payments Be able to find the interest rate on a loan Understand how loans are amortized or paid off Understand how interest rates are quoted


S in g le C a sh F lowC h a p te r 4

F V P V

D iffe re n tC a sh F lo w s

P e rpe tuities A n n u ities

E q u a l C a shF lo w s

A P R E A R

C o m p a ringR a tes

P u reD isc o u nt

In te re stO n ly

A m o rtiz edL o a n s

T y p e s o fL o a n s

M u ltip leC a sh F lo w s

D C FV a lu a tion



Multiple Cash Flows

An asset or a project typically embeds multiple cash flows. Since money has time value, the value of such an asset or project is not simply the sum of individual cash flows.

We need to “transform” all cash flows to one point in time before we can meaningfully add them together.


Example 1 – Future Value

You plan to deposit $4000 at the end of each next three years in a bank account paying 8 percent interest. You currently have $7000 in the account.

How much will you have in three years? In four years?


Example 1 - Key

0 2 3 41

$7000 $4000 $4000 $4000

Find the value at year 3 of each cash flow and add them together.

Today (year 0): FV = 7000(1.08)3 = 8,817.98 Year 1: FV = 4,000(1.08)2 = 4,665.60 Year 2: FV = 4,000(1.08) = 4,320 Year 3: value = 4,000 Total value in 3 years = 8817.98 + 4665.60 + 4320 +

4000 = 21,803.58 Value at year 4 = 21,803.58(1.08) = 23,547.87


Example 2 – Future Value

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 = 1616.26


Quick Check Suppose you plan to deposit $100 into an account in

one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?

FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97


Example 3 – Present Value

You are offered an investment that will pay you $200 in (the end) one year, $400 the next year, $600 the next year, and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?


Example 3 - key Find the PV of each cash flow and add them

Year 1 CF: 200 / (1.12)1 = 178.57 Year 2 CF: 400 / (1.12)2 = 318.88 Year 3 CF: 600 / (1.12)3 = 427.07 Year 4 CF: 800 / (1.12)4 = 508.41 Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93


Example 4 – Present Value

You are considering an investment that will pay you $1,000 in year 1, $2,000 in year 2 and $3,000 in year 3. If you want to earn 10% on your money, how much is the investment worth today?

PV = 1000 / (1.1)1 = 909.09 PV = 2000 / (1.1)2 = 1652.89 PV = 3000 / (1.1)3 = 2253.94 Total PV = 909.09 + 1652.89 + 2253.94 = 4815.93


Investment decisions

Net present value (NPV) NPV = PV(cash inflow) – PV(cash outflow) Cash inflows are earnings from investment,

sale of assets, etc, from the investment. Cash outflows are initial investment, future

maintenance cost, etc, due to the investment. What does the sign of NPV mean?

NPV measures the value created (or destroyed) by undertaking an investment.

Positive NPV suggests value is created and negative NPV suggests the opposite


Investment Decisions – Example Your broker calls you and tells you that he has this great

investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in the second year. You require a 15% return on investments of this risk (discount rate).

Should you take the investment?Step 1: Figure out how much should you invest if you can receive $40 in year 1 and $75 in year 2 and your annual return is 15%. => PV = 40/(1+15%) + 75/(1+15%)2 = $91.49Step 2: Compare with the offer ($100)

Use the formula of NPV NPV = -100 + 40/(1+15%) + 75/(1+15%)2

= -100 + 91.49 = -8.51 < 0 No – the broker is charging more than you would be willing to pay.

If you don’t invest in this project, you can earn 15% of return in other projects

(opportunity cost).


Multiple Cash Flows - A NoteThe cash flow timing is critically importantWithout explicitly told otherwise, it is assumed that the

cash flow occur at the end of each period A example--Suppose you are told a project has a first-

year cash flow of $100, a second cash-flow of $ 200, a third-year cash flow of $300.

Then, the time line should be:

21 3 4

Year = 0;Today

$100 $200 $300


Quick Quiz 1Suppose you are looking at the following possible

cash flows: Year 1 CF = $100 Years 2 and 3 CFs = $200 Years 4 and 5 CFs = $300. The required discount rate is 7%

a. What is the value of the cash flows at year 5?

b. What is the value of the cash flows today?

c. What is the value of the cash flows at year 3?(there are three ways of solving part c)


Quick Quiz 1 (cont’d)

1 2 3 4 5 100 200 200 300 300

r = 7%

a) What is the value of the cash flows at year 5?

FV 100 107 200 107 200 107 300 107 3004 3 2 1( . ) ( . ) ( . ) ( . )

1226 068.


Quick Quiz 1 (cont’d)b) What is the value of the cash flows today?

17.874)07.1(

300

)07.1(

300

)07.1(

200

)07.1(

200

)07.1(

100PV

543210

Method 1:

Method 2:

0 1 2 3 4 5

PV0 = ? FV5 = 1,226.068

0 1 2 3 4 50 100 200 200 300 300

PV0 = ?

PV0 = $1,226.068/(1.07)5 = $874.17



c) What is the value of the cash flows at year 3? Method 1: You can use FV5 from the first part

and compute PV3

FV3 = $1,226.068 / (1.07)2 = $1,070.89

0 1 2 3 4 5

FV3 = ? FV5 = 1,226.068


c)Method 2: You can use PV0 from the second part and compute FV3

FV3 = $874.17 (1.07)3 = $1,070.89

0 1 2 3 4 5

PV0 = 874.17FV3 = ?



Quick Quiz 1 (cont’d)c)

FV3 = $100(1.07)2 + $200(1.07) + $200 + $300/(1.07) + $300/(1.07)2

= $1,070.89

1 2 3 4 5 100 200 200 300 300

Method 3:

FV3 = ?


Go Back to Quick Survey in Ch.4

Today, you deposit $10,000 in a bank, which of the following option is better? (Suppose your required return is 4%.)

A. You get back $11,000 two years later.

B. You get $500 one year later and $10,500 two year later.

NPV(A) = $11,000/(1+4%)2 – 10,000 = $170.1 NPV(B) = $500/(1+4%) + 10,500 /(1+4%)2 – 10,000

= $188.6



F V P V




A P R E A R


P u reD isc o u nt

In te re stO n ly




D C FV a lu a tion



Special Cases of Multiple Cash Flows:Annuities and Perpetuities

Annuity – finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the

period, it is called an ordinary annuity If the first payment occurs at the beginning of

the period, it is called an annuity due Unless stated otherwise, all annuities are

assumed to be ordinary annuities Perpetuity – infinite series of equal payments


Annuities and Perpetuities

C

t=0 1 2 3 4

CCC C CC C CC CC

…….. T T+1 T+2 ……..

Annuity

Perpetuity


Annuities and PerpetuitiesApplicable when:

1. the values of cash flow are the same in all periods;

2. the discount rates are the same in all periods, i.e.

TrC

rC

rCPV

)1()1()1(....21

x

xaaxaxaxa

TT

1

1)...( 12

Recall how to add a geometric progression (GP) series


Annuities and Perpetuities

Annuities (t = T):

r

rC

rrrC

T

T

)1(1

1

1

11annuity of PV

(1 ) 1 of annuity

TrFV C

r


Annuity – Example 1 – buying a car

You can afford to pay $ 632 per month towards a car. The bank can loan you with 1% per month for 48 months. How much can you borrow?

Borrow money TODAY, so you need to compute the present value.

Using formula:54.999,23

01.

)01.1(

11

63248

PV


Annuity – Example 2 – winning a lottery

Suppose you win the Mark Six $10 million. 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?

Future money worth TODAY, so you need to compute the present value.

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

PV = PV(Rate, Nper,pmt,FV) =PV(0.05,30,-333333.33,0)


Annuity – Example 3 –Buying a House

You are ready to buy a house and you have HK$200,000 for a down payment (定金 ) and closing costs (借款手續费 ). Closing costs are estimated to be 4% of the loan value. You have an annual salary of HK$360,000 and the bank is willing to allow your monthly mortgage (抵押 ) payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (0.5% 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

How much loan you can get from the Bank?Monthly income = 360,000 / 12 = 30,000Maximum payment = .28(30,000) = 8400PV(total loan) = 8400[1 – 1/1.005360] / .005 = 1,401,050

How much can you afford to buy the house?Closing costs = .04(1,401,050) = 56,040

Closing costs are payments made to the bank, and therefore not to be included in the value of the house

Down payment = 200,000 – 56,040 = 143,960Total price = PV(total loan) + down paymentTotal Price = 1,401,050 + 143,960 = $1,545,010

You need to pay interest monthly.

1 period = 1 month


Annuities- Example- Finding the Payment

Suppose you want to borrow $20,000 for a new car now. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). If you take a 4 year loan, what is your monthly payment?

20,000 = C[1 – 1 / 1.006666748] / 0.0066667

C = 488.26Excel

C = PMT(rate,nper,pv,fv) =PMT(0.08/12,48,20000,0) = -488.26


Annuities – Example - Finding the Number of Payments

Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan?

2000 = 734.42(1 – 1/1.05t) / .05

.136161869 = 1 – 1/1.05t

1.157624287 = 1.05t

t = ln(1.157624287) / ln(1.05) = 3 yearsExcel

t = NPER(rate,pmt,pv,fv) =NPER(0.05,-734.42,2000,0) = 3


Annuities – Example - Finding the Rate

Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $500 per month for 30 months. What is the monthly interest rate?

Excel r = RATE(nper,pmt,pv,fv) = RATE(30,-500,10000,0) = 2.84%

rr)(1

11

5001000030

Verify r = 2.84%


Annuities– Finding the Rate Without a Financial CalculatorTrial and Error Process

Choose an interest rate and compute the PV of the payments based on this rate

Compare the computed PV with the actual loan amount

If the computed PV > loan amount, then the interest rate is too low

If the computed PV < loan amount, then the interest rate is too high

Adjust the rate and repeat the process until the computed PV and the loan amount are equal


Solving the interest rate by trial and error

rr)(1

11

5001000030

rr)(1

11

2030

r=3%: RHS = 19.6004 r=2%: RHS = 22.3965 By interpolation: r = 2% + {(22.3965-20)/(22.3965-19.6004)} x 1% r = 2% + 0.857% = 2.857% To get more precision: r=2.5% RHS = 20.9303 r = 2.5% + {(20.9303-20)/(20.9303-19.6004)} x 0.5% =

2.8498% …..


Annuities – Example - Finding the FV

Suppose you begin saving for your retirement by depositing $2000 per year in an retirement account. If the interest rate is 7.5%, how much will you have in 40 years?

FV = 2000(1.07540 – 1)/.075 = 454,513.04


Annuities Due

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

Timeline: Annuity due value

= Ordinary annuity value * (1+r) FV = 10,000[(1.083 – 1) / .08](1.08)

= 35,061.12

0 1 2 3

10k 10k 10k

32,464

35,061.12


Perpetuities

Recall the annuity formula:

Let t infinity with r > 0

r

CPV

rr)(1

11

CPVt


Perpetuity: Example

You are looking into an investment which will pay you $4 per year for the foreseeable future. If you require a 12% return, what is the most that you would pay for this investment?

PV = $4 / 0.12

= $33.33


Table 5.2


Answer to Exercises

1. Company A has identified an investment project with the following cash flows. If the discount rate is 10% what is the present value of these cash flows? What is the present value at 18%?

Year Cash Flow

1 $ 800

2 500

3 1,300

4 1,480


Answer to Exercises (cont’d)

PV = ∑[FVt / (1 + r)t ] @10%: - PV = $800 / 1.10 + $500 / 1.102 + $1,300 / 1.103 +

$1,480 / 1.104 = $3,128.07

@18%:- PV = $800 / 1.18 + $500 / 1.182 + $1,300 / 1.183 +

$1,480 / 1.184 = $2,591.65



2. An investment offers $7,000 per year for 15 years, with the first payment occurring 1 year from now. If the required return is 9%, what is the value of the investment? What would the value be if the payments occurred for 40 years? For 75 years? Forever?



PV = C({1 – [1/(1 + r)]t } / r ) For 15 yrs:

PV= $7,000{[1 – (1/1.09)15 ] / .09} = $56,424.82 For 40 yrs:

PV = $7,000{[1 – (1/1.09)40 ] / .09} = $75,301.52 For 75 yrs:

PV = $7,000{[1 – (1/1.09)75 ] / .09} = $77,656.48 Forever

PV = C/r = $7,000 / .09 = $77,777.78



3. If you put up $20,000 today in exchange for a 8.5%, 12-year annuity, what will the annual cash flow be?

PV = $20,000 = $C{[1 – (1/1.085)12 ] / .085}

=> C = $2,723.06



F V P V




A P R E A R

C o m p a r ingR a te s

P u reD isc o u nt

In te re stO n ly




D C FV a lu a tion



Annual Percentage Rate (APR)

Sometimes it is called stated annual interest rate. This is the annual rate that is quoted in industry.

APR = period rate * the number of periods per year

It is the annual interest rate without consideration of compounding.


Effective Annual Rate (EAR)

EAR is the actual rate paid (or received) after accounting for compounding that occurs during a year

EAR & APR

m = number of compounding per year If you want to compare two alternative investments with

different compounding periods, you need to compute the EAR and use that for comparison.

You should NEVER divide the EAR by the number of periods per year – it will NOT give you the period rate

1 m

m

APR 1 EAR


The Frequency of Compounding

You have a credit card that carries a rate of interest of 18% per year compounded monthly. What is the effective annual interest rate ?

18% per year compounded monthly is just code for 18%/12 = 1.5% per month

Monthly-compounded annual interest rate

= (1+0.015%)12 – 1 = 19.56% EAR = 19.56%; APR = 18% (Diff = 1.56%)


EAR and APR Two equal Annual Percentage Rates with different

frequency of compounding have different Effective Annual Rates. An example of an APR of 18%:

Annual Percentage rate (%)

Frequency of Compounding

Effective Annual Rate (%)

18 1 18.00

18 2 18.81

18 4 19.25

18 12 19.56

18 52 19.68

18 365 19.72


Continuous Compounding

What occurs as the frequency of compounding rises to infinity?

Example: The effective annual rate that’s equivalent to an annual

percentage rate of 18% = e 0.18 – 1 = 19.72%

(almost the same value with daily compounding)

1e1m

k1LimEAR k

m

m


Computing EAR and APR: Examples

Suppose you can earn 1% per month on $1 invested today.What is the APR?How much are you effectively earning?

FV = $1(1.01)12 = $1.1268Rate = $(1.1268 – 1) / $1 = .1268 = 12.68%EAR = $(1 + 0.01)12 – 1 = 12.68%

Suppose you can earn 3% per quarter if you put the money in another account.What is the APR? How much are you effectively earning per annum?

FV = $1(1.03)4 = $1.1255Rate = $(1.1255 – 1) / $1 = .1255 = 12.55%

1%(12) = 12%

3%(4) = 12%


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 rate to discounting monthly payments.

If you have payments other than monthly you need to adjust the interest rate into EAR of corresponding period.


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 = 147,089.22


Example

Suppose you have $10,000. You are looking at two savings accounts. One pays 5.25% per year, with daily compounding. The other pays 5.28% per year with semiannual compounding. (Suppose everything else is equal.) Which account should you put your money in? How much more can you earn from one account

than another? Use APR or EAR?We should use EAR! Because it takes compound interest into

consideration.


Example (cont’d)

Calculate EAR First account:

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

EAR = (1 + .0528/2)2 – 1 = 5.35% Suppose you invest $10000 in each account.

First Account: FV = $10000(1.0539) = $10,539 Second Account: FV = $10000(1.0535) = $10,535

You have $4 more money in the first account.


Present Value with Daily Compounding

You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? Daily rate = .055 / 365 = .00015068493 Number of days = 3(365) = 1095 PV = 15,000 / (1.00015068493)1095 = 12,718.56

To avoid rounding error, keep as many

decimal digits as possible.


Computing Payments: Example Suppose you want to buy a new stereo system and the

store is willing to allow you to make monthly payments. The stereo system costs $3500. The loan period is for 2 years and the interest rate is 16.9%. What is your monthly payment? Monthly rate = .169 / 12 = .01408333333 Number of months = 2(12) = 24 $3500 = C[1 – 1 / 1.01408333333)24] / .01408333333 C = $172.88

ALWAYS make sure that the interest rate and the time period match.


How to Use EARs?

A bank is offering 12% compounded quarterly. You put $100 in an account. What is the EAR?

12.55% How much will you have at the end of two years?

Method 1: $100 x(1+12%/4)8 =$126.68 Method 2: $100 x (1+12.55%)2 =$126.68


Quick Quiz 1 What is the APR if the monthly rate is .5%?

.5(12) = 6%

What is the APR if the semiannual rate is .5%? .5(2) = 1%

What is the monthly rate if the APR is 12% with monthly compounding? 12 / 12 = 1% Can you divide the above APR by 2 to get the effective

semiannual rate? NO!!! It should be (1+1%)6 – 1 = 6.15%


Quick Quiz 2

Here is an advertisement:

“We make getting credit easy with low monthly payments. For just $94.35* a month, you can have $3,000 today”

* 48 low monthly payments only.

What is the interest rate of the loan?

What is the APR of the loan?



Use formula :

Monthly Rate (by trial and error) = 1.8247%

[RATE(48,-94.35,3000,0) (Excel)]

Hence, APR = 12 * 1.8247% = 21.8967% And, Effective annual interest rate is

(1 + 1.8247%)12 - 1 = 24.2335%

r

rC

rrrC

T

T

)1(

11

1

11annuity of PV


Answer to Exercises

4. First National Bank charges 12.4% compounded monthly on its business loans. First United Bank charges 12.7% compounded semiannually. As a potential borrower, which bank would you go to for a new loan?



EAR = [1 + (APR / m)]m – 1 First National:

EAR = [1 + (.124 / 12)]12 – 1 = .1313 or 13.13%

First United:

EAR = [1 + (.127 / 2)]2 – 1 = .1310 or 13.10%

For a borrower, First United would be preferred since the EAR of the loan is lower.



5. Find the EAR in each of the following cases:

Stated Rate (APR)

Number of periods compounded

per year

Effective Rate (EAR)

6 % 1 　6 2 　6 4 　6 12 　6 52 　6 365 　



Stated Rate (APR)

Number of periods compounded per year

Effective Rate (EAR)

6 % 1 6%

6 2 6.090%

6 4 6.136%

6 12 6.168%

6 52 6.180%

6 365 6.183%


Loans and Loan Amortization


F V P V




A P R E A R


P u reD isc o u n t

In te r e stO n ly

A m or tiz edL o a ns

T y p e s ofL o a ns


D C FV a lu a tion


Types of LoansPure Discount Loans Interest-Only Loans Amortized Loans

No periodic interest payment

Principal repaid at end of period

Periodic payments are interest only

Principal paid at end of period.

Periodic payments include both interest & some principal repayment

Interest is paid on the declining balance of outstanding principle

Example:

Zero Coupon Bond;

Treasury bills

Example:

Coupon Bond

Examples:

Mortgage Loan

Car Loan


Pure Discount Loans: Example 5.11

The loan is sold at a discount. The principal amount is repaid at some future date, without any periodic interest payments.

If a 1-year loan promises to repay $10,000 in 12 months and the market interest rate is 7%, how much will this loan sell for in the market? Price = PV = $10,000 / 1.07 = $9,345.79 Interest = $(10,000 – 9,345.79) = $654.21


Interest-Only Loan: Example

Consider a 4-year, $10,000, interest only loan with a 7% interest rate. Interest is paid annually. The loan amount is $10,000. Principle is repaid at the end of year 4.

What would be the stream of cash flows of this loan? Years 1 – 3: Interest payments

= .07($10,000) = $700 Year 4: Interest + Principal = $10,700


Amortized Loan The borrower is required to repay part of the loan

amount over time The most common way of amortizing a loan is to

have the borrower make a single, fixed payment every period. Each payment covers the interest plus part of the principal

Example Consider a 4-year loan with annual repayments. The

interest rate is 8% and the principal amount is $5,000. What is the annual repayment?

$5,000 = C[1 – 1 / 1.084] / .08 C = $1,509.60


Amortization Table (Example, cont’d)

Year Beginning Balance

Regular

Payment

Interest Paid Principal Paid

Ending Balance

A B C D = B 8% E = C D F = B E

1 5,000.00 1509.60 400.00 1109.60 3890.40

2 3890.40 1509.60 311.22 1198.38 2692.02

3 2692.02 1509.60 215.36 1294.24 1397.78

4 1397.78 1509.60 111.82 1397.78 .00

Totals 6038.40 1038.40 5000.00


Fixed payment over periodsOutstanding Balance as a function of Time: A 30-Year Mortgage Loan (8%)

Find the excel file on our course website

Amorti zati on of Pri nci pal

0

50, 000

100, 000

150, 000

200, 000

250, 000

300, 000

350, 000

400, 000

450, 000

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

Month

Outs

tand

ing

Bala

nce


Fixed payment over periods Proportion of Payments in an Amortized Loan

Percent of I nterest and Pri nci pal

% I nterest

% Pri nci pal

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

Month

Find the excel file on our course website

At first, most of the payment is interest.


Summary

The periodic payment is fixed (the same) each year. The principal balance is reduced by the

different amount each period. Principal paid will increase each year

Interest paid is decreasing over time At the end of the last period, the principal is

reduce to zero.


Takeaways

Be able to compute the future value of multiple cash flows

Be able to compute the present value of multiple cash flows

Be able to compute loan payments Be able to find the interest rate on a loan Understand the effect of compounding periods

and how interest rates are quoted EAR APR

Understand how loans are amortized or paid off


Individual Homework

Critical Thinking and Concepts Review 5.1, 5.2

Questions and Problems 6, 10, 12, 20, 21, 34, 45, 55, 56

discounted cash flow valuation chapter 5 summer 2008

Documents

year cash flow

cash outflows

time value

total value

cash flow timing

yunling chen4 multiple

following possible cash

sum of individual cash