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Week 3

Lecture 3

Ross, Westerfield and Jordan 7e

Chapter 6

Discounted Cash Flow Valuation

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Last Week..

• Sources and Uses of Cash:• A or L and SE = Source• A or L and SE = Use

• Ratios – Most Used in Finance:• Profitability Ratios: PM, ROA(ROI), ROE• Market Value Ratios: P/E, M/B• Leverage Ratios: D/E, EM

• Simple vs Compound Interest

• Present & Future Values

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Chapter 6 Outline

• Future and Present Values of Multiple

Cash Flows

• Valuing Level Cash Flows: Annuities and

Perpetuities

• Comparing Rates: The Effect of

Compounding

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Multiple Cash Flows –Future Value Example 6.1

• If you have 7000 now to invest plus 4000 each year for 3 years and the rate is 8% p.a. what is the value of your investments in 3 years? In 4 years?

• First find the value at year 3 of each cash flow and add them together.• Year 0 (today): 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

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Multiple Cash Flows – Future Value Example 2

• 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?• First way: FV = 500(1.09)2 + 600(1.09) =

= 594 + 654 = 1248.05• Second way: FV = [500(1.09) + 600](1.09) =

= (545 + 600)(1.09) = 1145(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: FV = 1248.05(1.09)3 = 1616.26

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Multiple Cash Flows – FV Example 3

• 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

Year0 1 2 3 4 5

100 300 FV=?2 years

4 years

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FV of Multiple Cash Flows

• Formula:

FV = C1(1+r)n-1 + C2(1+r)n-2 + …+Ct(1+r)n-t

• where C is the cash flow generated at time t, and r is the discount rate, n = number of periods

t = current period.

• Note: if there is an immediate cash flow C0 at the start

then begin sum with t = 0.

• If there is a cash flow in the final year, don’t compound last cash flow (see example 6.1)

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Multiple Cash Flows – Present Value Example 6.3

• You are offered an investment that will pay you $200 in 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 now?

• Find the PV of each cash flows 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

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Example 6.3 Timeline

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1432.93 = Total PV

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PV of Multiple Cash Flows

• Formula:

PV = C1/(1+r) + C2/(1+r)2 + …+Ct/(1+r)t

• where Ct is the cash flow generated at time t, and r is the discount rate, t = current period

• Note: if there is an immediate cash flow (C0 ) at the start then begin sum with t = 0. Do not discount!!

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Multiple Cash Flows Using a Spreadsheet

• You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows

• Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas

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Quick Quiz

• Suppose 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%

• What is the value of the cash flows at year 5?

• What is the value of the cash flows today?

• What is the value of the cash flows at year 3?

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Annuities and Perpetuities Defined

• 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

• Perpetuity – infinite series of equal payments at regular intervals

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Annuities and Perpetuities – Basic Formulas

Ordinary Annuities:

Annuity interest factor

Perpetuity:Annuity Due Value = Ordinary Annuity Value x (1+r)

Other way of calculating FVannuity = PVannuity x (1+r)t

1r)(1r

CFV

r)(1

11

r

CPV

t

t

r

1r)(1CFV

rr)(1

11

CPV

t

t

or

r

CPV

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Annuity – Example 6.5 p158

• You can afford to pay $632 per month towards a new car. You go to

the bank and find out that the rate is 1% per month for 48 months.

How much can you borrow?

• You borrow money TODAY so you need to compute the present

value.

• t = 48, r = 1%, C = 632, PV = ?

• Formula:

54.999,2301.

)01.1(1

1632

48

PV

rr)(1

11

CPVt

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Buying a House

• You are ready to buy a house and you have $20,000 saved for a deposit and other fees. You enquire at the bank for a possible loan and they tell you the following:

• Loan fees 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 (0.5% per month) for a 30-year fixed rate loan.

• 1)How much money will the bank loan you?

• 2)How much can you offer for a house?

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Buying a House - Continued

• 1)How much is the bank loan? or.. what is the present value of the annuity?• Monthly income = 36,000 / 12 = 3,000• Maximum payment = .28(3,000) = 840• PV = 840[1 – 1/1.005360] / .005 = 140,105

• 2) How much can you offer for the house?• Loan fees = .04(140,105) = 5,604• Deposit = 20,000 – 5604 = 14,396• Offer Price = 140,105 + 14,396 = 154,501

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Quick Quiz

• You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value?

• You are 21 and want to receive 5000 per month in retirement. In you’re retirement you expect the interest to be 0.75% per month and you think you’ll need the income for 25 years. How much do you need to have in your account at retirement?

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Finding the Payment

• Suppose you want to borrow $20,000 to renovate your house. You can borrow at 8% per year. If you take a 4 year loan, what is your monthly payment?

• PV = 20000, r = 0.67%, t = 4820,000 = C[1 – 1 / (1.0067)48] / .0067

C = 20000/{[1-1/(1.0067)48]/.0067}

C = 488.64

rr)(1

11

PVC

r

r)(11

1CPV

t

t

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Finding the Number of Payments – Example 6.6 p.161

• You spend $1000 on your credit card and can afford to pay only the minimum payments of $20 per month. Knowing the interest rate is 1.5% per month, how long will it take you to pay $1000?

• Start with the equation to isolate t and use logarithms.• 1000 = 20[(1 – 1/1.015t) / 0.015]

• 1000/20 = (1 – 1/1.015t) / 0.015• 50 x 0.015 = 1 – 1/1.015t

• 0.75 = 1 – 1 / 1.015t

• 0.25 = 1 / 1.015t (negative signs cancel out) • 1 / 0.25 = 1.015t

• t = LN(1/0.25) / LN(1.015) = 93.111 months/12 = 7.76 years

• And this is only if you don’t charge anything more on the card!

r

r)(11

1CPV

t

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Finding “t” - 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) / 0.05• 2000/734.42 = (1 – 1/1.05t) / 0.05• 2.723237 = (1 – 1/1.05t) / 0.05• 2.723237 x 0.05 = (1 – 1/1.05t) • 0.136161869 = 1 – 1/1.05t

• 0.863838131 = 1/1.05t (negative signs cancel out)• 1/ 0.863838131 = 1.05t

• 1.157624287 = 1.05t • LN(1.157624287) = LN(1.05t )• LN(1.157624287) = t x LN(1.05)• t = LN(1.157624287) / LN(1.05) = 3 years

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Annuity – Finding the Rate

• Trial 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 given amount

• If the computed PV > given amount, then the interest rate is too low. Action: increase rate

• If the computed PV < given amount, then the interest rate is too high. Action: lower rate

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

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Example – Finding “r”

• An insurance company offers to pay $1000 per year for 10 years if you pay $6710 upfront. What is the implied rate of annuity?

• Solution: Trial and Error calculation• Step 1: pick a rate, say 10%, calculate PV

which is $6144.. Too low.. We need $6710• Step 2: start again with a different rate until

what you calculate is the same as the given value

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Future Values for Annuities• Suppose you begin saving for your retirement by

depositing $2000 per year in an investment 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

• Alternative Method:• FV Annuity: (PV Ordinary Annuity) x (1+r)t

PV = 2000[1-1/(1.075)40/0.075] = 25,188.82• FV Annuity: 25,188.82 x (1.075)40 = 454,513.04

r

1r)(1CFV

t

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Annuities on the Spreadsheet

• The present value and future value formulas for annuities in excel - fill in payment variable

• Finding the Annuity Payment: PMT(rate,nper,pv,fv)

• The same sign convention holds as for the PV and FV formulas

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Annuity 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?

• FV = 10,000[(1.083 – 1) / .08](1.08) =

32,464(1.08) = 35,061.12

• General relationship between Ordinary and Annuity Due:• Annuity Due = Ordinary Annuity x (1+r)

r)(1r

1r)(1CFV

t

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Perpetuity – Example 6.7

• A company wants to sell preferred stock at $100 per share. Similar shares sell for $40 with $1 dividend per quarter. What dividend will the new shares offer?

• Perpetuity formula: PV = C / r, C = ?• Current required return:

• 40 = 1 / r , r = 1/40• r = .025 or 2.5% per quarter

• Dividend for new preferred:• 100 = C / .025• C = 2.50 per quarter

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Table 6.2 - p167

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Effective Annual Rate (EAR)

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

• Is 10% p.a. = 10% compounded semiannually?• If you want to compare two alternative

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

• 10% p.a. ≠ 10.25% p.a. comp. semiannually

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Annual Percentage Rate (APR)

• This is the annual rate that is quoted by law• By definition APR = period rate x number of periods per year

• Period rate = APR / number of periods per year• You should NEVER divide the EAR by the number of periods per year –

it will NOT give you the period rate• What is the APR if the monthly rate is 0.5%?

• 0.005(12) = 6%• What is the APR if the semiannual rate is 0.5%?

• 0.005(2) = 1%• What is the monthly rate if the APR is 12% with monthly compounding?

• 0.12 /12 = 1%• Can you divide the above 12% APR by 2 to get the semiannual

rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate.

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APR & EAR - Example

• 1) Suppose you can earn 1% per month on $1 invested today.• What is the APR? 1% x (12) = 12%• How much are you effectively earning?

• FV = $1x(1.01)12 = $1.1268• EAR = ($1.1268 – $1) / $1 = .1268 = 12.68%

• 2) Suppose you put another $1 in another account, that earns 3% per quarter.• What is the APR? 3% x (4) = 12%• How much are you effectively earning?

• FV = $1x(1.03)4 = 1.1255• EAR = ($1.1255 – $1) / $1 = .1255 = 12.55%

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EAR - Formula

1 - EAR) (1 m APR m

1

1 m

APR 1 EAR

m

Remember that:

- APR is the quoted rate

-m is the number of compounding periods per year

Computing APRs from EARs:

-Rearrange the formula

- Example: Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?

11.39% or 2.1138655151.12)(112APR 1/12

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Continuous Compounding

• Sometimes investments or loans are figured based on continuous compounding

• EAR = eq – 1• q = rate• e = 2.71828, a mathematical constant• There is a special function on the calculator normally

denoted by ex

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

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Lecture 3 - Summary

• PV and FV of multiple cash flows• Uneven cash flows• Cash flows occurring at different time periods

• Annuities and Perpetuities• Equal cash flows• Equal time periods between the cash flows• Finding C, t, r

• EAR and APR

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End Lecture 3