Week 3
Lecture 3
Ross, Westerfield and Jordan 7e
Chapter 6
Discounted Cash Flow Valuation
6-0
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
6-1
Chapter 6 Outline
• Future and Present Values of Multiple
Cash Flows
• Valuing Level Cash Flows: Annuities and
Perpetuities
• Comparing Rates: The Effect of
Compounding
6-2
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
6-3
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
6-4
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
Year
0
1
100
2
3
4
300
5
FV=?
2 years
4 years
6-5
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)
6-6
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
6-7
Example 6.3 Timeline
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1432.93
= Total PV
6-8
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!!
6-9
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
6-10
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?
6-11
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
6-12
Annuities and Perpetuities –
Basic Formulas
Ordinary Annuities:
1

1  (1  r)t
P V  C
r









Annuity interest factor
 (1  r)  1 
FV  C 

r


C
Perpetuity: PV 
r
t
C
1 
PV  1
r 
(1 r)t 
or


C
FV 
(1 r)t  1
r
Annuity Due Value = Ordinary Annuity Value x (1+r)
Other way of calculating FVannuity = PVannuity x (1+r)t
6-13
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:
1

1  (1  r)t
P V  C
r







1

1  (1.01) 48
PV  632
.01




  23,999.54


6-14
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?
6-15
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
6-16
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?
6-17
Finding the Payment
• Suppose you want to borrow
1

$20,000 to renovate your
1  (1  r)t
house. You can borrow at 8% P V  C 
r

per year. If you take a 4 year

loan, what is your monthly
payment?
PV
C
1 
• PV = 20000, r = 0.67%, t = 48

1
48

20,000 = C[1 – 1 / (1.0067) ] / .0067
(1  r)t 


C = 20000/{[1-1/(1.0067)48]/.0067}
r


C = 488.64







6-18
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]
•
•
•
•
•
•
1

1000/20 = (1 – 1/1.015t) / 0.015
1


(1 r)t
t
50 x 0.015 = 1 – 1/1.015
PV  C 
r

0.75 = 1 – 1 / 1.015t

t
0.25 = 1 / 1.015 (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!
6-19
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
6-20
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
6-21
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
6-22
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?
 (1 r)t  1
FV  C

r


• 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
6-23
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
6-24
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?
 (1 r)t  1
FV  C
  (1 r)
r


• 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)
6-25
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
6-26
Table 6.2 - p167
6-27
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
6-28
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.
6-29
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%
6-30
EAR - Formula
m
 APR 
EAR  1 
1

m 

Remember that:
- APR is the quoted rate
-m is the number of compounding periods per year
Computing APRs from EARs:
1

APR  m (1  EAR) m - 1


-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?


APR  12 (1 .12)1/12  1  .1138655152
or 11.39%
6-31
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%
6-32
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
6-33
End Lecture 3
6-34