Topic 5 Valuation of Future Cashflows.ppt

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1. Acknowledgement Ross et al, 2011, Essentials of Corporate Finance, 7
th
Ed, McGraw-Hill Companies, Inc..
0
Topic 5
Valuation of
Future Cash
Flows
Taylor’s University
Dual Degree Program
Introduction
to Finance

2. 1-14-1
1
Learning Outcomes
At the end of the lesson, students should be
able to:
•compute present value and future value of
annuities;
•calculate perpetuity; and
•calculate APR (annual percentage rate) and
EAR (effective annual rate) on a loan

3. 1-24-2
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Topic Outline
•Valuing Level Cash Flows: Annuities
and Perpetuities
•Comparing Rates: The Effect of
Compounding Periods

4. 1-34-3
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Basic Definitions
•Interest rate –“exchange rate” between
earlier money and later money
–Discount rate
–Cost of capital
–Opportunity cost of capital
–Required return

5. 1-44-4
4
Perpetuities and Annuity
Defined
•Perpetuity–infinite series of equal
payments
•Annuity–finite series of equal
paymentsthat 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

6. 1-54-5
5
Perpetuities –Basic Formulas
PV=? PMT PMT PMT…. PMT
|______|_______|_______|____...____|
0 1 2 3 ∞
•Perpetuity formula: PV∞ = C / r
•Current required return:
$40 = $1 / r
r = .025 or 2.5% per quarter
•Dividend for new preferred:
$100 = C / .025
C = $2.50 per quarter

7. 1-64-6
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Annuities–Basic Formulas
•Annuities:




 
















r
r
CFV
r
r
CPV
t
t
1)1(
)1(
1
1
PVIFA
r;t
FVIFA
r;t

8. 1-74-7
7
Annuities and the Calculator
•You can use the PMT key on the
calculator for the equal payment
•The sign convention still holds
–Most problems are ordinary
annuities (first payment occur at the
end of each period)
Rule:
Interest(I/Y) and Periods(N) follow the
Payment(PMT)

9. 1-84-8
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Annuity –Example 1
•You plan to pay $632/mth for a new
car. The financing rate is 1%/mth for
48mths. How much can you borrow?
•You borrow money TODAY so you
need to compute the present value
•Using Formula:54.999,23
01.
)01.1(
1
1
632
48














PV

10. 1-94-9
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Time Line
PV=? 632 632 632 …. 632
|______|______|______|____...____| I/Y=1%
0 1 2 3 48
You are attempting to find the annuity value where PV is $3500
Practice

11. 1-104-10
10
Annuity –Example 1
•You plan to pay $632/mth for a new
car. The financing rate is 1%/mth for
48mths. How much can you borrow?
•Using financial calculator:
N = 48; I/Y = 1; PMT= –632; FV = 0;
CPT PV = 23,999.54 ($24,000)

12. 1-114-11
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Annuity –Example 2
•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?
•Using Formula:
PV = $333,333.33[1 –1/1.05
30
] / .05 =
$5,124,150.29;
•Using financial calculator:
N = 30; I/Y = 5; PMT = 333,333.33; FV = 0;
CPT PV = -5,124,150.29
Practice

13. 1-124-12
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Finding the Annuity Payment –Example 3
•Suppose you want to borrow $20,000 for a
new car. You can borrow at 8% per year,
compounded monthly (8/12 = .666666667%
per month). If you take a 4-year loan, what is
your monthly payment?
•Using formula:
$20,000 = C[1 –1 / 1.0066667
48
] / .0066667
C = $488.26
•Using financial calculator:
N = 4(12) = 48; PV = -20,000;
I/Y = 8/12 = 0.6667; CPT PMT = 488.26

14. 1-134-13
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Time Line
PV=20K PMT PMT PMT …. PMT=?
|______|______|______|____...____| I/Y=8/12%
0 1 2 3 N=4x12=48
You are attempting to find the annuity value where PV is $3500
Practice

15. 1-144-14
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Finding the Number of Payments –Example 4
•Youhave$1000onyourcreditcard
outstandingbutcanaffordpaymentofonly
$20permth.CardIRis1.5%permth.How
longdoesittaketopayoffthe$1000?
•UsingFormula:
$1,000 = $20(1 –1/1.015
t
) / .015
t = ln(1/.25) / ln(1.015)
= 93.111 months = 7.75 years
Using financial calculator:
I/Y = 1.5; PV = –1,000; PMT = 20; FV = 0
CPT N = 93.111 MONTHS = 7.75 years

16. 1-154-15
15
Time Line
PV=1K 20 20 20 …. 20
|______|______|______|____...____| I/Y=1.5%
0 1 2 3 N=?
You are attempting to find the annuity value where PV is $3500
Practice

17. 1-164-16
16
Finding the Rate
•Suppose you borrow $10,000 from
your parents to buy a car. You
agree to pay $207.58 per month for
60 months. What is the monthly
interest rate?
Sign convention matters!!!
N = 60
PV = –10,000
PMT = 207.58
CPT I/Y = 0.75% per mth

18. 1-174-17
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Quick Quiz: Part 3
Q1. You want to receive $5,000 per month for
the next 5 years. How much would you need to
deposit today if you can earn .75% per month?
•What monthly rate would you need to earn if
you only have $200,000 to deposit?
•Suppose you have $200,000 to deposit and
can earn .75% per month.
–How many months could you receive the
$5,000 payment?
–How much could you receive every month
for 5 years?

19. 1-184-18
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Quick Quiz: Part 3
Q1. You want to receive $5,000 per month
for the next 5 years. How much would you
need to deposit todayif you can earn .75%
per month?
Solutions: Using formula
Using financial calculator:
N = 5x12=60; PMT = 5000 ;I/Y =0.75;
CPT PV= –240,867 Practice??
075.
)0075.1(
1
1
5000
60125















x
PV

20. 1-194-19
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Quick Quiz: Part 3
Q2. You want to receive $5,000 per month
for the next 5 years. What monthly rate
would you need to earn if you only have
$200,000 to deposit?
Solutions:
Using financial calculator:
N =5x12=60; PMT =5000; PV = –200,000
CPT I/Y = 1.44% per month
Practice

21. 1-204-20
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Quick Quiz: Part 3
Q3. You want to receive $5,000 per month.
Suppose you have $200,000 to deposit and
can earn .75% per month. How many months
could you receive the $5,000 payment?
Solutions: Using formula:
PV = C[1 –1 / (1+r)
t
] / r
200,000 = 5,000(1 –1 / 1.0075
t
) / .0075
.3 = 1 –1/1.0075
t
1.0075
t
= 1.428571429
t = ln(1.428571429) / ln(1.0075) = 47.73 months
Practice

22. 1-214-21
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Quick Quiz: Part 3
Q3. You want to receive $5,000 per month.
Suppose you have $200,000 to deposit and
can earn .75% per month. How many months
could you receive the $5,000 payment?
Using financial calculator:
PMT =5000 ;I/Y =0.75%; PV = 200000;
CPT N= 47.73 months
Practice

23. 1-224-22
22
Future Values for Annuities
•Suppose you begin saving for your
retirement by depositing $2,000 per year in
an IRA. If the interest rate is 7.5%, how much
will you have in 40 years?
•Solutions: Using formula:
FV = C[(1 + r)
t
–1] / r
= $2,000(1.075
40
–1)/.075 = $454,513.04
Using financial calculator:
N = 40; I/Y = 7.5; PMT = 2,000;
CPT FV = 454,513.04

24. 1-234-23
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Table 5.2

25. 1-244-24
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Quick Quiz: Part 4
Q1.Youwanttohave$1milliontouse
forretirementin35years.Ifyoucan
earn1%permonth,howmuchdoyou
needtodepositonamonthlybasisif
thefirstpaymentismadeinone
month?
Q2.Youareconsideringpreferred
stockthatpaysaquarterlydividendof
$1.50.Ifyourdesiredreturnis3%per
quarter,howmuchwouldyoube
willingtopay?
Practice

26. 1-254-25
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Quick Quiz: Part 4
Q1.Youwanttohave$1milliontousefor
retirementin35years.Ifyoucanearn1%
permonth,howmuchdoyouneedto
depositonamonthlybasisifthefirst
paymentismadeinonemonth?
Solutions:Usingformula:
1,000,000 = C (1.01
35x12= 420
–1) / .01
C = $155.50
Using financial calculator:
N= 35x12 = 420; FV =1,000,000; I/Y = 1; CPT
PMT = $155.50
Practice

27. 1-264-26
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Quick Quiz: Part 4
Q2.Youareconsideringpreferredstock
thatpaysaquarterlydividendof$1.50.If
yourdesiredreturnis3%perquarter,
howmuchwouldyoubewillingtopay?
PV=? 1.50 1.50 1.50 …. 1.50
|______|_______|_______|____...____| I/Y=3%
0 1 2 3 ∞
Solutions:
RememberPerpetuityformula:
PV=C/rorPV=PMT/i
PV=1.50/.03=$50
Practice

28. 1-274-27
27
Effective Annual Rate (EAR)
•This is the actual rate paid (or
received) after accounting for
compounding that occurs during the
year
•If you want to compare two alternative
investments with different
compounding periods you need to
compute the EARand use that for
comparison.
**Compounded once a year

29. 1-284-28
28
AnnualPercentage Rate (APR)
•By definition APR = period rate times
the number of periods per year
•Consequently, to get the period ratewe
rearrange the APR equation:
–Period rate = APR / number of periods per
year
•You should NEVERdivide the effective
rateby the number of periods per year
–it will NOT give you the period rate
**Usually compounded more than once a
year e.g. monthly or quarterly

30. 1-294-29
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Computing APRs
•What is the APR if the monthly rate
(period rate) is .5%?
.5%x12 = 6%pa compounded
monthly
•What is the monthly rate if the APR is
12% with monthly compounding?
12% / 12 = 1% per month

31. 1-304-30
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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

32. 1-314-31
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EAR -Formula1
m
m
APR
1 EAR 







Remember that the APR is the quoted rate
(in decimals), and m is the number of
compounds per year

33. 1-324-32
32
Computing EARs -Example
•Suppose you can earn 1% per month on
$1 invested today.
–What is the APR? 1%(12) = 12%
–How much are you effectively earning
(EAR)?
•EAR = (1+0.12/12)
12
–1 = 12.68%
•Suppose if you put it in another account,
you earn 3% per quarter.
–What is the APR? 3%(4) = 12%
–How much are you effectively earning
(EAR)?
•EAR = (1+0.12/4)
4
–1 = 12.55%

34. 1-334-33
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Decisions, Decisions II
•You are looking at two savings accounts.
One pays 5.25%, with dailycompounding.
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?

35. 1-344-34
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Using Calculator
–First account:
2ndF RESET ENTER
2ndF ICONV NOM=5.25 C/Y=365
CPT EFF=5.39%
–Second account:
2ndF RESET ENTER
2ndF ICONV NOM=5.3 C/Y=2
CPT EFF=5.37%

36. 1-354-35
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Computing Payments with APRs
•Suppose you want to buy a new computer
system. The store will allow you to make
monthly payments. The entire computer
system costs $3,500. The loan period is
for 2 years and the interest rate is 16.9%
(annual) with monthly compounding. What
is your monthly payment?
Monthly rate = .169 / 12 = .01408333333
Number of months = 2(12) = 24
$3,500 = C[1 –1 / (1.01408333333)
24
] /
.01408333333
C = $172.88
N = 2(12) = 24; I/Y = 16.9/12 = 1.4083;
PV = 3,500; CPT PMT = -172.88
You are attempting to find the annuity value where PV is $3500
Practice

37. 1-364-36
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Time Line
3500 PMT PMT PMT …. PMT=?
|_____|_____|_____|___...___| I/Y=16.9%/12
0 1 2 3 24
You are attempting to find the annuity value where PV is $3500
Practice

38. 1-374-37
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Future Values with Monthly
Compounding
•Suppose you deposit $50 per 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[1.0075
420
–1] / .0075 =
$147,089.22
You are attempting to find the FV of an annuity stream
Practice

39. 1-384-38
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Future Values with Monthly
Compounding
•Suppose you deposit $50 per 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?
–N = 35(12) = 420
–I/Y = 9 / 12 = 0.75
–PMT = 50
–CPT FV = 147,089.22
You are attempting to find the FV of an annuity stream
Practice

40. 1-394-39
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Time Line
FV=?
50 50 50 …. 50
|_____|_____|_____|___...___| I/Y=9%/12
0 1 2 3 N=35x12
You are attempting to find the annuity value where PV is $3500
Practice

41. 1-404-40
40
Quick Quiz: Part 5
•What is the definition of an APR?
Period rate x No. of comp. per year
•What is the EAR? Rate after
accounting for compounding
•Which rate should you use to
compare alternative investments or
loans? EAR
•Which rate do you need to use in the
time value of money calculations?
Period rate Use APR to get it
Practice

42. 1-414-41
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Comprehensive Problem
•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? –671
•If you deposit those payments into an
account earning 8%, what will the future
value be in 10 years? 1448.66
•What will the future value be if you open the
account with $1,000 today, and then make
the $100 deposits at the end of each year?
3607.58
Practice

43. 1-424-42
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Reading
•Ross Chapter 6