Goal of the Lecture:
Understand how to properly
value a series of cash flows.
What is value?
OBJECTIVE VALUATION
EXAMPLE: C.C. Sabbathia’s contract averages 24 mm
while Mark Texiera’s averages 23 mm
years:
Sabbathia
Texiera
1
2
3
4
5
6
$20
$23
20
23
24
23
24
23
28
23
28
23
You can compare the two using "Present Value" which means
discounting dollars to be received in the future so that they are
equivalent to dollars received now. Maybe adjust for risk if pay
depends on uncertain performance. You are probably more familiar
with "Future Value" which is the number of dollars you expect to have
in the future if you leave a specific number of present dollars in a
bank account earning interest at k percent per year. Present value is
just the inverse of future value.
Note: Manny Ramirez 2 year contract with L.A. Dodgers for
20 and 25 million made him the “second highest paid” after
Alex Rodriguez, but couldn’t agree on deferral.
SETTING - MOST PEOPLE USE PRESENT VALUE
INTUITIVELY EVERY DAY
QUESTION: How do you decide whether to buy a snow
blower (lawn mower)?
ANSWER: Estimate present value of future plowing,
discount, and compare to snow blower price.
Other considerations may include time to shovel,
back-ache, plow digs up lawn/pavement and
doesn't come on time.
QUESTION: How do you decide how often to change the oil
or coolant in your car?
QUESTION: How do you decide whether to replace breaks,
tire, clutch etc. in your car?
QUESTION: How do you decide how much to save now for
retirement or a major future purchase ?
NEED TO CONSIDER THAT FUTURE CASH FLOWS ARE
WORTH LESS THAN PRESENT CASH FLOWS.
Discount - as in "discounting" what a person who
exaggerates says.
Compound - how the difficulty increases if you have three
finals in one day.
Time Value of Money
“A Dollar is Worth More Today Than a Dollar Tomorrow”
Illustration - Pie Concept
Maximize the Value of a Firm or the Size of the Pie
Three Basic Ideas Affect the Size of the Pie:
1.
Timing of Cash Flows - Ingredients in the Pie
2.
Valuation of Stocks/Bonds - Size of Pie
3.
Risk - Baking Conditions
Visualize as Follows:
V=S+B
V= S+ B
Factor
First Condition
Second Condition
Cash Flows:
More
Less
Timing of Cash Flows:
Sooner
Later
Riskiness of Cash Flows: Less
Produces Larger Pie
More
Produces Smaller Pie
Time Value of Money
I.
Time Value of Money - A Formula for Every Situation
II.
Future Value and Present Value of a Single Cash
Flow
III.
Present Value is the Inverse of Future Value
IV.
Future Value and Present Value of Multiple Cash
Flows
a. Equal annual cash flows (Annuities)
b. Infinite Annuities (Perpetuities)
c. Multiple Unequal Cash Flows
V.
Solving for an Unknown (Implied) Interest Rate
a. Given Present Value and Future Value, find
IRR (Internal Rate of Return)
Interest Rates
“Cost on Borrowed Funds or Rate Received on Money Lent”
I.
Nominal vs. Effective Annual Interest Rates
a. Adjust n for New Number of Periods
b. Adjust k for New (Lower) Rate for Portion of Year
Future Value
“Taking a Cash Flow Today and Determining What It will be
Worth Sometime in the Future at a Given Interest Rate, k”
I. Future Value
General Formula
= PV0(1+k)n
FVn
= PV0(1+k)n = [=FV(k, n, 0, PV)]
Note:
FVn = Future Value at the End of n Periods
PV0 = Present Value Now (Time = 0)
k
= the Interest (Discount or Compound) Rate
n
= the Number of Periods in the Future
[=FV()]= Excel Future Value Function (note that
Excel uses I instead of k and N instead of n)
II. Example: Suppose you have $5,000 and the interest rate
is 15% and you wish to invest for 4 years, how
much will you have at the end of 4 years?
FV4
=
PV0(1 + .15)4
=
$5,000(1.15)4
=
$5,000(1.75)
= [= FV(0.15, 4, 0, -5000)]
=
$8,745
PROBLEM: Suppose your company made $50 million this
year and you expect profit to grow by 15% per year
for the next 4 years. What will profit be at the end
of 4 years?
ANSWER: $50(1 + .15)4 = [= FV(0.15, 4, 0, -50)]
= $87.45 million
Note: This is the same basic problem as the previous
problem but with different language.
Present Value
“Inverse of Future Value and Gives the Value Today of
Money Received in the Future”
I. Present Value
General Formula
FVn
PV0
=
(1  k ) n
= FVn
1
= [= PV(k, n, 0, FV)]
(1  k ) n
[=PV()]= Excel Present Value Function (note that
Excel uses I instead of k and N instead of n)
II.
Example: What is the value of $100 to be received
at the end of 10 years if the k = 10%?
PV0
= $100/(1+.10)10
= $100/2.54
= [= PV(0.10, 10, 0, -100)]
= $38.6
PROBLEM: Suppose you need $10,000 to pay off a loan in 10
years. How much do you need to put in the bank
today at a 10% interest rate to have the $10,000
then?
PV0
= $3,860
Future Value of an Ordinary Annuity
“Future Value of a Series of Equal Cash Flows Received at
the End of Each Period for a Specified Number of Periods”
I. Future Value of an Ordinary Annuity
General Formula
FVAn
(1  k ) n  1
 PMT
k
= [= FV(k, n, PMT, PV)]
Note: FVAn = Future Value of an n-Period Annuity
PMT = Constant Annuity Payment
II. Example: Suppose you save $100 at the end of each
year for 3 years - k = 10%. How much will you
have in the account after 3 years?
FVA3 = $100 [((1+.10)3 - 1)/.10]
= $100(3.310)
= [= FV(0.10, 3, -100, 0)]
= $331.00
Future Value of an Annuity Due
“Future Value of a Series of Equal Cash Flows Received at
the Beginning of Each Period for a Specified Number of
Periods”
I. Future Value of an Annuity Due
General Formula
FVAn
(1  k ) n  1
 PMT
(1  k )
k
= [= FV(k, n, PMT, PV, 1)]
Note: Just multiply the ordinary annuity formula by (1 + k).
II. Example:
Suppose you save $100 at the beginning of each
year for 3 years - k = 10%. How much will you
have in the account after 3 years?
FVA3 = $100 [((1+.10)3 - 1)/.10](1 + .10)
= $100(3.310)(1 + .10)
= [= FV(0.10, 3, -100, 0, 1)]
= $364.10
Present Value of an Ordinary Annuity
“Present Value of a Series of Equal Cash Flows Paid at the
End of Each Period for a Specified Number of Periods”
I.
Present Value of an Ordinary Annuity
General Formula
1

1
PVAn = PMT  
n 
 k k (1  k ) 
= [= PV(k, n, PMT, FV)]
Note: PVAn = Present Value of an n-Period Annuity
PMT = Constant Annuity Payment
II. Example: You just won megabucks for $3.0M to be paid
over 20 yearly payments. How much did you really
win in present value dollars if k = 10%.
PMT = $150,000 (If Taxed at 30%, PMT = $105,000)
PVA20= $150,000[1/.10 - 1/(.1)(1+.1)20]
= $150,000(8.514)
= [= PV(0.10, 20, -150000, 0)]
= $1,277,100 (Less than Half)
If Taxed -> $105,000(8.514) = $894,000
Note: For the Present Value of an Annuity Due, just multiply
the above by (1 + k) because the money is earning interest for
one additional year. Or use [= PV(k, n, PMT, FV,1)].
For an Infinite Annuity the formula above reduces to
PVA = PMT [1/k] = PMT/k
Example: Redo the Lottery example except assume that you
receive payments for ever.
PVA = $150,000/.10 = $1,500,000
This is not much more than a receiving payments for
only 20 years because payments after 20 years are
not worth much - discounted heavily.
Note: The infinite annuity is often a good approximation to a
long annuity and is easy to calculate.
Multiple Unequal Cash Flows
“Present and Future Values are Additive so the Value of
Multiple Unequal Cash Flows is Just the Sum of the
Individual Present or Future Values”
Example: Present Value - Multiple Unequal Cash Flows
YEAR
CF
1
2
3
$200
$200
$300
PV = $200(1/1.10) + $200(1/1.10)2 + $300(1/1.10)3
= $200(.909) + $200(.826) + $300(.751)
= [= NPV(0.10, 200, 200, 300)]
= $572.5
Alternatively, you could find the value as a $200 annuity for 3
years plus a $100 cash flow in the third year.
Assume in the previous example that you receive 300 in year 3
and in each year forever afterward. The Present Value is
PV
= $200
 1

1
 0.1  0.1(1  0.1) 2 


+ 300/.10 [1/(1.10)2]
= $200[1.735] + $3000[.826]
= [= PV(0.10, 2, -200, 0)] + [= PV(0.10, 2, 0, -3000)]
= $2825
PROBLEM: Suppose someone offers you any of the
following 3 series of cash flows. Which do you
want if k = 20%
Year
1
2
3
4
5
1
$200
$200
$200
$200
$200
Cash Flows
.
2
3
0
0
0
0
$1000
$400
0
$400
0
$400
ANSWER: Choose 1. At high interest rates ignore 2.
at 20%
at 3%
1.
PV = 598
PV = 915.94
2.
PV = 579
PV = 915.14
3.
PV = 585
PV = 1066.5
QUESTION: What would you do if k = 3%? - choose 3.
NOTE: Low Japanese interest rates may explain their longterm view.Remember, behavior is driven by
financial incentives.
Determining Implied Interest Rates
Note: This is where a calculator or spreadsheat is helpful.
SINGLE CASH FLOW
Suppose we have one cash flow and the present value and
future value - we get the interest rate by solving for PVk,n.
PV0
PV0
FVn
= FVn
=
1
(1  k ) n
1
(1  k ) n
PROBLEM: Suppose you borrow $5,000 Today and agree
to payback $10,000 in 7 years. What was the
interest rate?
PV0/FV 7 = $5,000/$10,000
= .5 = 1/(1 + k)7
= [=Rate(n, PMT, PV, FV)]
= [=Rate(7, 0, -5000, 10000)]
=> k = 10.4%
ANNUITIES - SAME IDEA – but best to use Excel
PROBLEM: Suppose you borrow $5000 and agree to pay
$1500 per year for 7 years to pay off the loan.
What is the implied interest rate?
[=Rate(7, -1500, 5000, 0)] => k = 23%
QUESTION:In the previous problem you paid a total of
$10,000 while here you paid $10,500. Why the
large difference in implied interest rate?
ANSWER: Interim payments worth more than one balloon
payment in last year.
WHEN WE HAVE AN UNEVEN SERIES OF CASH FLOWS
ONE MUST USE IRR FUNCTION.
Example: If you pay $5000 to receive cash flows in the
future of (1) 2000, (2) 2000, (3) 3000, you get the
interest rate as follows;
Input -5000, 2000, 2000, 3000 into spreadsheet cells A1-A4
Then use [=IRR(A1:A4)] to get k =17.5%
Applications
GROWTH RATES - EARNINGS OR SALES GROWTH
PROBLEM: A company earned the following amounts in the
past 5 years - What was their earnings growth rate?
FIRST:
What formula is required?
Earn
$1000
$1500
$1725
$2250
$2500
$1000
Year
1
2
3
4
5
=
1
($2500)
(1  k ) n
$1000/$2500 = .40 = 1/(1 + k)4
= [=Rate(4, 0, -1000, 2500)]
=>k = 25.74%
Applications Continued
FUTURE SUMS
Suppose a company sells $1000 of bonds which mature in 5 years.
How much must be put in a sinking fund yearly to pay off the
bonds if k = 10%.
FIRST:
What formula?
(1  0.10)5  1
FV = $1000  PMT
0.10
= PMT(6.105)
= [=PMT(0.10, 5, 0, -1000)]
=> PMT = $163.80
TERM LOANS - CAPITAL
AMORTIZATION - MORTGAGE
RECOVERY
/
LOAN
If you take out a $10,000 car loan at 10% interest rate and
make 5 yearly payments. What will the payments be?
FIRST:
What formula?
PV = $10,000 = PMT
 1

1
 0.1  0.1(1  0.1)5 


=PMT[3.791]
= [=PMT(0.10, 5, -10000, 0)]
=> PMT = $2637.82
Effective vs. Nominal Interest Rates
“The Effective Rate is the True Rate of Interest per Year
and the Nominal Rate is the Quoted Rate per Year”
I.
Effective vs. Nominal
keff = (1+knom/m)m - 1
If interest is compounded “m” times annually, we must
adjust k and n in the formulas above, to take account
of the additional interest earned during a year.
II.
Example:
A nominal rate of 16% is compounded quarterly.
Thus, each quarter we earn 4% on the money in
the account.
k4 = k/m = 16%/4 = 4%
keff
= (1+knom/m)m - 1
= (1+.16/4)4-1
= .1698 => 17% (Effective Rate)
Therefore, although the nominal rate is 16%, we effectively
earn about 17%.
III. Adjusting formulas for compounding.
With “m” number of compound periods within a year, we can
use the earlier PV and FV formulas except that we replace
“n” with “mn” and replace “k” with “k/m”.
PROBLEM: Suppose you must make $500 monthly car
payments for 2 years. What is the present value
(Loan Amount) of the payments if knominal = 24%?
mn
= 12 * 2 = 24
k/m
= 24%/12 = 2%
PV0
= 500
 1

1
 0.02  0.02(1  0.02) 24 


= 500[18.914]
= [=PV(0.02, 24, -500, 0)]
= 9457