1
Present Value Methodology
Econ 422
E. Zivot 2006
R.W. Parks/L.F. Davis 2004
Investment, Capital & Finance
University of Washington
Eric Zivot
Last updated: April 11, 2010
Present Value Concept
• Wealth in Fisher Model:
W = Y
0
+ Y
1
/(1+r)
The consumer/producer’s wealth is their current endowment plus the future
endowment discounted back to the present by the rate of interest (rate at
which present and future consumption can be exchanged).
• Why do this?
– Purpose of comparison—apples to apples (temporal) comparison with
lti l t l t l i f i t t/ ti
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mu
lti
p
l
e agen
t
s or app
l
es
t
o app
l
es compar
i
son o
f
i
nves
t
men
t/
consump
ti
on
opportunities
• Uniform method for valuing present and future streams of
consumption in order for appropriate decision making by
consumer/producer
• Useful concept for valuing multiple period investments and
pricing financial instruments
Calculating Present Value
Present value calculations are the reverse of compound
growth calculations:
Suppose V
0
= a value today (time 0)
r = fixed interest rate (annual)
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T = amount of time (years) to future period
The value in T years we calculate as:
V
T
= V
0
(1+r)
T
(Future Value)
2
Example
• A $30,000 Certificate of Deposit with 5%
annual interest in 10 years will be worth:
•
V
T
=
V
0
(1 + r)
T
=
30 000
*
(1 + 0 05)
10
=
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R.W. Parks/L.F. Davis 2004
V
V
0
(1
+
r)
30
,
000
(1
+
0
.
05)
= $48,866.84
• Note: Computation is easy to do in Excel
= 30,000 *(1 + 0.05)^10
Present Value
In reverse:
V
0
= V
T
/(1+r)
T
(Present Value)
The present value amount is the future value discounted
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The
present
value
amount
is
the
future
value
discounted
(divided) by the compounded rate of interest
Example
: A $48,866.84 Certificate of Deposit received
10 years from now is worth today:
V
0
= $48,866.84/(1+0.05)
10
= $30,000
Exam Review
• Be able to calculate present and future
values
• For any three of four variables: (V
0
, r, T,
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V
T
) you should be able to determine the
value of the fourth variable.
• How do changes to r and T impact V
0
and
V
T
?
3
Example: Rule of 70
• Q: How many years, T, will it take for an initial
investment of V
0
to double if the annual interest
rate is r?
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• A: Solve V
0
(1 + r)
T
= 2V
0
• => (1 + r)
T
= 2
• => Tln(1 + r) = ln(2)
• => T = ln(2)/ln(1+r)
• = 0.69/ln(1 + r) ≈ 0.70/r for r not too big
Present Value of Future Cash Flows
• A cash flow is a sequence of dated cash amounts
received (+) or paid (-): C
0
, C
1
, …, C
T
C h t i d iti h h
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•
C
as
h
amoun
t
s rece
i
ve
d
are pos
iti
ve; w
h
ereas, cas
h
amounts paid are negative
• The present value of a cash flow is the sum of the
present values for each element of the cash flow
Discount factors: Intertemporal Price
of $1 with constant interest rate r
• 1/(1+r) = price of $1 to be received 1 year
from today
•
2
=
price of $1 to be received 2 years
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price
of
$1
to
be
received
2
years
from today
•1/(1+r)
T
= price of $1 to be received T years
from today
4
Present Value of a Cash Flow
•{C
0
, C
1
, C
2
, …C
T
} represents a sequence of cash
flows where payment
•C
i
is received at time i. Let r = the interest or
discount rate.
Q: What is the present value of this cash flow?
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A: The present value of the sequence of cash flows is
the sum of the present values:
PV = C
0
+ C
1
/(1+r) + C
2
/(1+r)
2
+ … + C
T
/(1+r)
T
Summation Notation
0
(1 )
T
t
t
t
C
PV
r
=
+
∑
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0
0
1
(1 )
(1 )
t
T
t
t
t
r
C
C
r
=
=
+
=+
+
∑
Example
You receive the following cash payments:
time 0: -$10,000 (Your initial investment)
time 1: $4,000
time 2: $4,000
time 3: $4,000
The discount rate = 0.08 (or 8%)
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PV = -$10,000 + $4,000/(1+0.08)
+ $4,000/(1+0.08)
2
+ $4,000/(1+0.08)
3
= -$10,000 + $3,703.70 + $3,429.36 + $3,175.33
= $308.39
See econ422PresentValueProblems.xls for Excel calculations
5
PV Calculations in Excel
Excel function NPV:
NPV(rate, value1, value2, …, value29)
Rate = per period fixed interest rate
value1 = cash flow in
p
eriod 1
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p
value 2 = cash flow in period 2
…
value 29 = cash flow in 29
th
period
Note: NPV function does not take account of
initial period cash flow!
Present Value Calculation Short-cuts
PERPETUITY:
A perpetuity pays an amount C starting next period
and pays
this same constant amount C in each period forever:
C
1
= C, C
2
= C, C
3
= C, C
4
= C, ….
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PV(Perpetuity)
12
2
11 1
(1 ) (1 ) (1 )
1
(1 ) (1 ) (1 )
t
t
t
tt t
tt t
C
CC
rr r
C
C
C
rr r
∞∞ ∞
== =
=+ ++ +
++ +
===
++ +
∑∑ ∑
""
PV of Perpetuity
Based on the infinite sum property, we can write PV
as:
PV = Initial Term/[1 – Common Ratio]
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= C/(1 + r)/[1 - (1/(1 + r))]
= C/r
Initial Term = C/(1 + r)
Common Ratio = 1/(1 + r)
6
PV(Perpetuity) = C/(1 + r) + C/(1 + r)
2
+ C/(1 + r)
3
+ . . .
+ C/(1 + r)
t
+ . . .
Let a = C/(1 + r) = initial term
x = 1/(1 + r) = common ratio
Rewriting:
PV = a (1 + x + x
2
+ x
3
+ …) (1.)
Post multiplying by x:
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PVx = a(x + x
2
+ x
3
+ …) (2.)
Subtracting (2.) from (1.):
PV(1 - x) = a → PV = a/(1 - x)
PV(1 - 1/(1 + r)) = C/(1 + r)
Multiplying through by (1 + r):
PV = C/r
Example
The preferred stock of a secure company will pay the
owner of the stock $100/year forever, starting next year.
Q: If the interest rate is 5%, what is the share worth?
A: The share should be worth the value to
y
ou as an
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y
investor today of the future stream of cash flows.
This share of preferred stock is an example of a perpetuity,
such that
PV(preferred stock) = $100/0.05 = $2,000
Example Continued
• Q: What if the interest rate is 10%?
•
PV(preferred stock) = $100/0 10 = $1 000
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•
PV(preferred
stock)
=
$100/0
.
10
=
$1
,
000
• Notice: That when the interest rate doubled,
the present value of the preferred stock
decreased by ½.
7
Example Continued
The preferred stock of a secure company will pay the owner of the
stock $100/year forever, starting this year
.
Q: If the interest rate is 5%, what is the share worth?
A: The share should be worth the value to you as an investor today
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A:
The
share
should
be
worth
the
value
to
you
as
an
investor
today
of the future stream of cash flows (perpetuity component) plus the
$100 received this year.
PV(preferred stock) = $100 + $100/0.05 = $100 + $2,000 = $2,100
GROWING PERPETUITY
Suppose the cash flow starts at amount C at time 1, but
grows at a rate of g thereafter, continuing forever:
C
1
= C, C
2
= C (1+g), C
3
= C(1+g)
2
, C
4
= C(1+g)
3
, …
21
(1 ) (1 ) (1 )
t
CCgCg Cg
−
++ +
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PV(Perpetuity) =
23
1
1
(1 ) (1 ) (1 )
(1 ) (1 ) (1 ) (1 )
(1 )
(1 )
t
t
t
t
CCgCg Cg
rr r r
g
C
r
−
∞
=
++ +
++ ++ +
++ + +
+
=
+
∑
""
GROWING PERPETUITY
Based on the infinite sum property, we can write this
as:
PV
=
Initial Term/[1
–
Common Ratio]
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PV
Initial
Term/[1
Common
Ratio]
= C/(1 + r)/[1- ((1 + g)/(1 + r))]
= C/(r - g)
Note:
This formula requires r > g.
8
Example
• Your next year’s cash flow or parental stipend will be
$10,000. Your parents have generously agreed to
increase the yearly amount to account for increases in
cost of living as indexed by the rate of inflation.
• Your parents have established a trust vehicle such
that after their death you will continue to receive this
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that
after
their
death
you
will
continue
to
receive
this
cash flow, so effectively this will continue forever.
• Assume the rate of inflation is 3%.
• Assume the market interest rate is 8%.
• Q: What is the value to you today of this parental
support?
Answer
This is a growing perpetuity with
C = $10 000 r = 0 08 g = 0 03
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C
=
$10
,
000
,
r
=
0
.
08
,
g
=
0
.
03
Therefore,
PV = $10,000/(0.08 – 0.03) = $200,000
FINITE ANNUITY
A finite annuity will pay a constant amount C
starting next period through period T, so that there
are T total payments (e.g., financial vehicle that
makes finite number of payments based on death of
owner or joint death or term certain number of
payments, etc.)
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C
1
= C, C
2
= C, C
3
= C, C
4
= C, …. C
T
= C
PV(Finite Annuity)
2
11
(1 ) (1 ) (1 )
1
(1 ) (1 )
T
TT
tt
tt
CC C
rr r
C
C
rr
==
=+ ++
++ +
==⋅
++
∑∑
"
9
Finite Annuity
Formula Result:
PV (Finite Annuity) = C*(1/r) [1
–
1/(1 + r)
T
]
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= C*PVA(r, T)
where
PVA(r, T) = (1/r) [1 – 1/(1+r)
T
]
= PV of annuity that pays $1 for T years
Value of Finite Annuity =
Difference Between Two Perpetuities
Consider the Finite Annuity cash flow: C
1
= C, C
2
= C, C
3
= C, C
4
= C, …. C
T
= C
Suppose you want to determine the present value of this future stream of cash.
Recall a perpetuity cash flow (#1):
C
1
= C, C
2
= C, C
3
= C, C
4
= C, …C
T
= C, C
T+1
= C, …
From our formula, the value today of this perpetuity = C/r
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Consider a second perpetuity (#2) starting at time T+1:
C
T+1
= C, C
T+2
= C, C
T+3
= C, …
The value today of this perpetuity starting at T+1:
= C/r [1/(1+r)
T
] (why?)
Note: The Annuity = Perpetuity #1 – Perpetuity #2
= C/r – C/r [1/(1+r)
T
]
= C/r [1 - 1/(1+r)
T
]
PV(Finite Annuity) = C/(1+r) + C/(1+r)
2
+ C/(1+r)
3
+ . . . +C/(1+r)
T-1
Let a = C/(1+r)
x = 1/(1+r)
Rewriting:
PV = a (1 + x + x
2
+ x
3
+ …+x
T-1
)(1.)
Multi
p
l
y
in
g
b
y
x:
Alternative Derivation
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py g y
PVx = a(x + x
2
+ x
3
+ …+x
T
)(2.)
Subtracting (2.) from (1.):
PV(1-x) = a (1-x
T
)
PV = a (1-x
T
) /(1-x)
PV = C/(1+r)[(1-1/(1+r)
T
)/(1-1/(1+r)]
Multiplying the (1+r) in the denominator thru:
PV (Finite Annuity) = C/r [1 – 1/(1+r)
T
]
10
Example
Find the value of a 5 year car loan with annual
payments of $3,600 per year starting next year (i.e, 5
payments of $3,600 in the future). The cost of capital or
opportunity cost of capital is 6%.
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PV = $3,600*PVA(5, 6%)
= $3,600*(1/0.06)[1 – 1/(1.06)
5
]
= $15,164.51
Example Continued
Suppose you had also made a down-payment
for the car of $5,000 to lower your monthly
loan payments. The total cost/value of the car
hdih
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you purc
h
ase
d
i
s t
h
en:
PV(down payment) + PV(loan annuity)
= $5,000 + $15,164.51
= $20,164.51
Computing Present Value of Finite
Annuities in Excel
Excel function PV:
PV(Rate, Nper, Pmt, Fv, Type)
Rate
=
per period interest rate
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Rate
per
period
interest
rate
Nper = number of annuity payments
Fv = cash balance after last payment
Type = 1 if payments start in first
period; 0 if payments start in initial
period
11
Example
• Borrow $200,000 to buy a house.
• Annual interest rate = 10%
• Loan is to be
p
aid back in 30
y
ears
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py
• Q: What is the annual payment?
• PV = $200,000 = C*PVA(0.10, 30)
• => C = $200,000/PVA(0.10, 30)
• PVA(0.10, 30) = (1/0.10)[1 – 1/(1.10)
30
] = 9.427
• => C = $200,000/9.427 = $21,215.85
Computing Payments from Finite
Annuities in Excel
Excel function PMT:
PMT(Rate, Nper, Pv, Fv, Type)
Rate
=
per period interest rate
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Rate
per
period
interest
rate
Nper = number of annuity payments
Pv = initial present value of annuity
Fv = future value after last payment
Type = 1 if payments are due at the
beginning of the period; 0 if payments
are due at the end of the period
Example
• You win the $5 million lottery!
• 25 annual installments of $200,000 starting
next
y
ea
r
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y
• Q: What is the PV of winnings if r = 10%?
• PV = $200,000 * PVA(0.10, 25)
• PVA = (1/0.10)[1 – 1/(1.10)
25
] = 9.07704
• => PV= $200,000 * (9.07704) = $1,815,408
< $5M!
12
Future Value of an Annuity
• Invest $C every year, starting next year, for T
years at a fixed rate r
• How much will investment be worth in year T?
•
Trick: FVA(r,T)
=
PVA(r,T)
*
(1
+
r)
T
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Trick:
FVA(r,T)
PVA(r,T) (1 r)
• = (1/r) [1 - 1/(1+r)
T
] *(1+r)
T
• = (1/r)[(1+r)
T
–1]
• Therefore
• FV = C*FVA(r, T)
• where FVA(r, T) = FV of $1 invested every year
for T years at rate r
Example
• Save $1,000 per year, starting next year, for 35 years
in IRA
• Annual rate = 7%
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• Q: How much will you have saved in 35 years?
• FV = $1,000*FVA(0.07, 35)
• FVA(0.07, 35) = (1/0.07)*[(1.07)
35
– 1] = 138.23688
• => FV = $1,000*(138.23688) = $138,236.88
Computing Future Value of Finite
Annuities in Excel
Excel function FV:
FV(Rate, Nper, Pmt, Pv, Type)
Rate
=
per period interest rate
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Rate
per
period
interest
rate
Nper = number of annuity payments
Pmt = payment made each period
Pv = present value of future payments
Type = 1 if payments start in first
period; 0 if payments start in initial
period
13
Finite Growing Annuities
• Similar to how we amended the Perpetuity formula for ‘Growing’
Perpetuities, we can amend the Annuity formula to account for a
‘Growing’ Annuity.
• The cash flow for a finite growing annuity pays an amount C, starting next
period, with the cash flow growing thereafter at a rate of g, through period
T:
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T:
PV = C/(1+r) + C(1+g)/(1+r)
2
+ C(1+g)
2
/(1+r)
3
+ . . . +C(1+g)
T-1
/(1+r)
T
= Σ C(1+g)
t-1
/(1+r)
t
for t = 1,…, T
` = C/(r-g) [1- (1+g)
T
/(1+r)
T
]
Class Example
• An asset generates a cash flow that is $1 next year,
but is expected to grow at 5% per year indefinitely.
• Suppose the relevant discount rate is 7%.
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Q: After receiving the third payment, what can you
expect to sell the asset for?
Q: What is the present value of the asset you held?
Compounding Frequency
• Cash flows can occur annually (once per annum),
semi-annually (twice per annum), quarterly (four
times per annum), monthly (twelve times per annum),
daily (365 times per annum), etc.
• Based on the cash flows
,
the formulas for
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,
compounding and discounting can be adjusted
accordingly:
General formula
: For stated annual interest rate r
compounded for T years n times per year:
FV =V
0
* [1 + r/n]
nT
14
Compounding Frequency
Effective Annual Rate (annual rate that gives
the same FV with compounding n times per
year):
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[1 + r
EAR
]
T
= [1 + r/n]
nT
=> r
EAR
= [1 + r/n]
n
-1
Example
• Invest $1,000 for 1 year
• Annual rate (APR) r = 10%
• Semi-annual compounding: semi-annual rate = 0.10/2 =
0.05
• FV = $1,000*(1 + r/2)
2*1
= $1,000*(1.05)
2
= $1102.50
• Note: $1,000*(1 + 0.05)
2
= $1,000*(1 + 2*(0.05) +
(0 05)
2
)
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(0
.
05)
)
• = $1,000 + $100 + $2.5
• = principal + simple interest + interest on interest
• Effective annual rate:
• (1 + r
EAR
) = (1 + APR/2)
2
• => r
EAR
= (1.05)
2
– 1 = 0.1025 or 10.25%
Example: The Difference In Compounding
Annual rate of Interest 5%
T = 1 Year
Compounding Times One plus
Frequency
Per Annum
Effective Rate
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Frequency
Per
Annum
Effective
Rate
Yearly 1 1.05
Semi-Annual 2 1.050625
Quarterly 4 1.050945337
Monthly 12 1.051161898
Daily 365 1.051267496
Hourly 8,760 1.051270946
By the minute 525,600 1.051271094
By the second 31,536,000 1.051271093
15
Example
• Take out (borrow) $300,000 30 year fixed rate
mortgage
• Annual rate = 8%, monthly rate = 0.08/12 =
0.0067
• 30*12 = 360 monthly payments
• Q: What is the monthly payment?
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• PV = $300,000 = C*PVA(0.08/12, 360)
• PVA(0.0067, 360) = 136.283
• => C = $300,000/136.283 = $2,201.30
• Note: total amount paid over 30 years is
• 360*$2,201.30 = $792,468
Example
• Consider previous 30 year mortgage
• Suppose the day after the mortgage is
issued, the annual rate on new mortgages
shoots up to 15%
•
Q: How much is the old mortgage worth?
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•
Q:
How
much
is
the
old
mortgage
worth?
• PV = $2,201*PVA(0.15/12, 360)
• PVA(0.15/12, 360) = 79.086
• => PV = $2,201*79.086 = $174,092 <
$300,000!
Continuous Compounding
Increasing the frequency of compounding to
continuously:
lim n→∞ [1 + r/n]
nT
= (2.718)
rT
= e
rT
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Effective Annual Rate:
[1 + r
EAR
]
T
= e
rT
=> r
EAR
= e
r
-1
16
Example
• r = annual (simple) interest rate = 10%, T =
1 year
• FV of 1$ with annual compounding:
• FV = $1(1+r) = $1.10
FV f 1$ ith ti di
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•
FV
o
f
1$
w
ith
con
ti
nuous compoun
di
ng:
• FV = $1*e
r
= 2.7180
0.10
= $1.10517
• Effective annual rate
•1 + r
EAR
= 1.10517 => r
EAR
= 0.10517 =
10.517%
Further Insight on Continuous Compounding
Example: Invest $V
0
for 1 year with annual rate r and continuous
compounding
1
1
10
0
rr
V
VVe e
V
×
⎛⎞
=⇒=
⎜⎟
⎝⎠
E. Zivot 2006
R.W. Parks/L.F. Davis 2004
1
0
10
ln
ln ln
V
r
V
VVr
⎛⎞
⇒=
⎜⎟
⎝⎠
⇒−=
Test/Practical Tips
• General formula will always work by may be
tedious
• Short-cuts exist if you can recognize them
E. Zivot 2006
R.W. Parks/L.F. Davis 2004
• Use short-cuts!
• Break down complicated problems into simple
pieces