Financial Economics Present Value
Two Equivalent Conditions
The traditional theory of present value puts forward two
equivalent conditions for asset-market equilibrium:
Rate of Return The expected rate of return on an asset equals
the market interest rate;
Present Value The asset price equals the present value of
expected future payments.
We explain these two conditions and show that they are
equivalent—either condition implies the other.
1
Financial Economics Present Value
Market Interest Rate
The rate-of-return condition says just that all assets share a
common expected rate of return. The market interest rate refers
to the expected rate of return common to all assets.
We assume that the market interest rate R > 0 is constant.
2
Financial Economics Present Value
Notation
Consider an asset with payment $
t
at time t. For a stock, the
payment would be the dividend. For a bond, the payment
would be interest or principal.
Let P
t
denote the asset price at time t.
3
Financial Economics Present Value
Expected Rate of Return
Definition 1 (Return) The return is the profit divided by the
amount invested.
Definition 2 (Expected Rate of Return) The expected rate of
return is the expected return divided by the length of the time
period.
4
Financial Economics Present Value
Disequilibrium
If the expected rate of return were greater than the market
interest rate, the security would be seen as a “good buy.”
Investors would like to buy the security; those holding the
security would not want to sell it. Demand would exceed
supply. The reverse inequality would lead to excess supply.
5
Financial Economics Present Value
Present Value
Definition 3 (Present Value) The present value of a payment
to be received in the future is the dollars attainable now by
borrowing against the future payment.
Definition 4 (Discount Factor) The present value is the future
payment multiplied by the discount factor.
6
Financial Economics Present Value
Discrete Time
With compound interest, a dollar borrowed at time 0 will
require a repayment of (1+ R)
t
at time t, the principal plus
interest.
Theorem 5 (Present Value) The present value at time 0 of $
t
dollars at time t is
$
t
(1+ R)
t
dollars. The discount factor is 1/(1+ R)
t
.
7
Financial Economics Present Value
The present-value equilibrium condition asserts that the asset
price at time 0 equals the present value of expected payments,
P
0
= $
0
+
E
0
($
1
)
1+ R
+
E
0
($
2
)
(1+ R)
2
+
E
0
($
3
)
(1+ R)
3
+ ⋅⋅⋅.
8
Financial Economics Present Value
Simple Example of Equivalence
Consider an asset paying P
1
at time 1 and paying nothing at
other times. Suppose that the interest rate is R. What would be
a fair price P
0
to pay for the asset at time 0?
9
Financial Economics Present Value
Rate-of-Return Condition
Using the rate-of-return condition, what would be a fair
price P
0
to pay for the asset at time 0? Setting the rate of return
equal to the market interest rate gives
P
1
− P
0
P
0
= R;
the profit is the capital gain. Solving for the price gives
P
0
=
P
1
1+ R
. (1
)
10
Financial Economics Present Value
Continuous Time
Here $
t
is the payment flow.
For an investment from time t to time t + dt, the profit is the
payment $
t
dt plus the capital gain dP
t
. The return during the
period is the profit divided by the beginning-of-period price,
$
t
dt + dP
t
P
t
.
12
Financial Economics Present Value
Rate-of-Return Equilibrium Condition
Condition 6 (Rate-of-Return Equilibrium Condition) The
expected rate of return equals the market interest rate,
$
t
dt + E
t
(dP
t
)
P
t
= Rdt. (2)
13
Financial Economics Present Value
Present Value
Consider an investment worth P
t
at time t. If the investment
earns the market interest rate R, then with continuous
compounding its value follows the differential equation
dP
t
= RP
t
dt,
with the solution
P
t
= P
0
e
Rt
.
One dollar invested at time 0 is worth e
Rt
dollars at time t.
Conversely, if one borrows e
−Rt
dollars at time 0, with interest
one will owe 1 dollar at time t.
14
Financial Economics Present Value
Theorem 7 (Present Value) The present value at time 0 of one
dollar at time t is e
−Rt
dollars, and the discount factor is e
−Rt
.
The present-value condition for asset-market equilibrium
asserts that the asset price equals the present value of expected
payments,
P
0
=
∞
0
e
−Rt
E
0
($
t
)dt. (3)
15
Financial Economics Present Value
Perpetual Bond
Consider a perpetual bond, a bond paying one-dollar interest
per period in perpetuity; the principal is never repaid. If the
interest rate is R, what is a fair price for the bond?
A fair price is
1
R
, (4
)
for the bond then has rate of return R. For example, if R = .10,
then the bond should sell for $10, so it will return 10%. A
lower price would give a higher yield, and a higher price would
give a lower yield.
17
Financial Economics Present Value
The present value is
∞
0
e
−Rt
$
t
dt =
∞
0
e
−Rt
1dt
= −
1
R
e
−Rt
∞
0
=
1
R
e
−R0
−
1
R
lim
t→∞
e
−Rt
=
1
R
,
in accord with the expected rate-of-return reasoning.
19
Financial Economics Present Value
Stock
Next we analyze a related but more complex example.
Consider a stock paying dividend D
t
, and the dividend grows at
the constant rate G. Here
$
t
= D
t
= D
0
e
Gt
.
What is a fair price P
0
for the stock at time 0?
20
Financial Economics Present Value
Rate-of-Return for the Stock
The rate of return is the dividend yield plus the rate of capital
gain. The dividend yield is D
t
/P
t
. Since the dividend grows
each period at rate G, intuitively the stock price should also
grow at this rate:
dP
t
P
t
= Gdt;
the rate of capital gain is G. The return is therefore
D
t
dt + P
t
Gdt
P
t
.
21
Financial Economics Present Value
Setting the rate of return equal to the market interest rate gives
D
t
P
t
+ G = R.
Solving for P
t
yields
P
t
=
D
t
R− G
. (5
)
The formula makes sense qualitatively: raising D
t
, reducing R,
or raising G should increase P
t
.
22
Financial Economics Present Value
This result requires R > G. Without this condition, the dividend
rises faster than the discount factor falls, and the present value
is infinite.
24
Financial Economics Present Value
General Equivalence
In general, the two conditions for equilibrium are equivalent. If
the price equals the present value at every moment, then the
rate of return equals the market interest rate at every moment;
and vice versa. We prove the equivalence.
25
Financial Economics Present Value
Condition 8 (Present-Value Equilibrium Condition) The
asset price equals the present value of expected payments,
P
t
=
∞
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
. (6)
Theorem 9 If the present-value equilibrium condition (6)
holds at every moment, then the rate-of-return equilibrium
condition (2) holds at every moment.
26
Financial Economics Present Value
Theorem 10 f the rate-of-return equilibrium condition (2)
holds at every moment, then
P
t
=
∞
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
+ lim
τ
→∞
e
−R(
τ
−t)
E
t
(P
τ
)
.
Condition 11 (No-Bubble) As the future time goes to infinity,
the present value of the expected future price goes to zero:
lim
τ
→∞
e
−R(
τ
−t)
E
t
(P
τ
)
= 0. (7)
Corollary 12 If the rate-of-return equilibrium condition (2)
holds at every moment, and the no bubble condition (7) holds,
then the present-value equilibrium condition (6) holds at every
moment.
27
Financial Economics Present Value
Bubble
A bubble refers to a situation in which the asset price is not set
by expected future payments but instead is driven by the
expectation of high capital gains. People pay a high price for
the asset because its price is rising and they hope for further
increases. The prospects for future payments are unimportant,
as the asset owner hopes to sell the asset to someone else at a
high price.
The terminology comes from the soap bubbles blown by
children. The bubbles have nothing inside, and soon they pop.
An asset bubble pops at some point, and the price falls.
28
Financial Economics Present Value
In the analysis here, bubble refers to a situation in which the
asset price exceeds the present value of expected payments.
The no-bubble condition rules out this possibility, as the price
and the present value are then equal.
29
Financial Economics Present Value
No Uncertainty
For simplicity, first we assume no uncertainty.
First, assume (6): at every moment the price equals the present
value.
We use the formula for the differentiation of an integral:
d
dt
b(t)
a(t)
f (
τ
,t) d
τ
= b
′
(t) f [b(t),t] − a
′
(t) f [a(t),t]+
b(t)
a(t)
∂
f (
τ
,t)
∂
t
d
τ
.
In the integral, t appears three times, so the derivative has three
terms.
30
Financial Economics Present Value
Uncertainty
We extend the proof to uncertainty.
First, assume (6): at every moment the price equals the present
value. Then
dP
t
= d
∞
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
=
−e
−R(t−
τ
)
E
t
($
τ
)∣
τ
=t
dt +
∞
t
e
−R(
τ
−t)
RE
t
($
τ
) d
τ
d
t
+
∞
t
e
−R(
τ
−t)
d[E
t
($
τ
)]
d
τ
=(−$
t
+ RP
t
)dt +
∞
t
e
−R(
τ
−t)
[E
t+dt
($
τ
) − E
t
($
τ
)] d
τ
.
34
Financial Economics Present Value
Take the expected value:
E
t
(dP
t
)=(−$
t
+ RP
t
)dt
+
∞
t
e
−R(
τ
−t)
E
t
[E
t+dt
($
τ
) − E
t
($
τ
)] d
τ
=(−$
t
+ RP
t
)dt.
Compared with the certainty case there is an extra term, but thi
s
extra term is zero:
E
t
[E
t+dt
($
τ
) − E
t
($
τ
)] = 0.
The change in the expected value is an innovation, and the
expected value of an innovation is zero. Rearranging
obtains (2).
35
Financial Economics Present Value
Alternative Derivation
P
t
=
∞
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
=
t+dt
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
+
∞
t+dt
e
−R(
τ
−t)
E
t
($
τ
) d
τ
= e
−R(t−t)
E
t
($
t
) dt + e
−Rdt
∞
t+dt
e
−R[
τ
−(t+dt)]
E
t
($
τ
) d
τ
= $
t
dt + e
−Rdt
E
t
∞
t+dt
e
−R[
τ
−(t+dt)]
E
t+dt
($
τ
) d
τ
,
since
E
t
[E
t+dt
($
τ
)] = E
t
($
τ
).
36
Financial Economics Present Value
Reverse Implication
By (2), then
dP
t
=(−$
t
+ RP
t
)dt + s
t
dz
t
,
in which s
t
dz
t
is the error term (the standard deviation s
t
is
stochastic).
Rearrange, multiply by the discount factor, and integrate:
∞
t
e
−R(
τ
−t)
($
τ
d
τ
− s
τ
dz
τ
)=
∞
t
e
−R(
τ
−t)
(RP
τ
d
τ
− dP
τ
).
Then take the expectation at time t.
38
Financial Economics Present Value
The expected value of the left-hand side is
E
t
∞
t
e
−R(
τ
−t)
($
τ
d
τ
− s
τ
dz
τ
)
=
∞
t
e
−R(
τ
−t)
E
t
($
τ
) d
τ
,
as
E
t
(s
τ
dz
τ
)=0.
39
