LECTURE 2
INTERTEMPORAL CONSUMPTION PART TWO
PRESENT VALUE AND ITS APPLICATIONS Question: How much does $1 worth in the furure?
Answer: It depends on the interest rate r.
Suppose the annual interest rate is r, then if we deposite $1 into a bank we will earn $1 + $r next year. Therefore, for a cash of
$1 deposite into a bank for t years, its value at the tth year (or the future value, F) is
F = (1 + r)t
Now if we reverse the question, how much does F worth today?
The answer is simple,
P = F (1 + r)t
We refer to P as the present value of F. In this example, P =1.
The present value P shows the current value of a future cash ‡ow F. We refer to the term 1= (1 + r)t as the discount factor from period t to the current period.
Suppose the interest rate r di¤ers across periods. Let rt be the in- terest rate of period t, the future value of $1 holding for T periods will be
F =
YT
t=1
(1 + rt)
Now consider a hypothetical case where we invest $1 and get an annual interest rate of 100%. At the end of the year, we will get
$1 + $1 = $2. We can write
V (1) = initial principal (1 + interest rate)
= 1 (1 + 1) = 2
If the interest is compounded semiannually (twice a year). At the end of the …rst six months, we get $1 + $0:5 (the interest rate is divided by two). For the second six months, we get the return of
$1:5 (1 + 0:5) = (1:5)2 or
V (2) = 1 1 + 1 2
2
Following the same steps, if the interest is compounded triannually,
V (3) = 1 + 13 3 : In general,
V (n) = 1 + 1 n
n
When the interest is compounded continuously, i.e., n ! 1, the value of a principal $1 at the end of the year becomes
n lim!1 1 + 1 n
n
2:7182818:::
We de…ne
e = lim
n !1 1 + 1 n
n
Therefore, a principal of $1 becomes $e at the end of the year.
If a principal is $A, an annual interest rate is r, and we hold this
principal A for t years,
V (n) = A h
1 + r n
int
= A 1 + 1 q
q rt
; q = n r
As n ! 1; q ! 1 and the term in the bracket becomes ert:
n lim!1A 1 + 1 q
q rt
= Aert
For a future value F, we can …nd a present value via the following formulas:
Discrete case : P = F (1 + r)t Continuous case : P = Fe rt
ASSETS
Assets are goods that provide a ‡ow of services over time. In case of …nancial assets, they provide a monetary ‡ow.
Consider an example of …nancial assets. An T -period bond promises to pay a …xed number of monetary return R (the coupon) each period until its maturity date T , at which the face value F of this bond is paid to the bond holder. If the interest rate r is constant, the present value of this bond is given by
P = R
(1 + r) + R
(1 + r)2 + : : : + R + F (1 + r)T
=
XT
t=1
R
(1 + r)t + F (1 + r)T
We know that
XT
t=1
1
(1 + r)t = 1 (1 + r) T r
Then,
P = R
"
1 (1 + r) T r
#
+ F
(1 + r)T
If this bond is traded at price P, we can solve the above equation for the expression of r. Denote r as the solution, r is known as e¤ective yield or rate of return. Notice that the bond price is inversely related to r.
Another kind of bond is a perpetuity or a consol in which it
makes …xed payments forever. Its present value is P = R
(1 + r) + R
(1 + r)2 + : : : P (1 + r) = R + R
(1 + r) + R
(1 + r)2 + : : : P (1 + r) = R + P
Pr = R or P = R r
Now consider a continuous version of a T-period bond that pays a coupon of R each period until date T. Its present value is
P =
Z T
0
Re rtdt
= R
Z T
0
e rtdt
= R ert r
T
0
= erT
r + 1 r
For T ! 1; P ! 1r which is the same as a discrete version of a perpetuity.
READING Varian Ch. 10
NEXT LECTURE
The next lecture will be the last one on the topic of consumers’
theory. It will deal with the choice the consumer has to make under risk and uncertainty. Some background in statistic will be helpful.
READING FOR THE NEXT LECTURE Varian Ch. 12; Pindyck and Rubinfeld Ch. 5
