Hand-out
Notes and Problem (Present Value) -- Class 1
A management
should maximize the value of the firm, which is the same as maximizing
the present value of future profits.
A. the
concept of Present value: what some
amount of money is worth today, given that it will be paid in the
future. The formula is PV = FV/(1+i)t
B. the
present value of some future profit is
C. PV
= Profitt / (1+i)t
where t is a year (e.g., 5)
and i is the applicable interest rate.
D. and
maximizing the present value of all future profits involves maximizing over
time(t) =0 to time = INF
SUM (πt / (1+i)t
E. this
also gives us the value of the firm (its price in sale). Thus, where π is profits, i is (constant) interest
rate, g is (const) growth rate of profits, and where g<i, and where the life
of the firm is very (inf.) long, the following holds:
1. PVfirm
= SUM [ (1+g)tπ0)
/ (1+i)t ] =
2. = π0[ SUM ((1+g)/(1+i))t]
3. since,
for an infinite series, where c is a constant with abs val. less than one,
SUM ct = 1/(1-c)
4. and
since (1+g)/(1+i) is a constant less than 1 in the above (since g<i),
5.#2 becomes:π0* 1 /
[1-((1+g)/(1+i))]= π0[(1+i)/(i-g)]
6. this
can be used to determine the value of a firm (useful in stock market activities
and acquisitions)
in class
Problem: Suppose the interest rate is 10 percent and the firm is expected to
achieve a growth rate of profits of 5 percent for the foreseeable future. If the current profits are 100 million, what
is the value of the firm (PV of all future profits).
PVfirm =
[(1+.1)/(.1-.05)]100 = 2200 (million)
7. Finally,
lets observe that the above final result in #5 indicates that (given those
assumptions) that maximizing short term (current) profits (π0)
is maximizes long term profits (PV of all future profits)
Problem: If
membership in an organization costs $300 now, vs. $125 now, plus $125 more in 1
year, plus $125 more in 2 yrs, which is best payment method for someone who
wants to join?
(use
int. rate of 10%)