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m)
mt
= V
0
(1+r
ear
)
t
⇒
(1 + r/m)
mt
= (1+r
ear
)
t
r/m) = t Ln(1+r
ear
)
⇒
Ln(1+r
ear
) = mLn(1 + r/m)
1 + r/m)
m
⇒
r
ear
= (1 + r/m)
m
− 1
Ec 173A
TIME VALUE OF MONEY
p. 1
Ec 173A – FINANCIAL MARKETS
LECTURE
NOTES
Foster, UCSD
13:14:47
A.
Interest Compounding, Growth and Discounting
1.
Compound Growth:
a)
Stocks and flows.
1) Stock  a quantity measured at an instant of time.
2) Flow  quantity measured over a period of time; the rate of change in some stock.
3) Stock may change level discretely
(interest paid at end of quarter) or continuously
(H
2
O
evaporating from lake).
NOTATION
Symbol
Definition
X
t
Level of stock X at end of period t
∆
X
t
1period change in X during period t:
∆
X
t
≡ X
t
− X
t−1
X(t)
Level of stock X at instant of time t
or X’(t)
Instantaneous rate of change of X at time t:
X’(t) ≡ dX(t)/dt
r
Interest or discount rate
g
Constant compound growth rate of stock X
b)
Discrete exponential growth model.
c)
Continuous exponential growth model.
2.
Future Values and Compound Interest:
a)
An interest rate (r) can be viewed 3 ways:
1) A cost of borrowing.
2) A growth rate of value of money invested in interestbearing debt securities.
3) A discount factor for finding the present equivalent value of payments to be made or
received in the future.
b)
Interest compounded annually.
1) If principal V
0
earns interest at annual rate r, compounded at the end of each year, its
value rises according to discrete exponential growth equation V
t
= V
0
(1+r)
t
.
2) Example  V
0
= $500, r = 0.04 (4%):
•
V
1
= V
0
(1+r) = 500 (1.04) = $520.00
•
V
2
= V
1
(1+r) = V
0
(1+r)
2
= 500 (1.04)
2
= $540.80
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View Full DocumentEc 173A
TIME VALUE OF MONEY
p. 2
•
V
3
= V
0
(1+r)
3
= 500 (1.04)
3
= 500(1.1249) = $562.43
c)
Interest compounded more frequently.
1) If V
0
earns interest at periodic
rate r/m, compounded m times per year, its value at the
end of t years
is given by V
t
= V
0
(1 + r/m)
mt
.
2) Example  V
0
= $500, r= 4%, m = 4, so r/m=1% (quarterly compounding):
•
In 3 months:
V
1/4
= V
0
(1 + r/m)
4*0.25
= 500 (1.01)
= $505.00
You won’t really need a financial calculator for any of these, but this is
the
sequence: N=mt=1, I/Y=r/m=1, PV=500, FV=?
•
In 6 months:
V
1/2
= V
0
(1 + r/m)
4*0.5
= 500 (1.0201) = $510.05
•
In 9 months:
V
3/4
= V
0
(1 + r/m)
4*0.75
= 500 (1.0303) = $515.15
•
In 2.5 years V
5/2
= V
0
(1 + r/m)
4*2.5
= 500 (1.1046) = $552.31
3) Example  V
0
= $500, r= 4%, m = 2, so r/m=2% (semiannual compounding):
•
In 3 months:
V
1/4
= V
0
(1 + r/m)
2*0.25
= 500 (1.0995)
= $504.9752
•
In 6 months:
V
1/2
= V
0
(1 + r/m)
2*0.5
= 500 (1.02) = $510
•
In 9 months:
V
3/4
= V
0
(1 + r/m)
2*0.75
= 500 (1.0301) = $515.07
•
In 2.5 years:
V
5/2
= V
0
(1 + r/m)
2*2.5
= 500 (1.1041) = $552.05
4) Example  V
0
= $500, r = 4%, m = 1 (annually), or 2 (semiannually), or 4 (quarterly), or
12 (monthly), or 365 (daily):
•
In 1 year, m=1:
V
1
= V
0
(1 + r/m)
1*1
= 500 (1.04) = $520
•
In 1 year, m=2:
V
1
= V
0
(1 + r/m)
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 Spring '09
 Foster

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