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Ec 173A – FINANCIAL MARKETS
LECTURE NOTES
Foster, UCSD
October 21, 2010
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
X
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.
1
1
1
0
)
1
(




=
∆
=
+
=
t
t
t
t
t
t
t
X
X
X
X
X
g
and
g
X
X
c)
Continuous exponential growth model.
)
(
)
(
)
0
(
)
(
t
X
t
X
g
and
e
X
t
X
gt
=
=
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
•
V
3
= V
0
(1+r)
3
= 500 (1.04)
3
= 500(1.1249) = $562.43
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PERSONAL FINANCE
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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
m
= 1%, m = 4:
•
In 3 months:
V
1/4
= V
0
(1 + r
m
)
= 500 (1.01)
= $505.00
•
In 6 months:
V
1/2
= V
0
(1 + r
m
)
2
= 500 (1.0201) = $510.05
•
In 9 months:
V
3/4
= V
0
(1 + r
m
)
3
= 500 (1.0303) = $515.15
•
In 1 year:
V
1
= V
0
(1 + r
m
)
4
= 500 (1.0406) = $520.30
d)
Continuous compounding.
1)
Some investments and bank accounts report annual rate of interest r, compounded
continuously (and deposited daily or weekly).
For continuous compounding, m
→
∞
and initial value V(0) increases as in a continuous exponential growth model.
The
value at the end of t years is given by V(t) = V(0)e
rt
.
2)
Example  V(0) = $500, r = 4%:
•
In 2.0 years:
V(2.0) = V(0)e
2.0r
= 500 e
.080
= $541.64
•
In 3.7 years:
V(3.7) = V(0)e
3.7r
= 500 e
.148
= $579.76
e)
Annuities  A stream of equal cash flows every year for T years.
There are two kinds:
•
Ordinary (“deferred”) annuity  cash flows occur at end
of each year
•
Annuity due  cash flows occur at beginning
of each year
f)
Future value of an annuity due (FVAD).
1)
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 Fall '09
 Foster

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