Ec 173A—FINANCIAL MARKETS
LECTURE NOTES
Foster, UCSD
12Jul08
TOPIC 4 – RISK & RETURN
A.
Risk Factors in Interest Rates
1.
Interest Compounding and Growth:
[Optional Review from Topic 2]
a)
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
b)
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
c)
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 1.0 years:
V(1.0) = V(0)e
1.0r
= 500 e
.040
= $520.41
•
In 3.7 years:
V(3.7) = V(0)e
3.7r
= 500 e
.148
= $579.76
d)
Finding annual interest rates from periodic rates.
1)
Annual percentage rate (APR).
•
(
29
m
m
r
r
apr
×
=
≡ the periodic rate
×
no. of compounding periods/year.
•
For a loan offered at 1%/month, r
apr
= 1%
×
12 = 12%/year.
2)
Effective (equivalent) annual rate (EAR).
•
r
ear
≡ annual rate resulting in same growth of principal as
quoted periodic rate. If r/m = quoted rate compounded m
times/year:
•
For a loan offered at 1%/month, r
ear
=
(1.01)
12
− 1 = 12.68%/year.
(
29
1
1

+
=
m
ear
m
r
r
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Ec 173A
RISK & RETURN
p. 2
2.
Building an Interest Rate:
a)
I am willing to give up 100 loaves of bread this year if I can have at least
102 loaves of
bread 1 year from now.
My brother in law wants to borrow $100 for a year.
What
interest rate should I charge?
1)
If bread costs $1/loaf and I think this price will remain constant, I might lend $100
for one year at r = 2%, having $102 when loan is paid off, which will buy me 102
loaves of bread.
(Of course, if I could get a higher rate on other loans, then I will
charge my brother in law the higher rate.)
2)
If I expect bread price to rise 4% to $1.04, I will charge r = (1.02)(1.04) − 1 = 6.08%.
I get $106.08 next year, with real purchasing power of 106.08/1.04 = 102 loaves.
3)
If I estimate a 5% risk
default, I charge r = (1.02)(1.04)/0.95 − 1 = 11.66%.
I get
back $111.66 if loan is repaid, and $0 if loan defaults, for an expected payoff of
0.95(111.66) + 0.05(0) = $106.08, with purchasing power of 102 loaves.
b)
Generalization.
1)
In the above 3 cases, we found the needed interest rate r by
the following formula, where:
•
r = nominal or actual interest rate
•
r
p
= minimum compensatory (pure) interest rate
•
i = predicted rate of price inflation
•
d = risk factor
2)
So (1 + r)(1 − d) = (1 + r
p
)(1 + i), or r = d + r×d + r
p
+ i + i×r
p
.
If we drop small
terms like r×d and i×r
p
, we get a linear approximation:
d
i
r
r
p
+
+
≈
.
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 Spring '09
 Foster
 Interest Rates

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