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Words: 9806
Pages: 40
Category: Business and Industry
Date Submitted: 01/27/2013 05:01 AM
Chapter
The
2
Basic
Theory
of
Interest
1. (A nice inheritance) Use the "72 rule". Years = 1994-1776 = 218 years.
(a) i = 3.3%. Years required for inheritance to double = Zf = 8 :'=! 1.8. Times
2
doubled= Hi = 10 times. $1 invested in 1776 is worth 210 :'=! 1,000 today.
$
(b) i = 6.6%. Years required to double = ~ :'=! 0.9. Times doubled = ~
1
times. $1 invested in 1776 is worth 220 :'=! 1, 000, 000 today.
$
2. (The 72 rule) Using (1 + r)n = 2 gives nIn (1 +r)
In2 = 0.69. We have nr :'=! .69 and thus n :'=!
0
~
= 20
= In2. Using In (1 + r) :'=! and
r
:'=! I.
P
Using instead In(1 + r) :'=! - !r2 = r(1 -!r)
r
we have nIn(1 + r) = In2 or
equivalently nr :'=! .
~
For r :'=! .08, we have (1 -r /2)-1 :'=! .042. Therefore,
0
1
n:'=! !(0.069)(1.042)
r
=~
r
=~
t
3. (Effective rates)
(a) 3.04%
(b) 19.56%
(c) 19.25%.
4. (Newton's method) We have
I(")
i
"k
0
1
1
2/3
2
13/21
3 0.618033
4
0.618034
I("k)
= -1 + " + " 2 , I , (,,) = 1 + 2" , "k+1 = "k -f'
I("k)
I' ("k)
1
3
1/9
7/3
0.00227 2.23810
-2.2 x 10-6 2.23607
0
2.23608
"k+1
2/3
13/21
0.618033
0.618034
0.618034
5. (A prize)
PV = $4, 682, 460.
1
2
CHAPTER. mE BASIC EORY OFINTEREST
2
m
6. (Sunk cost) The payment stream for apartment A is 1,000, 1,000, 1,000, 1,000
1,000, 1,000 while for B it is 1,900, 900, 900, 900, 900, 900. At any interest rate
PVA l1(x)
=
=
=
)'2X)'-1
)'2 ()' -1) X)'-2
1 -)'
Relative risk aversion coefficients, 11,are constant for both utility fW1ctions.
5. (Equivalency) If results are consistent, we have that V(x) = aU(x) + b, and since
V(A') = A' and V(B') = B' we must have
A'
=
aU(A') + b
B'
=
aU(B') + b
So solving both equations simultaneously we find parameters a and b:
a
=
A' -U(B')
U(A') -B'
b
=
B'U(A') -A'U(B')
U(A') -U(B')
6. (HARA) The hyperbolic absolute risk aversion function is given by:
U(x)
=
y
1-)'
(~ax...