Solution Manual for Investment Science by Luenberger

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...