Solution Manual for Investment Science by Luenberger
Chapter The Basic
2 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 :'=! 21.8. Times doubled= Hi = 10 times. $1 invested in 1776 is worth 210 :'=! $1,000 today. (b) i = 6.6%. Years required to double = ~ :'=! 10.9. Times doubled = ~ times. $1 invested in 1776 is worth 220 :'=! 000, 000 today. $1, 2. (The 72 rule) Using (1 + r)n = 2 gives nIn (1 +r) In2 = 0.69. We have nr :'=! 0.69 and thus n :'=! ~ = 20
= In2. Using In (1 + r) :'=! and r :'=! PI.
Using instead In(1 + r) :'=! r- !r2 = r(1 -!r) we have nIn(1 + r) = In2 or equivalently nr :'=! ~. For r :'=! 0.08, we have (1 -r /2)-1 :'=! 1.042. Therefore, n:'=! !(0.069)(1.042) r 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
= ~
r
= ~ t
I("k) = -1 + " + " 2 , I , (,,) = 1 + 2" , "k+1 = "k -f' I' ("k) 1 3 1/9 7/3 0.00227 2.23810 -2.2 x 10-6 2.23607
0 2.23608
I("k)
"k+1 2/3 13/21 0.618033 0.618034
0.618034
5. (A prize) PV = $4, 682, 460.
1
2
CHAPTER mE BASIC 2. mEORY OFINTEREST 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) = = = = = = ~ --IT x 1 )'2X)'-1 )'2 ()' -1) X)'-2 1 -)'
(b) U(x) = )'X)'-l
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' B' = = aU(A') + b aU(B') + b
So solving both equations simultaneously we find parameters a and b: a = A' -U(B') U(A') -B' B'U(A') -A'U(B') U(A') -U(B')
b
=
6. (HARA) The hyperbolic absolute risk aversion function is given by: U(x) = y 1-)'
(~+b ax
)' , b>O.
(a) Linear: We can write the HARA as: U(x) = l=l. 1 y ax(1 -)') )' + b(l-
2 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 :'=! 21.8. Times doubled= Hi = 10 times. $1 invested in 1776 is worth 210 :'=! $1,000 today. (b) i = 6.6%. Years required to double = ~ :'=! 10.9. Times doubled = ~ times. $1 invested in 1776 is worth 220 :'=! 000, 000 today. $1, 2. (The 72 rule) Using (1 + r)n = 2 gives nIn (1 +r) In2 = 0.69. We have nr :'=! 0.69 and thus n :'=! ~ = 20
= In2. Using In (1 + r) :'=! and r :'=! PI.
Using instead In(1 + r) :'=! r- !r2 = r(1 -!r) we have nIn(1 + r) = In2 or equivalently nr :'=! ~. For r :'=! 0.08, we have (1 -r /2)-1 :'=! 1.042. Therefore, n:'=! !(0.069)(1.042) r 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
= ~
r
= ~ t
I("k) = -1 + " + " 2 , I , (,,) = 1 + 2" , "k+1 = "k -f' I' ("k) 1 3 1/9 7/3 0.00227 2.23810 -2.2 x 10-6 2.23607
0 2.23608
I("k)
"k+1 2/3 13/21 0.618033 0.618034
0.618034
5. (A prize) PV = $4, 682, 460.
1
2
CHAPTER mE BASIC 2. mEORY OFINTEREST 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) = = = = = = ~ --IT x 1 )'2X)'-1 )'2 ()' -1) X)'-2 1 -)'
(b) U(x) = )'X)'-l
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' B' = = aU(A') + b aU(B') + b
So solving both equations simultaneously we find parameters a and b: a = A' -U(B') U(A') -B' B'U(A') -A'U(B') U(A') -U(B')
b
=
6. (HARA) The hyperbolic absolute risk aversion function is given by: U(x) = y 1-)'
(~+b ax
)' , b>O.
(a) Linear: We can write the HARA as: U(x) = l=l. 1 y ax(1 -)') )' + b(l-