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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 :'=!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 :'=!$1, 000, 000 today. 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) :'=!r and :'=!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

= ~

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 2. mE BASICmEORY 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

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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 :'=!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 :'=!$1, 000, 000 today. 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) :'=!r and :'=!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

= ~

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 2. mE BASICmEORY 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

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