108156.pdf

Solutions 267

2.3The timetsatisfies

(1 +t× 0 .09)×800 = 830,

which givest∼= 0 .4167 years, that is, 0. 4167 × 365 ∼= 152 .08 days. The return

will be

K(0,t)=^830800 −^800 =0. 0375 ,

that is, 3.75%.

2.4The principalPto be invested satisfies

(

1+ 36591 × 0. 08

)

×P=1, 000 ,

which givesP∼= 980 .44 dollars.

2.5The timetwhen the future value will be double the initial principal satisfies

the equation (

1+^0.^06

365

) 365 t

=2.

The solution ist∼= 11 .5534 years. Because no interest will be paid for a fraction

of the last day, this needs to be rounded up to a whole number of days, which

gives 11 years and 202 days. (We disregard leap years and assume for simplicity

that each year has 365 days.)

2.6The interest ratersatisfies the equation

(1 +r)^10 =2,

which givesr∼= 0 .0718, that is, about 7.18%.

2.7a) In the case of annual compounding the value after two years will be

V(2) =

(

1+^01.^1

) 2 × 1

100 = 121. 00

dollars.

b) Under semi-annual compounding the value will be

V(2) =

(

1+^02.^1

) 2 × 2

100 ∼= 121. 55

dollars, which is clearly greater than in case a).

2.8At 15% compounded daily the deposit will grow to

(

1+^0.^15

365

) 1 × 365

1 , 000 ∼= 1 , 161. 80

dollars after one year. If interest is compounded semi-annually at 15.5%, the

value after one year will be

(

1+^0.^1552

) 1 × 2

1 , 000 ∼= 1 , 161. 01

dollars, which is less than in the former case.