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Chapter 5: Q. 68 (page 326)

Time to Reach an Investment GoalThe formula t=lnA-lnPr

can be used to find the number of years t required for an investment P to grow to a value A when compounded continuously at an annual rate r.

(a) How long will it take to increase an initial investment of \(1000 to \)8000 at an annual rate of 10%?

(b) What annual rate is required to increase the value of a \(2000 IRA to \)30,000 in 35 years?

(c) Give a derivation of this formula.

Short Answer

Expert verified

(a) 20.79 years will take to increase an initial investment of $1000 to $8000 at an annual rate of 10%.

(b) The annual rate 7.74%is required to increase the value of a $2000 IRA to $30,000 in 35 years.

(c) The formular=lnA-lnPtis proved.

Step by step solution

01

Step 1. Given Information

We have

t=lnA-lnPrA=$8000P=$1000r=10%

02

Part (a) Step 1. Finding time to increase an initial investment  

t=lnA-lnPrt=ln8000-ln10000.10t=2.07940.10t=20.79

03

Part (b). Step 1. Finding annual rate

t=lnA-lnPrr=lnA-lnPtr=ln30000-ln200035r=2.70835=0.0774r=7.74%

04

Part (c). Derivation of the formula

We use continuous compound interest formula

A=PertAP=ert

Takinglnboth sides of the equation

lnAP=lnertlnA-lnP=rtt=lnA-lnPr

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