Question

Waiting period.  Jamal is waiting to be a millionaire. He wants to know how long he must wait if

a. he invests ​$23,806.07 at 20​% ​today?

b.he invests ​$48,435.33 at 15​% ​today?

c. he invests ​$127,469.15 at 10​% ​today?

d. he invests $294,279.91 at 6​% ​today?

Answer 1

(a)

Present Value (PV) = 23,806.07$

Interest Rate (r) = 20%

Future Value(FV) = 1,000,000$ (Million)

Number of years (n) = ?

FV = PV*(1+r)^n

1000000 = 23806.07*(1+0.20)^n

1000000/23806.07 = (1.20)^n

42.0061 = (1.20)^n

in order to find n, we can use the function log or try it on calculator till (1.20)^n = 42.0061

log (42.0061) = log (1.20)^n

1.6233 = n *log (1.20) (Since log(x^n) implies = n*log x)

1.6233 = n*(0.07918)

n = 1.6233/0.07918

n = 20.5 years

you can check it by keeping 20.5 years in place of n

(b)

Present Value (PV) = 48,435.33$

Interest Rate (r) = 15%

Future Value(FV) = 1,000,000$ (Million)

Number of years (n) = ?

FV = PV*(1+r)^n

1000000 = 48435.33*(1+0.15)^n

1000000/48435.33 = (1.15)^n

20.6461 = (1.15)^n

in order to find n, we can use the function log or try it on calculator till (1.15)^n = 20.6461

log (20.6461) = log (1.15)^n

1.3148 = n *log (1.15) (Since log(x^n) implies = n*log x)

1.3148 = n*(0.06070)

n = 1.3148/0.06070

n = 21.66 years

you can check it by keeping 21.66 years in place of n

(c)

Present Value (PV) = 127,469.15$

Interest Rate (r) = 10%

Future Value(FV) = 1,000,000$ (Million)

Number of years (n) = ?

FV = PV*(1+r)^n

1000000 = 127469.15*(1+0.10)^n

1000000/127469.15 = (1.10)^n

7.8450 = (1.10)^n

in order to find n, we can use the function log or try it on calculator till (1.10)^n = 7.8450

log (7.8450) = log (1.10)^n

0.8946 = n *log (1.10) (Since log(x^n) implies = n*log x)

0.8946 = n*(0.0414)

n = 0.8946/0.0414

n = 21.61 years

you can check it by keeping 21.61 years in place of n

(d)

Present Value (PV) = 294,279.91$

Interest Rate (r) = 6%

Future Value(FV) = 1,000,000$ (Million)

Number of years (n) = ?

FV = PV*(1+r)^n

1000000 = 294279.91*(1+0.06)^n

1000000/294279.91 = (1.06)^n

3.3979 = (1.06)^n

in order to find n, we can use the function log or try it on calculator till (1.06)^n = 3.3979

log (3.3979) = log (1.06)^n

0.5312 = n *log (1.06) (Since log(x^n) implies = n*log x)

0.5312 = n*(0.02531)

n = 0.5312/0.02531

n = 20.9878 years

you can check it by keeping 20.99 years in place of n

Note: You can even excel formula nper and insert values in order to get answer.

nper(rate,pmt,pv,fv,type)

Rate = Interest rate, PMT = 0, PV = Present Value of investment, FV = Future Value of Investment, type = end