Question

(Related to Checkpoint 5.4)  (Present value)  Sarah Wiggum would like to make a single investment and have

$2.0

million at the time of her retirement in

35

years.  She has found a mutual fund that will earn

4

percent annually.  How much will Sarah have to invest today?  If Sarah earned an annual return of

14

percent, how soon could she then retire?  

Answer 1

Present value = Future value / (1+ rate per period)^ no. of periods

= 2200000/1.04^25

= 2200000/2.66583633149

= 825256.96

Future value = Present value * (1+ rate per period)^ no. of periods

2200000 = 825256.96 * 1.15^n

1.15^n = 2200000/825256.96

= 2.66583634751

n = log 2.66583634751/ log 1.15

= 0.42583348508/0.06069784035

= 7.02 years

That is Sarah can retire in 7.02 years. ie;17.98 years sooner.

Answer 2

SOLTION :

FV = 2 m$ = 2000000 ($) (given)

Interest rate , r = 4% annually = 0.04 (given)

=> 1 + r = 1.04 

Total period , n = 35 years.(given) 

Annual  compounding is applicable here. 

We have to find amount to be invested today.

So,

Amount to be invested today 

= PV

= FV / (1 + r)^n

= 2000000 / 1.04^35

= 506830.94 ($) (ANSWER).

Let time needed be n years if the rate of interest, r is 14% = 0.14.

=> 1 + r = 1.14

FV = PV (1 + r)^n

=> (1 + r)^n = FV / PV

=> 1.14^n = 2000000 / 506830.94 = 3.9461

Taking log :

=> n log(1.14) = log(3.9461)

=> n = log(3.9461) / log(1.14) 

=> n = 10.476 = 10.48 years (ANSWER)