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Erika and Kitty, who are twins, just received $30,000 each for their 25th birthday.They both have aspirations to beco e i!!ionaires. Each p!ans to ake a $5,000annua! contribution to her "ear!y retire ent fund# on her birthday, be innin ayear fro today. Erika opened an account with the %afety &irst 'ond &und, a utua!fund that invests in hi h( )ua!ity bonds whose investors have earned *+ per year inthe past. Kitty invested in the ew -ssue 'io(Tech &und, which invests in s a!!,new!y issued bio(tech stocks and whose investors have earned an avera e of 20+ peryear in the fund s re!ative!y short history. a. -f the two wo en s fund earn thesa e returns in the future as in the past, how o!d wi!! each be when she beco es a

i!!ionaire/ b. ow !ar e wou!d Erika s annua! contributions have to be for her tobeco e a i!!ionaire at the sa e a e as Kitty, assu in their e1pected returns arerea!i ed/ c. -s it rationa! or irrationa! for Erika to invest in the bond fundrather than in stocks/

The future va!ue & of such pay ents that are ade on a year!y basis and have theirinterest co pounded annuities4 is iven by the fo!!owin for u!a.

])1()1(

[1

r

r r P F

n++

=

+

4

where 6 is the year!y pay ent $5,000 in this case4, r is the annua! interest rate,and n is the nu ber of years.

-n this case & is iven one i!!ion4 and we need to so!ve for the nu ber of yearsn. The above for u!a can be written in ter s of n as

( )

( ) 1

1log

1log

+

++

=r

r P

Fr

n24

a4 &or Erika s case & 7 ,000,000, r 7 0.0*, 6 7 5,000 which, usin the e)uation24 above, ives n 7 83. years. Thus Erika wi!! be just over 68 years old by the

ti e she beco es i!!ionaire.

&or Kitty s case & 7 ,000,000, r 7 0.2, 6 7 5,000 which, usin the e)uation 24above, ives n 7 9.8 years. Thus Kitty wi!! beco e a i!!ionaire by the ti e sheis about 45.5 years old , i.e. a! ost 20 years before Erika.

b4 :sin e)uation 4 above we can deter ine how uch Erika wi!! have to contributeeach year to beco e a i!!ionaire in 9.8 years. & is ,000,000, r 7 0.0*, n is

9.8, 6 is unknown. %o!vin for 6 we et $2;,000. Thus Erika wi!! have to invest$27,000 each year to beco e a i!!ionaire at the sa e ti e as Kitty.

c4 &ro the ana!ysis above it is not rationa! for Erika to invest in the bond fund.owever, it shou!d be kept in ind that stock funds are historica!!y un!ike!y to

yie!d steady 20+ returns for 9< years and it wi!! o up and down a !ot. Erika sinvest ent wi!! row ore steadi!y, so she is ettin ore enta! peace.