Present and future value tables of $1 at 3% are presented below: Debbie has $320

Question

Present and future value tables of $1 at 3% are presented below:
  

  
Debbie has $320,788 accumulated in a 401K plan. The fund is earning a low, but safe, 3% per year. The withdrawals will take place annually starting today. How soon will the fund be exhausted if Debbie withdraws $40,000 each year?

N FV $1 PV $1 FVA $1 PVA $1 FVAD $1 PVAD $1 1 1.03000 0.97087 1.0000 0.97087 1.0300 1.00000 2 1.06090 0.94260 2.0300 1.91347 2.0909 1.97087 3 1.09273 0.91514 3.0909 2.82861 3.1836 2.91347 4 1.12551 0.88849 4.1836 3.71710 4.3091 3.82861 5 1.15927 0.86261 5.3091 4.57971 5.4684 4.71710 6 1.19405 0.83748 6.4684 5.41719 6.6625 5.57971 7 1.22987 0.81309 7.6625 6.23028 7.8923 6.41719 8 1.26677 0.78941 8.8923 7.01969 9.1591 7.23028 9 1.30477 0.76642 10.1591 7.78611 10.4639 8.01969 10 1.34392 0.74409 11.4639 8.53020 11.8078 8.78611 11 1.38423 0.72242 12.8078 9.25262 13.1920 9.53020 12 1.42576 0.70138 14.1920 9.95400 14.6178 10.25262 13 1.46853 0.68095 15.6178 10.63496 16.0863 10.95400 14 1.51259 0.66112 17.0863 11.29607 17.5989 11.63496 15 1.55797 0.64186 18.5989 11.93794 19.1569 12.29607 16 1.60471 0.62317 20.1569 12.56110 20.7616 12.93794

Explanation / Answer

Present value of annuity due=(1+interest rate)*Annuity[1-(1+interest rate)^-time period]/rate

320788=1.03*40,000[1-(1.03)^-n]/0.03

320788=1373,333.33[1-(1.03)^-n]

[1-(1.03)^-n]=(320788/1373,333.33)

1.03^-n=1-(320788/1373,333.33)

(1/1.03)^n=0.766416504

Taking log on both sides;

n*log (1/1.03)=log 0.766416504

Hence n=log 0.766416504/log (1/1.03)

=9 years.

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