Assume (a) that during your 30 years of retirement you plan to consume C^pi = $1

Question

Assume (a) that during your 30 years of retirement you plan to consume C^pi = $100,000 per year and (b) that during this entire period you will earn R^pi = 8% on your money. However, instead of retiring with the appropriate value of PV(C, R, D) to fund your retirement, you have only 75% of PV(C, R, D). In other words, you are 25% underfunded at retirement. This obviously means that if you continue spending C^pi then you will run out of money well before the age of death at period D. Compute the period during which you will run out of money. Derive a general expression for the "ruin period" if you retire with z

Explanation / Answer

To solve for the actual period when we run out of money :

Present value at death / Present value at 30 years = 0.75

which means [(earnings - 100,000){1-(1/1.08)^n}/0.08]/[(earnings - 100,000){1-(1/1.08)^30}/0.08] = 0.75

which when solved gives n = 26.287 years

For a general expression, equate the formulas again but instead of 0.75, put z as the ratio.

So, the equation becomes (R-z)[1-(1/1.08)^n]/R[1-(1/1.08)^30] = z

and we solve for n as the ruin period.

n = log[(R-z)/0.033(R-z-0.9Rz)]

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