Question

On the first day of your new job, your employer presents you with the following two plans regarding retirement contributions: • Plan A: The company will make an annual contribution of $20,000 into your retirement account for the next 30 years. The first payment starts 1 year from now. • Plan B: If you stay with the company for 3 years, then the company makes annual contributions starting from $20,000 and grow at 5% per year. The first payment starts 3 years from now, and there will be 28 payments in total, from year 3 to year 30. If you leave within 3 years, then the company makes no payment. Assume that the annual interest rate is 3%. For simplicity, also assume that you will either quit within 3 years or stay with the company for the rest of your career.

c) If the probability of quitting in 3 years is p. How high does p need to be so that you'd prefer plan A to plan B?

d) Suppose you need to pay a 20% tax on those contributions, how does that change your analysis in c)?

Answer 1

c) Present value of Plan A = 20000/1.03 + 20000/1.03^2+.....+20000/1.03^30

=20000/0.03*(1-1/1.03^30)

=$392008.83

Present value of Plan B = 20000/1.03^3 + 20000*1.05/1.03^4+.....+20000*1.05^27/1.03^30

=20000/1.03^3*(1-(1.05/1.03)^28)/(1-1.05/1.03)

=$672445.39

If p is the probability of quitting in 3 years ,

expected value of Plan B = p*0+(1-p)*672445.39 = (1-p)*672445.39

For one to select plan A,  

(1-p)*672445.39 < 392008.83

(1-p) < 0.58296

p> 0.41703

So, for one to select plan A, probability of quitting within 3 years (p) must be atleast 41.70%

d) If 20% tax is paid on the contributions, it would reduce the present value of both plan A and B by 20%

Thus,

For one to select plan A,  

(1-p)*672445.39*0.8 < 392008.83*0.8

(1-p) < 0.58296

p> 0.41703

So, the value of p will not change