Question

Assume the following scenario: Bob plans to retire in 20 years from now and wants to have the following stream of CFs after retirement. Monthly payments of $4,000 for 15 years starting right after retirement (The first payment will be at the end of the first month in year 21). He then needs an extra 50000$ with the final payment (final month of year 35). Starting from year 36, he wants the monthly payments to be 6000$ for 10 years (The first payment will be at the end of the first month in year 36). And finally, starting from year 46, he wants the monthly payments to grow at 0.5% per month forever (first payment will be at the end of the first month in year 46). The APR is 12% with quarterly compounding. What is present value (at t = 0) of this retirement plan? options:

a) 33444

b) 55905

c) 45044

d) 65888

show your calculations. and formulas used

Answer 1

Answer -

Option C - 45,044

Explanation -

Rate of Interest = PVAF (r%, n)

= [1 + 12%/4]^1/3 - 1

= [1.03]^1/3 - 1

= 0.99%

Periodic Interest Rate Factor = 1 + 0.99% = 1.0099

1. Monthly Payment (P) = $ 4,000

n = 15 x 12 = 180 months

Present Value of Annuity at t = 20

= P x PVAF (r%, n)

= 4000 x PVAF (0.99%, 180)

= 4000 x 83.85

= $ 335,400

2. Present Value at t=20 of extra $50,000 final payment

= 50,000/(1.0099)¹⁸⁰

= $8,487

3. Monthly Payment (P) = $ 6,000

n = 12 x 10 = 120 months

Present Value of Annuity at t = 20

= 6,000 x PVAF (0.99%, 120) x 1/(1.0099)¹⁸⁰

= 6,000 x 70.03 x 0.1697

= $ 71,318

4. Monthly Payment = 6,000 x 1.005 = $6,030

Growth Rate = 0.5%

Periodic Interest Rate = 0.99%

Present Value of perpetuity at t = 20,

= [6030 / (0.0099 - 0.005)] x 1/(1.0099)³⁰⁰

= [6030/0.0049] x 0.0520

= $64,011

Total all the above computations = 335,400 + 8,487 + 71,318 + 64,011

= $ 479,216

Present Value at t=0 of this retirement plan = 479,216/(1.0099)²⁴⁰

= $45,044

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