Solve the following problems involving varying annuities a) Every year Fred make

Question

Solve the following problems involving varying annuities a) Every year Fred makes an investment into an account on December 31, starting with a payment of SP on December 31, 2008. Every year he increases his investment by $500. His account earns 6% interest and his accumulated value on December 31, 2017 is $53,000. Find P An employer sponsors a pension plan with a $2m unfunded liability. The employer wants to eliminate the unfunded liability using payments made at the end of each year for 5 years, starting in one year. The employer plans to increase their payment by 10% each year and the pension plan uses a valuation interest rate of 5%. How much should they pay in the first year? A perpetuity starts with a payment of S1 in one year. Every year the payment amount increases by S1. At an interest rate i, the perpetuity has a present value of $360. Findi b) c)

Explanation / Answer

a) p*1.06^9 + (p+500)*1.06^8 + (p+1000)*1.06^7 + ..... = 53000

p*(1.06^9 +1.06^8 + ....1) + 500*(1.06^8+.....1.06) = 53000

P*1.06(1.06^9-1)/(1.06-1) + 500*1.06*(1.06^8-1)/(1.06-1) = 53000

Solving for P gives = 3920

b) suppose initial amount is P

P((1.05^5) + 1.1*(1.05^4) + 1.1^2*1.05^3 + (1.1^3)*1.05^2 + (1.1^4)*1.05) = 2 Mn

P = 0.285 Mn

c) 360 = 1/i

i = 0.278%

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