Question

1. A perpetuity-due has monthly payments in this pattern: Q, 2Q, 3Q, Q, 2Q, 3Q, Q, 2Q, 3Q, . . . The present value of the perpetuity is $700,000 and the effective annual discount rate is 6%. Find Q.

2. A 30 year annuity-immediate has first payment $1200 and each subsequent payment increases by 0.5%. The payments are monthly and the annual effective rate is 8%. Find the accumulated value of the annuity at the end of 30 years.

3. A 20 year annuity has annual payments which increase by $500 each year. The first payment is $10,500 on Jan. 1, 2018. The annual effective interest is 1%. What is the value of the annuity on Oct. 1, 2017?

Answer 1

Question 1)

Given

PV of Perpetuity P=$700,000

Effective annual discount rate =6%

Monthly Discount rate r=6%/12=0.5%

Let annuity due schedule is Q,2Q,3Q,Q,2Q,3Q....

So PV of annuity due P= Q + 2Q/(1+r)+ 3Q/(1+r)^2 + Q/(1+r)^3 + 2Q/(1+r)^4+ 3Q/(1+r)^5 +Q/(1+r)^6 + 2Q/(1+r)^7+ 3Q/(1+r)^8 ….

P=Q + 2Q/(1+r)+ 3Q/(1+r)^2 + {1/(1+r)^3}*(Q + 2Q/(1+r)+ 3Q/(1+r)^2 + Q/(1+r)^3 + 2Q/(1+r)^4+ 3Q/(1+r)^5 +Q/(1+r)^6 + 2Q/(1+r)^7+ 3Q/(1+r)^8 ….)

P=Q + 2Q/(1+r)+ 3Q/(1+r)^2 +P/(1+r)^3

700000=Q+2Q/(1+0.5%) + 3Q/(1+0.5%)^2 + 700000/(1+0.5%)^3

5.96Q=10395.87

Q=$1744.19