Question

Find the present value of a 20 year annuity due where payments are $1, 000 at the beginning of the first year, third year, etc. and payments are $1, 500 at the beginning of the second year, fourth year, etc. Here effective annual interest is 5%. Hint: Draw a time diagram!!!

Answer 1

Answer :

As per the information given in the question,

Annuity payments in the 1st year, 3rd year and so on is $1,000

Annuity payments in the 2nd year, 4th year and so on is $1,500

Period of payments (N) = 20 years

Effective annual interest (i) = 5%

The annuity payment cycle continues as in the 1st year is = $1,000 and in the 2nd year is = $1,500 and this annuity payment cycle repeat till 20 years.

Therefore, the growth of annual payments from 1st year to 2nd year (G) = 1,500 - 1,000 = $500

Annuity payments in the 1st year is (A1) = $1,000

payment cycle (N) = 2 years

Equal uniform annuities (A) = A1 + G(A/G,i,N)

Equal uniform annuities (A) = 1,000 + 500 ( A/G,5%,2 )

= 1,000 + 500 (0.4878)

= 1,000 + 243.9

= $1243.9

Present worth (PW) of annuities = A(P/A,i,N)

= $1243.9 (P/A,5%,20)

Present worth (PW) of annuities = $1243.9 (12.46221)

= $15,501.74

Therefore, the Present value of annuities = $15,501.74