Question

Q3) Maria Garcia has an (arithmetic) annuity immediate that will make 10 annual payments. The first payment is P = $1000 and payment increases by Q = $100 from the payment before. The effective annual interest rate is i = 2.75%.

a) Compute both the present and future value of Maria Garcia’s annuity by showing it is equivalent to the following 2 annuities:

• Annuity A: Level pay, $900 for 10 years
• Annuity B: Arithmetic increasing annuity immediate: starts today, lasts 10 years, first payment P’ = $100, increment Q’ = $100.

b) Write down formulas for the PV and FV of any similar arithmetic annuity immediate with first payment P, increment Q, n periods, and effective rate per period i.

Answer 1

a) Present value of the Annuity

= 1000/1.0257+ 1100/1.0275^2+1200/1.0275^3+......+1900/1.0275^10

=(900/1.0275+100/1.0275)+(900/1.0275^2+200/1.0275^2)+.......+ (900/1.0275^10+1000/1.0275^10)

=(900/1.0275+900/1.0275^2+.....+900/1.0275^10) + (100/1.0275+200/1.0275^2+....+1000/1.0275^10)

=present value of Level annuity of 900 (Annuity A) + Present value of arithmetic annuity starting with 100 and annual increment of 100 (Annuity B)

=900/0.0275*(1-1/1.0275^10)+ 100/0.0275*(1-1/1.0275^10) + 100*((1-1/1.0275^10)/0.0275-10/1.0275^10)/0.0275

=7776.07+3694.90+864.01

=$12334.97

Future value = 12334.97 *1.0275^10 = $16179.18

b) Formula for PV of arithmetic annuity with first payment P , increment Q, n periods and effective rate i

= P/i*(1-1/(1+i)^n) + Q* ((1+i)^n-ni-1)/(1+i)^n*i^2)

Formula for FV of arithmetic annuity with first payment P , increment Q, n periods and effective rate i

= P/i*((1+i)^n-1) + Q* ((1+i)^n-ni-1)/i^2