Question

What minimum initial amount will sustain a 25-year annuity paying $1000 at the end of each year if the initial amount can be invested to earn:

a) 6% compounded annually?
b) 6% compounded semiannually?
c) 6% compounded quarterly?
d) 6% compounded monthly?

Answer 1

Given in the question,                                                         

Annual Instalment (P) = 1000

Rate of interest(i) = 6% = 0.06 (Semi-annually 0.06/2 = 0.03, Quarterly = 0.06/4 = 0.015, Monthly = 0.06/12 = 0.005)

Period(n) = 25 (Semi-annually 25 X 2= 50, Quarterly = 25 X 4=100, Monthly = 25 X 12=300)

Initial Amount (V) = ?

a) V = P/i[1-(1+i)^-n]

               = 1000/0.06[1-(1+0.06)^-25]                       (Substituting the value from above)                    

               = 1000/0.06[1-(1.06)^-25]

               = 1000/0.06[1-(1/1.06)²⁵]

               = 1000/0.06[1-(0.9434)²⁵]

              = 1000/0.06[1-0.233]

              = 1000/0.06[0.767]

              = $12,783.33

b) V = P/i[1-(1+i)^-n]

               = 1000/0.03[1-(1+0.03)^-50]                 (Substituting the value from above)                    

               = 1000/0.03[1-(1.03)^-50]

               = 1000/0.03[1-(1/1.03)⁵⁰]

               = 1000/0.03[1-(0.9709)⁵⁰]

              = 1000/0.03[1-0.2284]

              = 1000/0.03[0.7716]

              = $25,720

c) V = P/i[1-(1+i)^-n]

               = 1000/0.015[1-(1+0.015)^-100]                          (Substituting the value from above)                    

               = 1000/0.015[1-(1.015)^-100]

               = 1000/0.015[1-(1/1.015)¹⁰⁰]

               = 1000/0.015[1-(0.9852)¹⁰⁰]

              = 1000/0.015[1-0.2251]

              = 1000/0.015[0.7749]

              = $51,660

d) V = P/i[1-(1+i)^-n]

               = 1000/0.005[1-(1+0.005)^-300]                       (Substituting the value from above)                    

               = 1000/0.005[1-(1.005)^-300]

               = 1000/0.005[1-(1/1.005)³⁰⁰]

               = 1000/0.005[1-(0.995)³⁰⁰]

              = 1000/0.005[1-0.2223]

              = 1000/0.005[0.7777]

              = $155,540

Please give likes for the work. Thanks & Regards.