Sandy purchases a perpetuity immediate that makes annualpayments. The first paym

Question

Sandy purchases a perpetuity immediate that makes annualpayments. The first payment is 100, and each payment thereafterincreases b 10. Danny purchases a perpetuity due which makes annualpayments of 180. Using the same effective annual interest rate,i > 0, the present value of both perpetuities areequal. Show that i = .102 Sandy purchases a perpetuity immediate that makes annualpayments. The first payment is 100, and each payment thereafterincreases b 10. Danny purchases a perpetuity due which makes annualpayments of 180. Using the same effective annual interest rate,i > 0, the present value of both perpetuities areequal. Show that i = .102

Explanation / Answer

The first perpetuity is 100V + 110V^2 + 120V^3+....., which may be rewritten as PV1=100K + 10*K^2, where K = V+V^2+V^3....=1/I.

The second perpetuity is PV2=180*(1+K). Setting these equal gives the quadratic equation 10K^2-80K-180=0, or K^2 - 8K - 18 = 0. The positive solution is K = .5*(8+sqrt(136)) = 9.83, and I = 1/K = .102 = answer C.

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