Essential Mathematics of Money: A Self-Teaching Guide by Hill Tim

Author:Hill, Tim [Hill, Tim]
Language: eng
Format: mobi
Tags: Mathematics
Publisher: Questing Vole Press
Published: 2018-06-19T16:00:00+00:00

For postponing to give a higher PV, we need

Rearranging this inequality leads to 0.039725xn < 0.070325x10 − 0.0306 = 0.01257; that is, xn < 0.3165, or n > 23.58 years. Alice needs to live to about 84 for postponing to give a higher PV.

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To compensate for inflation, you might want your annual annuity income to rise steadily. The following derivation shows the adjustment that’s necessary to recalculate the new payments.

We’ve seen that the formula for calculating fixed annuity payments is the same as that used for fixed mortgage repayments. Suppose that the insurance company will earn interest at rate γ > 0 per year, whereas you want the annuity to increase at rate β > 0 per year. We can adapt the argument leading to equation (4.2) to cover this case, finding the amount of the (initial) annual annuity payment to you, assuming γ ≠ β. By fixing γ, and letting β approach γ, we can use L’Hôpital’s Rule to find the initial annuity amount when γ = β.

Let R be the initial annuity amount, and Cn be the sum the insurance company has left at the end of year n. Then

C0 = C,

C1 = C0(1 + γ) − R,

C2 = C1(1 + γ) − R(1 + β),

and, for N ≥ 3,

CN = CN − 1(1 + γ) − R(1 + β)N − 1,

giving

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