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Determine the present value of an age- $\newline$ $90$ lifetime annuity with payments of $80 at the end of each year with an interest rate of$ $8$ \% $per year if known$ \ell_{ $90$ } = $1000$ $;$ \ell_{ $91$ } = $850$ $;$ \ell_{ $92$ } = $520$ $;$ \ell_{ $93$ } = $300$ $;$ \ell_{ $94$ } = $0$ $.

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Exponential growth and decay: word problems

Full solution

Q. Determine the present value of an age-

\newline

90

lifetime annuity with payments of

80 at the end of each year with an interest rate of

8

\%

per year if known

\ell_{

90

} =

1000

;

\ell_{

91

} =

850

;

\ell_{

92

} =

520

;

\ell_{

93

} =

300

;

\ell_{

94

} =

0

$.

Calculate PV $90$ : Calculate the present value of the first payment at age $90$ . $\newline$ $\text{PV}90 = \frac{80}{(1 + 0.08)^1}$ $\newline$ $\text{PV}90 = \frac{80}{1.08}$ $\newline$ $\text{PV}90 = 74.07$
Calculate PV $91$ : Calculate the present value of the second payment at age $91$ . $\newline$ $PV_{91} = \frac{80 \times 850}{1000} / (1 + 0.08)^2$ $\newline$ $PV_{91} = \frac{80 \times 0.85}{1.1664}$ $\newline$ $PV_{91} = \frac{68}{1.1664}$ $\newline$ $PV_{91} = 58.29$
Calculate PV $92$ : Calculate the present value of the third payment at age $92$ . $\newline$ $PV_{92} = \frac{(80 \times 520/1000)}{(1 + 0.08)^3}$ $\newline$ $PV_{92} = \frac{(80 \times 0.52)}{1.259712}$ $\newline$ $PV_{92} = \frac{41.6}{1.259712}$ $\newline$ $PV_{92} = 33.02$
Calculate PV $93$ : Calculate the present value of the fourth payment at age $93$ . $\newline$ $PV_{93} = \frac{80 \times 300}{1000} / (1 + 0.08)^4$ $\newline$ $PV_{93} = \frac{80 \times 0.3}{1.36048896}$ $\newline$ $PV_{93} = \frac{24}{1.36048896}$ $\newline$ $PV_{93} = 17.64$
Calculate $PV_{94}$ : Calculate the present value of the fifth payment at age $94$ . Since $l_{94} = 0$ , there are no survivors to make payments to, so the present value is $0$ . $PV_{94} = 0$
Add PV for Total: Add up the present values to get the total present value of the annuity. $\newline$ Total $PV = PV_{90} + PV_{91} + PV_{92} + PV_{93} + PV_{94}$ $\newline$ Total $PV = 74.07 + 58.29 + 33.02 + 17.64 + 0$ $\newline$ Total $PV = 183.02$

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