Question

Christina purchased an annuity from an insurance company. Christina paid $156,300 for the annuity, and in exchange she will receive $20,100 per year for the next 10 years.

How much of the first $20,100 payment should Christina exclude from gross income?_________

Please explain how you got each number in your solution and/or where the number comes from.

Answer 1

In this annuity we need to find out the internal rate of return i.e., interest rate payable by the insurance company.

To find the interest rate we need to use trial and error method for best results.

lets take an interest rate of 5% and calculate the annuity value

Year	1	2	3	4	5	6	7	8	9	10	Total
PV factor for 5%	0.952381	0.907029	0.863838	0.822702	0.783526	0.746215	0.710681	0.676839	0.644609	0.613913	7.721735

[PV factor formulae is {1/(1+r)}ⁿ-1 ] here 'r' is interest rate and 'n' is no. of years.

Annuity value = $20,100 * 7.721735 = $155,206.87 (Rounded Off)

Since the total amount invested is $156,300 which is higher than the annuity value claculated above we need to take interest rate lower than the above rate.

Now, lets take interest rate of 4% and check it,

Year	1	2	3	4	5	6	7	8	9	10	Total
PV factor for 4%	0.961538	0.924556	0.888996	0.854804	0.821927	0.790315	0.759918	0.73069	0.702587	0.675564	8.110896

[PV factor formulae is {1/(1+r)}ⁿ-1 ] here 'r' is interest rate and 'n' is no. of years.

Annuity value = $20,100 * 8.110896 = $163,029.01 (Rounded Off)

Therefore the amount invested lies in between the interest rates 4 % and 5%

Change in annuity values calculated above for 1%(5%-4%) change = $163,029.01 - $155,206.87 = $7,822.13

Change required in annuity values = $163,029.01 - $156,300 = $6,729.01

Interest rate = 4% + (6729.01/7822.13)*(1) = 4.86%

Therefore insurance company is paying interest @ 4.86% P.a

The stream of cashflows on the amount invested is as follows:

Year	Amount Invested (Opening balance) (a)	Amount received per year (b)	Interest Amount received (c ) = (a)*4.86%	Invested amount received (d) = (b)-(c )	Closing balance (e ) = (a) - (d)
1	$              156,300.00	$             20,100.00	$               7,596.18	$               12,503.82	$ 143,796.18
2	$              143,796.18	$             20,100.00	$               6,988.49	$               13,111.51	$ 130,684.67
3	$              130,684.67	$             20,100.00	$               6,351.28	$               13,748.72	$ 116,935.95
4	$              116,935.95	$             20,100.00	$               5,683.09	$               14,416.91	$ 102,519.04
5	$              102,519.04	$             20,100.00	$               4,982.43	$               15,117.57	$    87,401.46
6	$                 87,401.46	$             20,100.00	$               4,247.71	$               15,852.29	$    71,549.17
7	$                 71,549.17	$             20,100.00	$               3,477.29	$               16,622.71	$    54,926.46
8	$                 54,926.46	$             20,100.00	$               2,669.43	$               17,430.57	$    37,495.89
9	$                 37,495.89	$             20,100.00	$               1,822.30	$               18,277.70	$    19,218.19
10	$                 19,218.19	$             20,100.00	$                   934.00	$               19,218.19	$ 0.00

(Note: for last year it is rounded off to closing balance as zero)

Therefore for the first payment of $20,100, christina should exclude $12,503.82 from gross income since it is the amount invested by her and only the interest income is her income for the year.

Thank you.