Question

An investor invests $53,030.00 and receives an annuity of $7,000.00 at the end of each year for 12 years and an additional payment of $15,000.00 at the end of the 13th year. Each time he gets a $7,000.00 payment, he immediately deposits $4,000.00 in a savings account that earns 9%. Find the annual yield received by the investor over the 13 year period.

Please provide a mathematical explanation to the answer if possible. I am more concerned with semantics.

Answer 1

The annual yield would be equal to IRR (Internal Rate of Return) of the cash flows.

For IRR calculation, all values with signs intact (-ive for cash outflow) are put in sequencial order & IRR function is used selelcting all the values from 1st to last.

We assume that $4000 savings put in the bank gives a 9% return per annum. The savings are getting added by $4000 every year and interest is getting accumulated to be withdrawn at the end of 13 years.

We would calculate cash flows & then IRR function in excel to calculate the yield

The second column is the cash flow from annuity of $7000 per year.

The third column is the cash flow from annuity of $7000 and the accumulated value of $4000 put in savings account, growing at 9% pa, drawn at the end of 13 years, which equals $87813.54

The yield or return is 15.474%

Year	Cashflow from Annuity	Cashflow from Annuity including Cash Flow from Bank Deposit
0	(53,030.00)	(53,030.00)	This value is a Cash Outflow & Therefore we use a -ive sign. It is essential
1	7,000.00	7,000.00
2	7,000.00	7,000.00
3	7,000.00	7,000.00
4	7,000.00	7,000.00
5	7,000.00	7,000.00
6	7,000.00	7,000.00
7	7,000.00	7,000.00
8	7,000.00	7,000.00
9	7,000.00	7,000.00
10	7,000.00	7,000.00
11	7,000.00	7,000.00
12	7,000.00	7,000.00
13	15,000.00	102,813.54	Additional Cash flow from $4000 pa in savings account, which is $87813.54
IRR	9.658%	15.474%

The working of $4000 put in savings bank and earning 9% interest works like below.

The final value in year 13 includes 12 payment periods and 12 interest periods = $87813.54

Year	Opening Balance A	Deposits in Savings A/c (B)	Interest Earned C = (A+B) x 9%	Closing Balance (D) = A + B + C	Explanation
2	4,000.00	-	360.00	4,360.00	$7000 earned at the end of year 1, $4000 put in the savings account. We start year 2 with an opening balance of $4000, earn interest and close the year2 at $4360
3	4,360.00	4,000.00	752.40	9,112.40	$7000 earned at the end of year 2, $4000 put in the savings account. We start year3 with an opening balance of $4360, earn interest and close the year3 at $9112.40
4	9,112.40	4,000.00	1,180.12	14,292.52	Same as above
5	14,292.52	4,000.00	1,646.33	19,938.84
6	19,938.84	4,000.00	2,154.50	26,093.34
7	26,093.34	4,000.00	2,708.40	32,801.74
8	32,801.74	4,000.00	3,312.16	40,113.90
9	40,113.90	4,000.00	3,970.25	48,084.15
10	48,084.15	4,000.00	4,687.57	56,771.72
11	56,771.72	4,000.00	5,469.45	66,241.17
12	66,241.17	4,000.00	6,321.71	76,562.88
13	76,562.88	4,000.00	7,250.66	87,813.54	This Closing balance is 12 periods of deposits and 12 periods of interest earned