5.7 Deferred Annuities

You borrow $10,000 for your business to purchase new computer software.  The loan agreement requires you make quarterly payments for 5 years at 3% compounded quarterly.  The first loan payment is due in 2 years.  Calculate the size of the quarterly loan payments.

Solution:

Step 1:  Draw the timeline for the deferred annuity.  Because the payment setting is not specified, the payments occur at the end of the payment interval.

Some notes about the timeline.

1. The $10,000 is the present value at the start of the interest period.
2. There is a 2 year time gap between the start of the interest period and first payment.  Because the payments are at the end of the payment interval, the first payment does NOT occur at the same time as the present value of the annuity, which occurs at the beginning of the first payment interval.
3. Because the payments are at the end of the payment interval, the present value of the annuity occurs one payment interval before the first payment.  The payments are quarterly (or 3 months), so the present value of the annuity occurs 3 months before the first payment.
4. The $10,000 at the start of the interest period must be moved (with a future value calculation) to the location of the present value of the annuity.  Because the present value of the annuity occurs 3 months before the first payment and it is 2 years from the start of the interest period to the first payment, the $10,000 must move 1 year and 9 months (2 years minus 3 months) to get the location of the present value of the annuity.
5. The payments for the annuity last for 5 years.

Step 2:  Calculate the future value of the $10,000.  As noted above, the $10,000 must move to the location of the present value of the annuity, which is 1 year and 9 months from the start of the timeline.

[latex]FV=\$10,536.961...[/latex]

Step 3:  Calculate the payment for the annuity.  The future value calculated in the previous step becomes the present value for the annuity:  [latex]PV=\$10,536.961...[/latex]

[latex]PMT=\$569.32[/latex]

Your quarterly loan payments are $569.32.

You borrow $10,000 for your business to purchase new computer software.  The loan agreement requires you make quarterly payments for 5 years at 3% compounded quarterly.  The first loan payment is due in 2 years.  Calculate the size of the quarterly loan payments.

Solution:

Step 1:  Draw the timeline for the deferred annuity.  Treat the annuity as an annuity due with the payments at the beginning of the interval.

Some notes about the timeline.

1. The $10,000 is the present value at the start of the interest period.
2. Because the payments are at the beginning of the payment interval, the present value of the annuity and the first payment occur at the same time, at the start of the annuity.  Note that this eliminates the gap between the present value of the annuity and the first payment seen in the previous example.
3. The $10,000 at the start of the interest period must be moved (with a future value calculation) to the location of the present value of the annuity.  Because the present value of the annuity occurs at the time of the first payment, the $10,000 must move 2 years to get the location of the present value of the annuity.
4. The payments for the annuity last for 5 years.

Step 2:  Calculate the future value of the $10,000.  As noted above, the $10,000 must move to the location of the present value of the annuity, which is 2 years from the start of the timeline.

[latex]FV=\$10,615.988...[/latex]

Step 3:  Calculate the payment for the annuity.  The future value calculated in the previous step becomes the present value for the annuity:  [latex]PV=\$10,615.988...[/latex]

[latex]PMT=\$569.32[/latex]

Your quarterly loan payments are $569.32.

You want to invest some money today to help fund your retirement.  When you retire, you want to receive $5,000 at the end of every month for 15 years.  You want to receive your first payment in 20 years.  If your retirement fund earns interest at 5% compounded semi-annually, how much money do you need to invest?

Solution:

Step 1:  Draw the timeline for the deferred annuity.  Using the trick identified above, treat the annuity as an annuity due with the payments at the beginning of the interval (despite the wording in the question that identifies the payments at the end of the interval).

Some notes about the timeline.

1. The present value at the start of the interest period is unknown.
2. Because the payments are at the beginning of the payment interval, the present value of the annuity and the first payment occur at the same time, at the start of the annuity.  There is a 20 year gap from the start of the interest period and the present value of annuity/future value of interest/first payment.
3. The $5,000 payments for the annuity last for 15 years.
4. The present value of the annuity becomes the future value of the interest period.  Then, the future value of the interest period will be moved back 20 years to the start of the interest period.

Step 2:  Calculate the present value of the annuity.

PMT Setting	BGN
N	[latex]12 \times 15=180[/latex]
PV	?
FV	[latex]0[/latex]
PMT	[latex]5,000[/latex]
I/Y	[latex]5[/latex]
P/Y	[latex]12[/latex]
C/Y	[latex]2[/latex]

[latex]PV=\$637,034.376...[/latex]

Step 3:  Calculate the present value for the interest period.  The present value calculated in the previous step becomes the future value for the interest period:  [latex]FV=\$637,034.376...[/latex]

N	[latex]2 \times 20=40[/latex]
PV	?
FV	[latex]637,034.376...[/latex]
PMT	[latex]0[/latex]
I/Y	[latex]5[/latex]
P/Y	[latex]2[/latex]
C/Y	[latex]2[/latex]

[latex]PV=\$237,251.11[/latex]

You need to invest $237,251.11 today to fund your retirement.

Bashir has $50,000 to invest.  Eventually, Bashir wants the annuity to pay him $2,500 at the beginning of every month for 10 years. When can Bashir receive his first payment if the investment earns 8.25% compounded quarterly?

Solution:

Step 1:  Draw the timeline for the deferred annuity.

Some notes about the timeline.

1. The $50,000 is at the start of the interest period.
2. Because the payments are at the beginning of the payment interval, the present value of the annuity and the first payment occur at the same time, at the start of the annuity.
3. The time from the start of the interest period to the present value of the annuity/future value of the interest period/first payment is unknown.
4. The $2,500 payments for the annuity last for 10 years.

Step 2:  Calculate the present value of the annuity.

[latex]PV=\$205,719.154...[/latex]

Step 3:  Calculate the number of compounding periods for the interest period and convert to years and months.  The present value calculated in the previous step becomes the future value for the interest period:  [latex]FV=\$205,719.154...[/latex]

[latex]N=69.286...[/latex]

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{N}{C/Y} \\ & = & \frac{69.286...}{4} \\ & = & 17.321... \\ & \rightarrow & 17 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.321... \times 12 \\ & = & 3.858... \\ & \rightarrow & 4 \mbox{ months}  \end{eqnarray*}[/latex]

Bashir can receive his first payment 17 years and 4 months after he invests the $50,000.

Emile received a $25,000 one-time bonus from his employer today, and he immediately invested it at 3.25% compounded semi-annually. Fourteen years from now, he plans to withdraw $2,300 every quarter to use as his retirement income. How long can Emile receive these payments?  

Solution:

Step 1:  Draw the timeline for the deferred annuity. Treat the annuity as an annuity due with the payments at the beginning of the interval (despite the wording in the question that identifies the payments at the end of the interval).

Some notes about the timeline.

1. The $25,000 is at the start of the interest period.
2. Because the payments are at the beginning of the payment interval, the present value of the annuity and the first payment occur at the same time, at the start of the annuity.  There is a 14 year gap from the start of the interest period and the present value of annuity/future value of interest/first payment.
3. The $2,300 payments for the annuity last for an unknown number of years.

Step 2:  Calculate the future value of the $25,000.

[latex]FV=\$39,260.484...[/latex]

Step 3:  Calculate the number of payment periods for the annuity and convert to years and months.  The future value calculated in the previous step becomes the present value for the annuity:  [latex]PV=\$39,260.484...[/latex]

[latex]N=18.284...\rightarrow 19 \mbox{ payments}[/latex]

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{19}{4} \\ & = & 4.75 \\ & \rightarrow & 4 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.75 \times 12 \\ & = & 9  \end{eqnarray*}[/latex]

Emile can receive payments for 4 years and 9 months.