Reading: Annuities
Any retirement-age adult with a mandatory full-time job will likely tell you that the one thing they would change is to have planned for retirement. Does that mean a person falling into this category forgot to put away one large deposit a long time ago? Perhaps, but we often plan for retirement by making smaller payments at regular intervals. This is known as an annuity.
On a smaller scale, let's suppose you decide to put away $100 for each of the next 12 months, at which time you'll take that money to make a purchase. Additionally, suppose that you find a rate of 12% compounded monthly. The time diagram will look like this:
You'll notice that the first payment is actually not made until the
end of month 0 and that the last payment is made at the time of withdrawal. This is known as an annuity-immediate. For simplicity, we assume this is the case. Alternatively, it is possible to have payments made at the beginning of each month, which is known as an annuity-due.
To compute the future, 12-month value of each of these 12 payments, we note that the first payment will earn interest for 11 months, since it wasn't made until the end of the initial month. Thus, for this payment, the future value is:
A = 100(1.01)11
We note that each additional payment has one month fewer of interest-earning capabilities thant the previous payment. In other words, the payment made at the end of month 1 has 10 months to go, or
A = 100(1.01)10
We have to do this for each of the remaining payments, which we show in the table below:
Importantly, notice that the final payment made at the end of month 11 does not earn any interest, since the account is closed out and the balance is withdrawn. To find the total, we sum these values up and get $1,268.25. Since 12 payments of $100 were made, the total comes out to be $1,200 in outflows. The remainder is earned interest, which comes out to be $1,268.25 – $1,200 = $68.25.
Clearly, making these computations is no pleasant matter. Let's do a little bit of manipulation. The sum can be written in the following way:
[latex-display]\displaystyle{A}={100}{\left({1.01}\right)}^{{11}}+{100}{\left({1.10}\right)}^{{10}}+{100}{\left({1.01}\right)}^{{9}}+{100}{\left({1.01}\right)}^{{8}}+\cdot\cdot\cdot+{100}{\left({1.01}\right)}^{{0}}[/latex-display]
Is there a quick way to write this all out? Here's a trick: multiply both sides of the equation by 1.01, so that we get:
[latex-display]\displaystyle{1.01}{A}={1.01}\times{\left[{100}{\left({1.01}\right)}^{{11}}+{100}{\left({1.10}\right)}^{{10}}+{100}{\left({1.01}\right)}^{{9}}+{100}{\left({1.01}\right)}^{{8}}+\cdot\cdot\cdot+{100}{\left({1.01}\right)}^{{0}}\right]}[/latex-display]
Distributing 1.01 results in increasing all exponents by 1:
[latex-display]\displaystyle{1.01}{A}={100}{\left({1.01}\right)}^{{12}}+{100}{\left({1.01}\right)}^{{11}}+{100}{\left({1.01}\right)}^{{10}}+{100}{\left({1.01}\right)}^{{9}}+\cdot\cdot\cdot+{100}{\left({1.01}\right)}^{{1}}[/latex-display]
Notice that
A and 1.01A have right sides that are similar. The only terms that are different are that 1.01A has an extra term with a power of 12 and that A has a term with a power of 0 that 1.01A is missing. Thus, if we take the difference between 1.01A and A, we get:
[latex-display]\displaystyle{1.01}{A}-{A}={100}{\left({1.01}\right)}^{{12}}-{100}{\left({1.01}\right)}^{{0}}[/latex-display]
We need not write out the terms with powers between 1 and 11, since they zero each other out by the operation of subtraction. We now need to solve for A, which is the total future value of all payments:
[latex-display]\displaystyle{.01}{A}={100}{\left({1.01}\right)}^{{12}}-{100}{\left({1}\right)}[/latex-display]
[latex-display]\displaystyle{.01}{A}={100}{\left[{\left({1.01}\right)}^{{12}}-{1}\right]}[/latex-display]
[latex-display]\displaystyle{A}=\frac{{{100}{\left[{\left({1.01}\right)}^{{12}}-{1}\right]}}}{{.01}}[/latex-display]
Since the payment is constant, and the duration of time is irrelevant, since all but the
power and the 0 power terms will cancel out, we can generalize this result:
Using the annuity formula, we have:
[latex-display]\displaystyle{A}=\frac{{{99}{\left[{\left({1.00583}\right)}^{{360}}-{1}\right]}}}{{.00583}}\approx\${120},{681.85}[/latex-display]
She put in a total of
out-of-pocket, so she earned $88,281.85 in interest.
Thus, the future value according to the calculator is $120,777.13. Why the difference? This occurs simply because we rounded the interest rate in our calculations by hand. The calculator keeps track of more significant digits. Our error in calculating by hand is $95.28, which is a relative error of
[latex]\displaystyle\frac{{95.28}}{{1220777.13}}\approx{.07}%[/latex], which is insignificant.
What does the graph of balances after each payment look like? Not surprisingly, the graph is exponential:
Notice that we indicate regular payments in the boxes, but that we additionally have question marks in the inflows and outflows. This is because when the rates change, we have already accumulated an overall balance for the given period. For the first 5 months, we calculate the ending balance:
We have an accumulated balance of $1,520.13 after 5 months. Note that this amount is principle and interest.
Even though the rate changes, we proceed to reinvest the $1,520.13, along with recurring payments of $300 at the new interest rate. This is where it becomes handy to use PV for lump sum amounts and payments for recurring values. First, we re-label the time diagram:
We calculate with these values in TVM Solver:
Relabeling the time diagram gives:
Similarly to before, the current reinvested balance at the new rate of 7% for 240 months is $6,747.40 with regular recurring payments of $300. We adjust P/Y and C/Y to be 365 for daily compounding, and must multiply 365 by 20 to get the number of daily compound periods in 20 years. This gives:
Updating the time diagram, we have that the ending balance of the retirement account will be $4,805,712.62.
CAUTION: It is simple to overlook the fact that payments are based on deposits per compounding period. In the last 240 months, it was the case that compounding was taking place daily. Thus, the $300 payment is also assumed to be made on a daily basis! So, don't assume that it is this easy to be a multi-millionaire!
It is often useful or necessary to determine the amount of interest earned on an investment account. Since money accumulates with each additional payment for
each payment made prior, the amount of earned interest grows exponentially. Since every payment is assumed to have been made at the end of the month, we know that only the accumulation of payments and interest prior to that most recent payment earns interest.
For example, in the last example we noticed that after 5 months the balance was $1,520.13. This means that interest for the next month, month 6, will be
[latex]\displaystyle\frac{{.15}}{{12}}[/latex], multiplied by the account balance to get in interest dollars.
We can use the TVM Solver to expedite this. Since, month 6 is actually month 1 of a new calculation, we will have to reference this as month 1. We first enter the FV calculation:
This step is CRUCIAL! Without it, the interest reported will not be valid because the calculator relies on the current financial computation. Now, we exit out of TVM Solver by pressing 2nd, then MODE.
Putting $640,126.48 into the safe account will yield
[latex]\displaystyle\frac{{.03}}{{12}}[/latex]. Thus, each month you will earn .0025(640126.48) = $1,600.32, assuming you remove the interest each month, thus leaving only the $640,126.48 that you had at the close of the retirement account.
Milos Podmanik, By the Numbers, "The Mathematics of Finances," licensed under a CC BY-NC-SA 3.0 license. MathIsGreatFun, "4 2 P1 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P2 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P3 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P4 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P5 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P6 MAT217," licensed under a Standard YouTube license.
You'll notice that the first payment is actually not made until the
end of month 0 and that the last payment is made at the time of withdrawal. This is known as an annuity-immediate. For simplicity, we assume this is the case. Alternatively, it is possible to have payments made at the beginning of each month, which is known as an annuity-due.
To compute the future, 12-month value of each of these 12 payments, we note that the first payment will earn interest for 11 months, since it wasn't made until the end of the initial month. Thus, for this payment, the future value is:
A = 100(1.01)11
We note that each additional payment has one month fewer of interest-earning capabilities thant the previous payment. In other words, the payment made at the end of month 1 has 10 months to go, or
A = 100(1.01)10
We have to do this for each of the remaining payments, which we show in the table below:
| End of Month | Balance |
| 0 | 100(1.01)11 = 111.57 |
| 1 | 100(1.01)10 = 110.46 |
| 2 | 100(1.01)9 = 109.37 |
| 3 | 100(1.01)8 = 108.29 |
| 4 | 100(1.01)7 = 107.21 |
| 5 | 100(1.01)6 = 106.15 |
| 6 | 100(1.01)5 = 105.1 |
| 7 | 100(1.01)4 = 104.06 |
| 8 | 100(1.01)3 = 103.03 |
| 9 | 100(1.01)2 = 102.01 |
| 10 | 100(1.01)1 = 101 |
| 11 | 100(1.01)0 = 100 |
Formula for Future Value of an Annuity with Regular Payments
For an annuity with APR r, compounded n times per year for t years, lasting n × t periods, and with a constant payment, PMT, that is made each compounding period, then the resulting balance, A, is given by [latex-display]\displaystyle{A}=\frac{{{P}{M}{T}{\left[{\left({1}+\frac{{r}}{{n}}\right)}^{{{n}\times{t}}}-{1}\right]}}}{{\frac{{r}}{{n}}}}[/latex-display]Example 1
A company offers its employees a 401(k) plan and offers to match its employees' contributions at 10% for up to a $600 employer contribution per year. An employee decides to use her 401(k). She makes payments of $90 per month to an account that pays an average of 7%, compounded monthly. If she retires in 30 years, how much can she retire on from this account?Solution
The employee's contributions will be matched by .10($90) = $9 per month, which will only be $108 per year and so will be well below the allowable contribution. Thus, the employee will have $99 going to her 401(k) fund every month at a monthly rate of[latex]\displaystyle\frac{{.07}}{{12}}\approx{.00583}[/latex] months. On a time diagram, we have:
Using the annuity formula, we have:
[latex-display]\displaystyle{A}=\frac{{{99}{\left[{\left({1.00583}\right)}^{{360}}-{1}\right]}}}{{.00583}}\approx\${120},{681.85}[/latex-display]
She put in a total of
out-of-pocket, so she earned $88,281.85 in interest.
Example 2
Confirm your answer from Example 1 by using TVM Solver.Solution
We enter all the values. Since there is no lump-sum deposit, we put 0 for PV. To account for regular payments, we put –99 for PMT to represent her regular monthly outflow. After calculating, we get:
Thus, the future value according to the calculator is $120,777.13. Why the difference? This occurs simply because we rounded the interest rate in our calculations by hand. The calculator keeps track of more significant digits. Our error in calculating by hand is $95.28, which is a relative error of
[latex]\displaystyle\frac{{95.28}}{{1220777.13}}\approx{.07}%[/latex], which is insignificant.
What does the graph of balances after each payment look like? Not surprisingly, the graph is exponential:
Example 3
When planning for retirement and opening up a 401(k) or 403(b), which are simply names for annuities based on corresponding IRS tax codes, you have the option of choosing how aggressive you want your fund to be. (A 401(k) is typically seen within private sector companies. A 401(b) is specific to government employers, such as the military or public educational institutions.) The riskier you are willing to be, the more of the potential return, but the higher the risk of loss and major fluctuations. Suppose you open a 401(k) with your current employer and decide to contribute $300 each month. For the first 5 months, the average return is 8%, and then jumps to 15% for the next 15 months. For the 20 years following, the rate remains fairly stable at about 7%. Assume the first two rates are compounded monthly and that the last is compounded daily. How much will be in your 401(k) at that time?Solution
We must evaluate three separate time periods—one for each of the three rate periods. Following is the time diagram we consider:
Notice that we indicate regular payments in the boxes, but that we additionally have question marks in the inflows and outflows. This is because when the rates change, we have already accumulated an overall balance for the given period. For the first 5 months, we calculate the ending balance:
We have an accumulated balance of $1,520.13 after 5 months. Note that this amount is principle and interest.
Even though the rate changes, we proceed to reinvest the $1,520.13, along with recurring payments of $300 at the new interest rate. This is where it becomes handy to use PV for lump sum amounts and payments for recurring values. First, we re-label the time diagram:
We calculate with these values in TVM Solver:
Relabeling the time diagram gives:
Similarly to before, the current reinvested balance at the new rate of 7% for 240 months is $6,747.40 with regular recurring payments of $300. We adjust P/Y and C/Y to be 365 for daily compounding, and must multiply 365 by 20 to get the number of daily compound periods in 20 years. This gives:
Updating the time diagram, we have that the ending balance of the retirement account will be $4,805,712.62.
CAUTION: It is simple to overlook the fact that payments are based on deposits per compounding period. In the last 240 months, it was the case that compounding was taking place daily. Thus, the $300 payment is also assumed to be made on a daily basis! So, don't assume that it is this easy to be a multi-millionaire!
It is often useful or necessary to determine the amount of interest earned on an investment account. Since money accumulates with each additional payment for
each payment made prior, the amount of earned interest grows exponentially. Since every payment is assumed to have been made at the end of the month, we know that only the accumulation of payments and interest prior to that most recent payment earns interest.
For example, in the last example we noticed that after 5 months the balance was $1,520.13. This means that interest for the next month, month 6, will be
[latex]\displaystyle\frac{{.15}}{{12}}[/latex], multiplied by the account balance to get in interest dollars.
We can use the TVM Solver to expedite this. Since, month 6 is actually month 1 of a new calculation, we will have to reference this as month 1. We first enter the FV calculation:
This step is CRUCIAL! Without it, the interest reported will not be valid because the calculator relies on the current financial computation. Now, we exit out of TVM Solver by pressing 2nd, then MODE.
Calculator Clinic
Calculating interest on an annuity between time t1 and t2:- Press
.
- Go to 1: Finance
- Scroll down to A: ΣInt and press ENTER. (Σ is the Greek letter sigma. In mathematics, this symbol means "sum" or "add up.")
- A parenthesis is automatically entered. In the parenthesis, we specify the two points in time during which we would like the interest. The format of your inputs will be ΣInt(t1,t2). If you want to find the interest during a single period, t1 and t2 will be the same value. Note that the first number must be 1 or greater. Think of these as the payment number. There is no such thing as the 0th payment!
Example 4
Suppose you decide today that you will put away $400 per month for the next 35 years, at which point you will retire. You are confident you will be able to find a rate that averages around 6.5% for the duration of the investment. At the time you retire, you transfer your money to a safe account that is invested namely in bonds and savings. This account returns about 3% per year, compounded monthly. How much money will you be able to withdraw each month so that you never touch the accumulation of money you made over the 35 years?Solution
The only way to never touch the balance of an account is to survive entirely off interest. We first calculate the future value of this investment:
Putting $640,126.48 into the safe account will yield
[latex]\displaystyle\frac{{.03}}{{12}}[/latex]. Thus, each month you will earn .0025(640126.48) = $1,600.32, assuming you remove the interest each month, thus leaving only the $640,126.48 that you had at the close of the retirement account.
Practice Problems
- Give possible stories for the following scenarios. Be sure to clearly describe the investment taking place.
- Suppose you open an IRA. You have an automatic withdrawal of $250 each month from your checking account into the IRA, which pays an average of 9.1% compounded monthly. Ignoring fees and taxes, answer the following questions.
- How much will you have saved up at retirement in 35 years?
- How long would it take you to save up $3 million?
- Suppose you were able to find a rate that earns you $3 million in 35 years. What would the interest rate have to be to earn this amount exactly?
- 3.Chevy is thinking about purchasing a car next year. He prefers to pay in cash whenever possible so as not to take out a loan and incur interest. The car he is considering is currently running for $26,995 (with a sunroof for his Jack Russell Terrier!). Having done his homework, he knows that this particular make and model depreciates by 24% over the first year. He is able to find a 12-month CD at an APY of 3.4%, which is the result of monthly compounding. What would he have to put away each month in order to pay cash for the depreciated car next year?
- Suppose you open a 401(k) through your employer. You decide to use a recent bonus of $1000 given to you to get the account started and you plan on making regular deposits of $150 per month. Suppose you are able to get a rate of 7.88% compounded monthly.
- What will you be able to accumulate in 20 years?
- How much interest will be accumulated during the last year?
- How much larger would your payment need to be in order to reach a goal of $1,000,000 by retirement? Assume that you will still make the lump-sum deposit of $1000.
- Suppose you are set on making the $150 monthly payment along with your initial $1000 lump-sum deposit.
- What would your interest rate need to be to accomplish your $1,000,000 goal at retirement?
- Keeping the original rate, how much longer would you need to make $150 payments in order to accomplish your goal of $1,000,000 by retirement?
- 5.Homer wants to retire by the time he is 50. Today he turned 18. Homer wants to be able to withdraw $3,000 each month (for as long as he lives) once he closes his annuity at the time of retirement. Currently, he is earning an average of 8.05% compounded monthly. When he retires, he puts the balance into a safer, less volatile investment account that only earns 2% compounded monthly. He does not make additional payments to the account after retirement.
- What accumulation of money should he have in his annuity at retirement for the above scenario to be possible? (Keep in mind that in order to withdraw payments as long as he lives, he must live off interest alone.)
- What would be his required monthly annuity payment for the accumulation of money to occur?
- How much would he have put in out of his pocket over the duration of his annuity?
- How much interest will he have earned on the annuity?
- As people say, "hindsight is always 20/20." This is to mean that looking at the past makes it seem as if the events that occurred should have been predictable. Suppose that two twin sisters, Darla and Thelma, are of retirement age. Both took different routes throughout their annuity investing careers. They each put away $450 per month since they were 20 and are now both 61. Darla started out with an account that paid 11.06% for the first 18 months, 1% for the next 30 months, and then averaged 6.01% for the remaining time. Thelma was a bit more conservative and went with an account that paid 5.2% for the first 10 months, but then jumped to 16.1% for the next 50 months, and then dropped to 5.4% for the remainder of the time. Who is better off now that they're both retired, and by how much?
Milos Podmanik, By the Numbers, "The Mathematics of Finances," licensed under a CC BY-NC-SA 3.0 license. MathIsGreatFun, "4 2 P1 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P2 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P3 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P4 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P5 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 2 P6 MAT217," licensed under a Standard YouTube license.
