5.3: Savings Annuities
- Page ID
- 106255
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Most of us aren’t able to put a large sum of money in the bank today for retirement in the future. Instead, we save for the future by depositing a smaller amount of money from each paycheck into a type of account called an annuity. Then, upon retirement, we begin to withdraw from the annuity for income.
In this section we will consider one type of way to save for retirement or some other large purchase called a savings annuity. A typical example of this type of annuity account is a retirement plan or an IRA plan into which someone makes a regular deposit from each paycheck. This type of annuity is also known as a savings plan. In the next section we will look at how the funds in an annuity will be paid out at retirement.
A savings annuity is used to save money by depositing equal amounts periodically so the money deposited and the interest accumulates to some value for future use.
Mathematical formulas for a savings annuity are given below. As in the last section, we will emphasize using the TVM Solver application but will calculate each example with a formula as well.
\(A = \dfrac{d\left[\left(1+\dfrac{r}{n}\right)^{n t}-1\right]}{\left(\dfrac{r}{n}\right)}\)
\(d = \dfrac{A\left(\dfrac{r}{n}\right)}{\left[\left(1+\dfrac{r}{n}\right)^{n t}-1\right]}\)
where
- \(A\) is the balance in the account after \(t\) years
- \(d\) is the regular deposit amount (or payment) each month, quarter, year, etc.
- \(r\) is the annual interest rate (APR) in decimal form
- \(n\) is the number of compounding periods in one year
- \(t\) is the number of years
If the compounding frequency is not stated, assume there is the same number of compounds in a year as there are deposits made in a year.
- If you make your deposits every year, then use yearly compounding: \(n=1\).
- If you make your deposits every quarter, then use quarterly compounding: \(n=4\).
- If you make your deposits every month, then use monthly compounding: \(n=12\).
- If you make your deposits every week, then use weekly compounding: \(n=52\).
- And so on.
A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA at 6% interest, what is the balance in your account after 20 years? How much interest will you have earned? What percentage of the balance is interest?
Solution
To use a formula to solve this example, use the formula solved for \(A\) since we want to know the balance at some point in the future. Here, we know
\(\begin{array}{ll} d = \$100 & \text{the monthly deposit} \\ r = 0.06 & 6\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly deposits, we’ll compound monthly} \\ t = 20 & \text{we want the amount after 20 years} \end{array}\)
Putting these values into the equation:
\(A = \dfrac{d\left[\left(1+\dfrac{r}{n}\right)^{n t}-1\right]}{\left(\dfrac{r}{n}\right)} \; \; = \; \; \dfrac{100\left[\left(1+\dfrac{0.06}{12}\right)^{20 \cdot 12}-1\right]}{\left(\dfrac{0.06}{12}\right)} \; \; = \; \; \dfrac{100\left[\left(1.005\right)^{240}-1\right]}{\left(0.005\right)} \; \; = \; \; $46,204.09\)
The account will grow to $46,204.09 after 20 years.
To find the amount of interest earned, we first need the total of all deposits that are made during the 20 years: \($100 \times 20 \text{ years } \times 12 \text { months per year } = $24,000.\) The difference between the balance and the total deposits made is \($46,204.09 - $24,000 = $22,204.09.\)
Therefore, the total amount of interest earned is $22,204.09.
To find the percentage of the balance that is interest, divide the interest earned by the total balance: \(\frac{$22,204.09}{$46,204.09} \approx 0.48056 \approx 48.1\%\).
After 20 years, 48.1% of the balance is from accrued interest.
We can also use the TVM Solver on the TI calculator to solve this problem. Recall to access the TVM Solver press the APPS key and then choose the Finance... option from the menu. Similar to using the TVM Solver to determine the future value for a compound interest problem, we will solve for \(FV\).
\(N\) is the number of times a deposit is made and interest is compounded during the entire period. The big difference is that now \(PMT\) will show amount of the regular deposit. The \(PMT\) amount must include a \('-'\) sign in front so that it identifies the direction of cash flow as away from you.
- \(N = 240 \): The number of deposits and times compounding is done is \(20 \text{ years } \times 12 \text { months } = 240\).
- \(I\%=6\): Remember to leave this as a percentage. Do not change to a decimal.
- \(PV =0\): There is no money in the account before the first deposit is made.
- \(PMT = -100\): This is the amount deposited into the account with the negative sign showing cash flow away from you.
- \(P\text{/}Y = C\text{/}Y = 12\) because deposits are made and the interest is compounded 12 times per year.
Solve for future value \(FV\) by moving the cursor back to the value for \(FV\) and pressing ALPHA and then ENTER.
The TVM Solver agrees with the result using the formula in Example 1. The balance of the IRA in 20 years will accumulate to $46,204.09.
Professor Mirtova saves for retirement by depositing $100 from each of her biweekly paychecks into a savings annuity with an annual interest rate of 3%. She has been doing this for the past 15 years she has worked at PGCC. How much money has accumulated in her account during this period? How much money did the professor deposit into her annuity during this period? How much interest did Professor Mirtova earn?
- Answer
-
The balance in the account is now $49,212.47. Professor Mirtova has deposited a total of $39,000. The amount of interest earned is $10,212.47.
Sometimes, a "lump sum" may be deposited into a savings plan initially and then followed by regular smaller deposits. The next example will illustrate this type of scenario.
You want to jumpstart your savings for retirement by depositing $1,500 from your tax refund into an account. You will continue to deposit $150 every month into the same account. The savings account earns 5.5% interest compounded monthly. How much will be in the account after 30 years?
Solution
To solve this example with a formula, separately use the compound interest formula from Section 5.2 to find future value of the initial $1,500 deposit and then use the annuity savings formula from this section to find future value of the deposits made during the next 30 years. Finally, add these two future value amounts to find the value of the account.
Solving the problem is much less complicated using the TVM Solver and can be done in one step.
- \(N = 360 \): The number of deposits and times compounding is done is \(30 \text{ years } \times 12 \text { months } = 360\).
- \(I\%=5.5\): Remember to leave this as a percentage. Do not change to a decimal.
- \(PV =-1,500\): This is the amount that is initially deposited in the account from the tax refund. Note the negative sign represents cash flow away from you.
- \(PMT = -150\): This is the amount deposited into the account with the negative sign showing cash flow away from you.
- \(P\text{/}Y = C\text{/}Y = 12\) because deposits are made and the interest is compounded 12 times per year.
Solving for future value \(FV\), the account will grow to $144,822.87 after 30 years.
You deposit $50 into an account and plan to invest $5 every week into the same account. The account pays interest of 3.75% annually. If you keep up these same deposits for 25 years, how much will accumulate in the account? What is total of the deposits? What interest has accrued?
- Answer
-
The balance in the account is $10,893.21. The total of the deposits is $6,550. The interest that has accrued is $4,343.21.
Another typical application of a savings annuity is to find the amount that should be deposited, \(d\), each term to reach a desired future value. This is especially important in financial planning for retirement or saving to make a large purchase. For example, if you know how much money you need for retirement, how much should you be saving each month?
You want to have a half million dollars in your retirement account when you retire in 30 years. Your account earns 8% interest. How much should you deposit each month to meet your retirement goal?
Solution
To solve this with a formula, use the second formula solved for \(d\), the regular deposit (payment) amount, where
\(\begin{array}{ll} A = \$500,000 & \text{the retirement goal (future value)} \\ r = 0.08 & 8\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly deposits, we’ll compound monthly} \\ t = 30 & \text{we want the amount after 30 years} \end{array}\)
\(d = \dfrac{A\left(\dfrac{r}{n}\right)}{\left[\left(1+\dfrac{r}{n}\right)^{n t}-1\right]} \; \; = \; \; \dfrac{500,000\left(\dfrac{0.08}{12}\right)}{\left[\left(1+\dfrac{0.08}{12}\right)^{12 \cdot 30}-1\right]} \; \; \approx \; \; $335.49\)
You will need to deposit $335.49 each month to have $500,000 in 30 years if the account earns 8% interest.
To find the answer using the TVM Solver, we solve for \(PMT\) when \(FV = $500,000\) and \(N = 30 \times 12 = 360\).
Again, monthly deposits of $335.49 are needed for 30 years to accumulate a half million dollars for retirement
During the next 5 years you will save for the down payment on a house. The houses you are considering cost around $250,000. To approve your mortgage, the bank requires a 10% down payment. Find the down payment amount. Then, find how much you must deposit monthly into a savings annuity that earns 4.2% interest so you will have the required down payment in 5 years.
- Answer
-
The required down payment is $25,000.
To save for this down payment, monthly deposits of $375.17 must be made into the savings annuity.
