4.6: Annuities - A Stream of Payments Over Time
- Page ID
- 132024
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Define annuities as a stream of regular payments over time.
- Differentiate between types of annuities (ordinary vs. due) and their use cases.
- Apply annuity formulas to calculate present and future value.
Regular Payments
So far, we’ve explored single payments—like your grandparent setting aside money for a one-time graduation gift. But what if instead of a single lump sum, they gave you a smaller gift every year for the next four years? That’s no longer just one decision about a lump sum of money. It’s a series of payments made over time, and we refer to this as an annuity.
An annuity is simply a stream of equal payments made at regular intervals over time. Think of it as this:
- Your grandparents are giving you $1,000 every year for four years to help you with college costs.
- A subscription service where you pay $10 a month for access to a streaming platform.
- A savings plan where you deposit $100 into your account at the end of every year.
The key here is that the payments are:
- Equal payments - each payment is the same amount.
- Regular intervals - the payments happen on a fixed schedule (monthly, annually, etc.).
Annuities in Real Life
Annuities might sound like a fancy financial term, but they’re everywhere. For example, these are annuities:
- Saving for a Goal: You decide to save $1,000 each year for four years to buy a car.
- Receiving Payments: When you retire, you might receive equal monthly payments from your retirement fund to cover your expenses.
Whether you’re the one saving or receiving, annuities are all about managing money over time - a predictable stream of payments that builds toward a goal.
Why Time Matters
Here’s the catch: Even though each payment is equal, time still influences the value of those payments. Money received today is worth more than the same amount received later, thanks to inflation and opportunity cost.
To compare these payments or determine their current value, we need tools to calculate the future value (FV) and present value (PV) of annuities.
Future Value of an Annuity: Saving for a Goal
Imagine your grandparent decides to help you save for a big trip after graduation. They deposit $1,000 at the end of every year into an account that earns 5 percent interest. You’ll receive four payments. How much money will you have in total after the fourth year?
To find the future value of an annuity (FVA), we add up the value of each payment as it grows with interest:
- Year 1: The first $1,000 earns interest for three years.
- Year 2: The second $1,000 earns interest for two years.
- Year 3: The third $1,000 earns interest for one year.
- Year 4: The fourth $1,000 payment doesn’t earn interest; it’s the final payment.
Each payment grows using the same formula for future value:
\(\displaystyle{FV = PV \times (1+r)^{n}}\)
But we’re summing the future values of all four payments. Instead of doing this manually, there’s a formula that calculates the total Future Value of an Annuity (FVA):
\(\displaystyle{FV_{A} = C \times \frac {(1 + r)^{n} - 1}{r}}\)
Where:
C = Payment amount (e.g., $1,000)
r = Interest rate (e.g., 5% = 0.05)
n = Number of periods (e.g., 4 years)
Plugging in the numbers:
\(\displaystyle{FV_{A} = 1,000 \times \frac {(1 + 0.05)^{4} - 1}{0.05}}\)
\(\displaystyle{FV_{A} = 1,000 \times \frac{1.2155}{0.05}}\)
\(\displaystyle{FV_{A} = 1,000 \times 4.31}\)
\(\displaystyle{FV_{A} = 4,310}\)
Result: After four years, you’ll have approximately $4,311 saved for your big trip.
Present Value of an Annuity: Bringing Future Payments to Today
Now let’s flip the problem. Suppose your grandparent wants to give you $4,000 total over four years by making equal annual payments of $1,000 at the end of each year. Instead of setting aside $4,000 upfront, they want to know this: How much money do they need to set aside today to make these payments?
To answer this, we calculate the Present Value of an Annuity (PVA). This brings each payment back to today’s value, or what we call “discounting”:
The formula for the PVA is:
\(\displaystyle{PV_{A} = \frac{C}{r} \times \left(1 - \frac{1}{(1 + r)^{n}}\right)}\)
Where:
C = Payment amount (e.g., $1,000)
r = Interest rate (e.g., 5 percent = 0.05)
n = Number of periods (e.g., four years)
Plugging in the numbers:
\(\displaystyle{PV_{A} = \frac{1,000}{0.05} \times \left(1 - \frac{1}{(1 + 0.05)^{4}}\right)}\)
\(\displaystyle{PV_{A} = \frac{1,000}{0.05} \times \left(1 - 0.8227\right)}\)
\(\displaystyle{PV_{A} = 20,000 \times 0.1773}\)
\(\displaystyle{PV_{A} = 3,546}\)
Result: To make four equal payments of $1,000 over four years, your grandparent would need to set aside approximately $3,546 today.
Why This Matters
Annuities are everywhere in personal finance: saving for a car, receiving retirement income, or repaying a loan. By understanding the future and present value of a stream of payments, you can plan for your goals, make better financial decisions, and compare options over time.
Future Value illustrates how regular payments accumulate into a larger amount over time, such as saving for a significant goal.
Present Value shows how to calculate the amount needed today to fund future payments, such as planning for a series of expenses.
You can use online calculators or Excel spreadsheets to solve for FV and PV of annuities:
Future Value of an Annuity:
Excel formula: =FV(rate, nper, pmt, 0, 0)
Example: =FV(5%, 4, -1000, 0, 0) → Returns 4,311
Present Value of an Annuity:
Excel formula: =PV(rate, nper, pmt, 0, 0)
Example: =PV(5%, 4, -1000, 0, 0) → Returns 3,546
Tip: In Excel, payments are entered as negative numbers because they’re outgoing cash flows.
What If You’re the One Making the Payments?
So far, we’ve explored how money grows when you receive regular payments, like deposits into a savings account or gifts spread over time. But what if the payments go the other way?
Imagine you’re not the one saving or receiving. Imagine you’re the one repaying a loan, like for a car or a house. You’re still making regular payments, but this time they work to reduce a debt.
This brings us to amortization—the process of breaking a loan into equal payments over time. While the math may look familiar, what’s happening behind the scenes is a little different and fascinating.
Annuities are time-based contracts that provide regular, predictable payments in or out. Whether it’s paying off a loan or receiving retirement income, annuities show up wherever money flows repeatedly. This section introduces the idea of valuing a stream, not just a lump sum, and reinforces the notion that time affects every dollar.
- An annuity is a series of equal payments over equal time intervals.
- Common examples include loan payments, lease agreements, and withdrawals from retirement accounts.
- Time and frequency affect value: an annuity due (payments at the start) is more valuable than an ordinary annuity (payments at the end).
This section highlights how to calculate annuity value using formulas, but the emphasis is still on why structure matters and how time and payment position affect outcomes.
- Have you ever made or received regular payments over time (e.g., subscriptions, rent, loans)? How does it feel different than a single transaction?
- Why might a landlord prefer rent due at the start of the month rather than the end?
- You receive $200/month for three years from an ordinary annuity. If the interest rate is 6 percent annually, what’s the present value of those payments?