Welcome to Level Annuities!

Hello there! If you’ve ever paid rent, made a car payment, or received a monthly allowance, you’ve already encountered the world of annuities. In Financial Mathematics, an annuity is simply a series of level (equal) payments made at equal intervals of time for a finite term (a set period).

Don't worry if the formulas look scary at first—we’re going to break them down piece by piece. By the end of these notes, you'll see that these formulas are just shortcuts to help us avoid doing the same calculation twenty times in a row!

1. What Exactly is a Level Annuity?

Imagine you win a small lottery prize that pays you \$100 every year for the next 5 years. That is a level annuity.

\nThere are two main things to look for:\n

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  1. Level: The payment amount doesn't change.
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  2. Finite Term: It has a specific end date (e.g., 5 years, 10 months).
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\nPrerequisite Concept Check: Remember that to find the value of money at different times, we use the interest rate \( i \) and the discount factor \( v = \frac{1}{1+i} \). We use \( v \) to "pull" future money back to the present.

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Quick Review:\n

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  • Present Value (PV): What the stream of payments is worth today.
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  • Future Value (FV): What the stream of payments will be worth at the end.
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2. Annuity-Immediate (Payments at the END)

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An Annuity-Immediate is a series of payments where each payment is made at the end of each period. Think of this like a paycheck—you work the month first, then you get paid at the end.

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The Formula for Present Value

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The symbol for the Present Value of an annuity-immediate of \$1 for \( n \) periods at interest rate \( i \) is \( a_{\overline{n}|i} \).
\( a_{\overline{n}|i} = \frac{1 - v^n}{i} \)

Step-by-Step Breakdown:

  1. Find your interest rate per period (\( i \)).
  2. Calculate the discount factor \( v = (1+i)^{-1} \).
  3. Raise \( v \) to the power of the number of payments (\( n \)).
  4. Plug it into the formula!

The Formula for Future Value

The symbol for the Future Value (the value right after the last payment) is \( s_{\overline{n}|i} \).
\( s_{\overline{n}|i} = \frac{(1 + i)^n - 1}{i} \)

Analogy: Imagine you are standing at Time 0. \( a_{\overline{n}|} \) is the value of all payments looking forward. If you stand at Time \( n \), \( s_{\overline{n}|} \) is the value of all those same payments looking backward.

Summary: Annuity-Immediate = Payments at the End. (Mnemonic: The "I" comes before "E" in the alphabet, but in annuities, Immediate means you wait until the End!)

3. Annuity-Due (Payments at the BEGINNING)

An Annuity-Due is when payments are made at the beginning of each period. Rent is the perfect example—you pay on the 1st of the month to live there for the upcoming month.

The Formula for Present Value

The symbol for an annuity-due has two little dots (called a diaeresis) over the 'a': \( \ddot{a}_{\overline{n}|i} \).
\( \ddot{a}_{\overline{n}|i} = \frac{1 - v^n}{d} \)
(Note: We use \( d \), the discount rate, instead of \( i \), where \( d = \frac{i}{1+i} \).)

The Formula for Future Value

\( \ddot{s}_{\overline{n}|i} = \frac{(1 + i)^n - 1}{d} \)

Memory Trick: Due = Dots. If you see the dots, it’s an annuity-due, and you usually use d in the denominator!

Did you know? An annuity-due is always worth more than an annuity-immediate if everything else is the same. Why? Because you get your money sooner, so it has more time to earn interest!

4. The Relationship Between Them

Don't let the different formulas overwhelm you. They are all related! You can move between them easily:

  • The "One Period" Rule: \( \ddot{a}_{\overline{n}|} = a_{\overline{n}|} \times (1+i) \)
  • The Future Value Connection: \( s_{\overline{n}|} = a_{\overline{n}|} \times (1+i)^n \)
Basically, if you have the Present Value, you can just multiply by \((1+i)\) to "shift" the payments by one period or multiply by \((1+i)^n\) to find the Future Value.

Key Takeaway: If you forget the formula for \( \ddot{a}_{\overline{n}|} \), just calculate \( a_{\overline{n}|} \) and multiply by \((1+i)\)!

5. Deferred Annuities

Sometimes, an annuity doesn't start right away. This is called a deferred annuity. For example, you might buy a plan today that starts paying you \$500 a month, but not until 10 years from now.

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How to solve these:\n

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  1. Calculate the Present Value of the annuity at the time it would have started.
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  2. Discount that single "lump sum" value back to Time 0.
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\nExample: An annuity-immediate with 5 payments of \$1, but the first payment is at Time 4.
The formula \( a_{\overline{5}|} \) gives the value one period before the first payment. If the first payment is at Time 4, \( a_{\overline{5}|} \) gives us the value at Time 3. To get the value at Time 0, we multiply by \( v^3 \).
PV = \( v^3 \times a_{\overline{5}|} \)

Common Mistake: Students often discount by the wrong number of years. Always draw a timeline! Mark when the first payment happens, then remember that \( a_{\overline{n}|} \) lands you exactly one period before that first payment.

6. Finding the Payment (R), Term (n), or Interest (i)

Often, the exam will give you the Present Value and ask you to solve for something else.

  • Solving for R (Payment): Just rearrange: \( R = \frac{PV}{a_{\overline{n}|}} \).
  • Solving for n (Term): You will likely need to use logarithms (\( \ln \)) to get \( n \) out of the exponent.
  • Solving for i (Interest): This is the hardest one to do by hand. Use your Financial Calculator!

Calculator Tip (BA II Plus): Most successful students use the TVM (Time Value of Money) buttons:
[N] = Number of periods
[I/Y] = Interest rate per period
[PV] = Present Value
[PMT] = Payment amount
[FV] = Future Value
Note: Set your calculator to "BGN" mode for Annuities-Due and "END" mode for Annuities-Immediate.

7. Summary and Final Tips

Key Points to Remember:

  • Annuity-Immediate: Payments at end of period; starts at Time 1; uses \( i \).
  • Annuity-Due: Payments at start of period; starts at Time 0; uses \( d \).
  • Finite Term: There are exactly \( n \) payments.
  • Timelines: When in doubt, draw it out! It prevents mistakes with deferred annuities.

Encouragement: If you feel confused by the notation \( a_{\overline{n}|} \), just remember it's just a label for a specific math formula. Practice using your calculator's TVM buttons—it will make these problems much faster and reduce the chance of manual calculation errors. You've got this!