Notes on Annuities: Ordinary vs Annuity Due (Chapter 4 Part 1)

Time value of money and annuities

  • The lecture focuses on two main concepts: time value of money and annuities (a sequence of equal payments over time).
  • Key motivation: the quicker you receive money, the more valuable it is today because you can invest it sooner.
  • Homework and class notes revolve around Chapter 1 concepts and Chapter 4 Part 1 in-class exercises related to annuities, including annuities due and ordinary annuities.

Two main types of annuities

  • Ordinary annuity (end-of-period payments): each payment occurs at the end of each period.
    • Example: 100 dollars paid at the end of years 1, 2, 3, …
  • Annuity due (beginning-of-period payments): each payment occurs at the beginning of each year.
    • Example: 100 dollars paid today (time 0), then at the beginning of each subsequent year.
  • Core intuition: switching from annuity due to ordinary annuity is equivalent to pushing every payment back by one period. Conversely, annuity due accelerates cash flows by one period relative to ordinary annuity.

Present value (PV) and future value (FV) intuition

  • Present value (PV): the value today of a stream of future payments, discounted back at rate r.
  • Future value (FV): the value at a future time of a stream of payments, compounded forward at rate r.
  • If two cash flows have the same amount but different timing, the sooner payment has the higher value (PV and FV).
  • Real-world relevance: pension/retirement payments often come as annuities; some systems provide annuity due (payments start immediately) vs ordinary annuity (payments start after a delay).

Formulas (LaTeX)

  • Present value of an ordinary annuity (payments at end of each period):
    PVextord=PMT 1−(1+r)−nrPV_{ ext{ord}} = PMT \,\frac{1 - (1+r)^{-n}}{r}
  • Future value of an ordinary annuity (payments at end of each period):
    FVextord=PMT (1+r)n−1rFV_{ ext{ord}} = PMT \,\frac{(1+r)^{n} - 1}{r}
  • Present value of an annuity due (payments at beginning of each period):
    PV<em>extdue=PMT 1−(1+r)−nr (1+r)=PV</em>extord (1+r)PV<em>{ ext{due}} = PMT \,\frac{1 - (1+r)^{-n}}{r} \, (1+r) = PV</em>{ ext{ord}} \, (1+r)
  • Future value of an annuity due (payments at beginning of each period):
    FV<em>extdue=PMT (1+r)n−1r (1+r)=FV</em>extord (1+r)FV<em>{ ext{due}} = PMT \,\frac{(1+r)^{n} - 1}{r} \, (1+r) = FV</em>{ ext{ord}} \, (1+r)
  • Relationship checks:
    • FVdue = FVord \cdot (1+r)
    • PVdue = PVord \cdot (1+r)
  • Sign convention (Excel): PV/FV inputs may be signed depending on whether they are inflows or outflows. When FV is a positive inflow, PV often appears negative in Excel unless you flip signs.

Worked example: one-year comparison (illustrative, simple cash flows)

  • Setup: compare two cash-flow patterns with the same amount, $100, but different timing.
    • Pattern A (ordinary): $100 paid at the end of year 1.
    • Pattern B (present): $100 paid today (time 0).
  • Given rate: $r = 10\%$.
  • For Pattern A (end-of-year $100$): Present value is
    PV=100(1+0.10)1=1001.10≈90.91.PV = \frac{100}{(1+0.10)^{1}} = \frac{100}{1.10} \approx 90.91.
  • For Pattern B (paid today): Present value is $100$ (cash is received now).
  • Conclusion: a cash flow today is worth more than the same amount received one year from now, because you can reinvest today’s cash immediately.
  • In this context, the “present value” of $100 at t=1$ is about $90.91; the value today of $100 at t=0$ is $100$.

A more formal multi-period example (three-year horizon)

  • Ordinary annuity (end-of-year payments) with
    • PMT = $100, n = 3, r = 10\%$.
    • Present value: PVextord=1001−(1+0.10)−30.10≈100×2.487=248.7.PV_{ ext{ord}} = 100 \frac{1 - (1+0.10)^{-3}}{0.10} \approx 100 \times 2.487 = 248.7.
    • Future value at end of year 3: FVextord=100(1+0.10)3−10.10=100×3.31=331.FV_{ ext{ord}} = 100 \frac{(1+0.10)^3 - 1}{0.10} = 100 \times 3.31 = 331. (Note: (1.10)^3 = 1.331)
  • Annuity due (beginning-of-year payments) with the same PMT, n, r:
    • PVdue = PVord \cdot (1+r) ≈ 248.7 \cdot 1.10 ≈ 273.6.
    • FVdue = FVord \cdot (1+r) ≈ 331 \cdot 1.10 ≈ 364.1.
  • Practical takeaway: for the same cash flows, annuity due has higher PV and higher FV than ordinary annuity because payments are received earlier.

Excel usage for PV/FV calculations

  • Excel function: PV(rate, nper, pmt, [fv], [type])
    • rate: interest rate per period, r.
    • nper: number of periods, n.
    • pmt: payment per period (cash flow each period).
    • fv: future value after the last payment (optional).
    • type: when payments are due: 0 = end of period (ordinary annuity), 1 = beginning of period (annuity due).
  • Important notes:
    • If you omit type, Excel assumes type = 0 (ordinary annuity).
    • Sign convention: Excel typically returns a negative PV if FV is positive, due to cash flow direction. You can flip signs to compare more easily.
    • For annuity due, you must set type = 1 in the PV function to reflect payments at the beginning.
  • Example patterns from the class discussion:
    • One-year example with a single $100 payment at end of year, r = 10\%, n = 1, PMT = 0:
    • PV = PV(0.10, 1, 0, 100, 0) ≈ -90.91.
    • If the payment is today, the PV is 100 (or PV(0.10, 1, 0, 0, 1) would reflect a cash inflow today, depending on setup).
    • For a three-year ordinary annuity with PMT = 100, r = 10\%, PV ≈ 248.69; FV ≈ 331.
    • For a three-year annuity due with PMT = 100, r = 10\%, PVdue ≈ 273.56; FVdue ≈ 364.10.
  • Practical Excel tips discussed in class:
    • When calculating ordinary annuity, leaves type as 0 (or omit) in Excel.
    • When calculating annuity due, explicitly set type = 1.
    • If you know the ordinary annuity FV, you can obtain the due FV by multiplying by (1+r): FV<em>extdue=FV</em>extord⋅(1+r).FV<em>{ ext{due}} = FV</em>{ ext{ord}} \cdot (1+r).
    • Similarly, PV<em>extdue=PV</em>extord⋅(1+r).PV<em>{ ext{due}} = PV</em>{ ext{ord}} \cdot (1+r).
  • Common pitfalls and troubleshooting:
    • Forgetting to set type to 1 for annuity due leads to the wrong values.
    • Misinterpreting PV vs FV signs in Excel; flip signs if needed for consistent comparison.
    • Avoid excessively long horizons for intuition; 1–2 years often makes the conclusion clear.

Conceptual and real-world implications

  • If a product pays you earlier, its value (PV and FV) is higher, given the same cash flows and interest rate.
  • Pricing in the market reflects this: an annuity due is typically priced higher than an ordinary annuity with the same PMT, n, and r, because you receive money sooner.
  • Pension and retirement systems sometimes mandate annuity due payments (first payment today on retirement) vs ordinary annuities (first payment after a delay). This affects the pricing and perceived value of the product.
  • If you must choose between two products with the same cash flows but different timing, the one with earlier payments is worth more today and will have a higher future value at the horizon, all else equal.
  • Economic intuition: the same dollar amount has different present value depending on when it is paid; you can reinvest earlier payments to generate additional wealth over time.
  • Real-world analogy: stock pricing and expected returns; high-value stocks (e.g., well-known tech giants) may have higher prices today because investors value immediate benefits; this parallels the idea that earlier payments are worth more.

Real-world context discussed in the class

  • The instructor emphasized practical questions in class exercises (Chapter 4 Part 1) focused on annuities and their calculations.
  • In-class exercises and questions (e.g., questions 6–9) involve computing future value and present value for ordinary annuities and annuities due using specified parameters (e.g., 8% or 10% rates, PMT = 3000, n = 5).
  • The solutions and problem set materials are organized in a course folder (Chapter 4 Part 1) with Word documents and Excel solutions for reference.
  • Submission deadlines and process: answers must be submitted in Excel format by a stated deadline (e.g., 12:30); students are reminded to refresh pages to access updated questions (e.g., 6–9 in Chapter 4 Part 1).
  • The instructor also noted that mistakes in understanding annuities can be productive learning moments and stressed the importance of clear timing (beginning vs end) to avoid exam errors.

Worked exercise references (from the lecture)

  • Question 6 (Future value of ordinary annuity):
    • PMT = 3000, n = 5, rate = 0.08, PV = 0, type = 0.
    • FV_ord = 3000 \frac{(1+0.08)^5 - 1}{0.08}.
  • Question 7 (Present value today for ordinary annuity):
    • PMT = 3000, n = 5, rate = 0.08, FV = 0, type = 0.
    • PV_ord = 3000 \frac{1 - (1+0.08)^{-5}}{0.08}.
  • Question 8 (Present value for annuity due):
    • PMT = 3000, n = 5, rate = 0.08, FV = 0, type = 1.
    • PV_due = 3000 \frac{1 - (1+0.08)^{-5}}{0.08} \cdot (1+0.08).
  • Question 9 (Future or present value for annuity due using conversion):
    • If given FVord, you can obtain FVdue via FVdue = FVord \cdot (1+r); similarly for PVdue = PVord \cdot (1+r).

Quick practical tips for exam readiness

  • Always identify the timing of payments first: is it ordinary (end) or due (beginning)? Set type accordingly in Excel (0 for ordinary, 1 for due).
  • For multiple-period annuities, you can use the standard formulas or rely on Excel once you set the correct type.
  • Use the relationships between ordinary and due forms to sanity-check results:
    • PVdue = PVord \cdot (1+r)
    • FVdue = FVord \cdot (1+r)
  • When comparing two cash-flow patterns with the same total amount but different timing, compute PV (and FV) to determine which is more valuable today or in the future.
  • Practice with the given parameters in the chapter exercises (e.g., 5-year horizon, PMT = 3000, r = 8% or 10%) to become comfortable with PV and FV calculations in Excel.
  • Remember to include the type parameter explicitly for annuity due in Excel; leaving it blank defaults to ordinary annuity (end of period).

Resources mentioned in class

  • In-class exercises folder: Chapter 4 Part 1 with Word document questions and Excel solutions.
  • A single source of truth for questions 6–9 (and updated 7–9 as needed) in the Word document.
  • Instructions for submitting Excel-based solutions before the class deadline.

Summary of takeaways

  • The essential difference between ordinary annuity and annuity due is timing of payments; this changes both PV and FV.
  • Annuity due yields higher PV and FV than ordinary annuity for the same PMT, n, and r, because payments are received earlier.
  • Use the PV function in Excel with the correct type parameter and be mindful of sign conventions when interpreting results.
  • For quick checks, use the relationships FVdue = FVord \cdot (1+r) and PVdue = PVord \cdot (1+r).
  • Practice with the given numeric examples (e.g., PMT = 100 or PMT = 3000, n = 3 or 5, r = 8–10%) to build fluency before the exam.