CHAPTER 9: FV and PV of Annuities (part 1)

Annuity Basics

  • Annuity: a series of equal payments in equal time periods; time period is usually 1 year.
  • Annuity due: payments at the beginning of each period; ordinary annuity: payments at the end; most annuities are ordinary.
  • Present value (PV) and future value (FV) extend to annuities: FV is the value of future payments; PV is the value today of future payments.

Future Value of Annuity

  • Future value of an ordinary annuity (payments at end of each period):
    FVOA=A(1+r)n−1r.FVOA = A \frac{(1+r)^n - 1}{r}.
  • Future value of an annuity due (payments at beginning):
    FVAD=A(1+r)n−1r×(1+r).FVAD = A \frac{(1+r)^n - 1}{r} \times (1 + r).
  • Relationship between FV for due and ordinary:
    FV<em>AD=FV</em>OA×(1+r)FV<em>{AD} = FV</em>{OA} \times (1 + r)
    and
    FV<em>OA=FV</em>AD1+r.FV<em>{OA} = \frac{FV</em>{AD}}{1 + r}.
  • Difference between FVAD and FVOA:
    FVAD−FVOA=A((1+r)n−1).FV_{AD} - FVOA = A\bigl((1+r)^n - 1\bigr).
  • Timing note: the last payment of an ordinary annuity earns no interest; the last payment of an annuity due earns interest during the last period.
  • Example (ordinary annuity): deposit $100 per period, r = 0.06/12 per month, n = 120 months →
    FVOA≈16,387.93.FVOA \approx 16{,}387.93.
  • Example (annuity due): beginning-of-period deposits yield an additional amount relative to the ordinary case (due to one extra period of interest).

Present Value of Annuity

  • Present value of an annuity (payments at end):
    PVA=A1−(1+r)−nr.PVA = A \frac{1 - (1+r)^{-n}}{r}.
  • Present value of an annuity due:
    PVAAD=A1−(1+r)−nr×(1+r).PVA_{AD} = A \frac{1 - (1+r)^{-n}}{r} \times (1 + r).
  • Examples:
    1) Lottery paid as installments: $50{,}000$ per year for 20 years discounted at 5%:
    PV=50,0001−(1+0.05)−200.05≈623,110.52.PV = 50{,}000 \frac{1 - (1+0.05)^{-20}}{0.05} \approx 623{,}110.52.
    2) Mortgage loan amount (end-of-period payments): with monthly rate $r = 0.05/12$ and $n = 360$,
    PV = A \frac{1 - (1+r)^{-n}}{r},
    \quad A = PV \cdot \frac{r}{1 - (1+r)^{-n}}.
    Example yields about $186{,}281.62$ for a $1$-monthly payment, given the loan amount.
    3) Monthly mortgage example: $200{,}000$ loan at 6% annual rate for 30 years (monthly):
    A=PVr1−(1+r)−n,r=0.06/12,n=360,A = PV \frac{r}{1 - (1+r)^{-n}}, \quad r = 0.06/12, \quad n = 360,
    giving about \$1{,}199.10$ per month.

Time Value, Inflation, and Real Values

  • Real (today’s) value of a future amount given inflation i:
    Real FV=FV(1+i)n.\text{Real FV} = \frac{FV}{(1+i)^n}.
  • Example: $1{,}000{,}000$ in 50 years at 3% inflation is worth
    1,000,000(1+0.03)50≈228,107.08.\frac{1{,}000{,}000}{(1+0.03)^{50}} \approx 228{,}107.08.
  • If you save $2,000 per year for 50 years at 5% nominal, the nominal FV is
    FVOA=2,000(1+0.05)50−10.05≈418,695.99.FVOA = 2{,}000 \frac{(1+0.05)^{50} - 1}{0.05} \approx 418{,}695.99.
  • Real value in today’s dollars (3% inflation):
    418,695.99(1+0.03)50≈95,507.52.\frac{418{,}695.99}{(1+0.03)^{50}} \approx 95{,}507.52.
  • Extra value of starting earlier (annuity due vs ordinary):
    • Nominal extra in FV: FVAD−FVOA=A((1+r)n−1).FV_{AD} - FVOA = A\bigl((1+r)^n - 1\bigr).
    • Real extra in today’s dollars: (\frac{A\bigl((1+r)^n - 1\bigr)}{(1+i)^n}).

Present Value, NPV, and Internal Rate of Return (IRR)

  • Mixed streams: Present value is the sum of the PV of each payment:
    PVA<em>mixed=∑</em>tCFt(1+r)t.PVA<em>{mixed} = \sum</em>t \frac{CF_t}{(1+r)^t}.
  • Net Present Value (NPV):
    NPV=PV of inflows−PV of outflows.\text{NPV} = \text{PV of inflows} - \text{PV of outflows}.
  • Decision rule in capital budgeting:
    • Hurdle rate (discount rate) DR; if NPV > 0 or IRR >= DR, invest.
  • IRR: the discount rate that sets NPV to 0; if IRR ≥ DR, invest; otherwise, reject.
  • IRR formula (cash flows CF):
    NPV(IRR)=∑<em>t=0nCF</em>t(1+IRR)t=0.\text{NPV}(\text{IRR}) = \sum<em>{t=0}^{n} \frac{CF</em>t}{(1+\text{IRR})^t} = 0.
  • Note: IRR can be difficult to solve algebraically; spreadsheets provide IRR/NPV tools.

Using Excel (PV, NPV, FV) for Investments

  • Key functions:
    • Present Value: PV(rate,nper,pmt,fv,type)\text{PV}(rate, nper, pmt, fv, type)
    • Future Value: FV(rate,nper,pmt,pv,type)\text{FV}(rate, nper, pmt, pv, type)
    • Net Present Value: NPV(rate,value1,value2,…)\text{NPV}(rate, value1, value2, …)
  • Type argument:
    • 0 = payments at end of period; 1 = payments at beginning.
  • Important input notes:
    • Payments are entered as negative numbers if you pay them; positive if you receive.
    • If there is no series of payments, leave that input blank and provide FV or PV instead.
  • Examples from practice:
    • PV of $1{,}000{,}000 to be received at age 65 with 3% inflation over 35 years:
      PV=PV(0.03,35,,−1000000)≈355,383.40.\text{PV} = \text{PV}(0.03, 35, , -1000000) \approx 355{,}383.40.
    • Saving $4{,}000 annually for 40 years at 5%:
    • End of year payments:
      FV(0.05,40,−4000,,)≈483,199.10.\text{FV}(0.05, 40, -4000, , ) \approx 483{,}199.10.
    • Beginning of year payments:
      FV(0.05,40,−4000,,1)≈507,359.05.\text{FV}(0.05, 40, -4000, , 1) \approx 507{,}359.05.
    • With $10{,}000 already saved (PV input):
      FV(0.05,40,−4000,−10000,1)≈577,758.94.\text{FV}(0.05, 40, -4000, -10000, 1) \approx 577{,}758.94.

Quick Reference (Key Formulas)

  • Ordinary annuity FV: FVOA=A(1+r)n−1r.FVOA = A \frac{(1+r)^n - 1}{r}.
  • Annuity due FV: FVAD=A(1+r)n−1r(1+r).FVAD = A \frac{(1+r)^n - 1}{r} (1 + r).
  • PV of annuity: PVA=A1−(1+r)−nr.PVA = A \frac{1 - (1+r)^{-n}}{r}.
  • PV of annuity due: PVAAD=A1−(1+r)−nr(1+r).PVA_{AD} = A \frac{1 - (1+r)^{-n}}{r} (1 + r).
  • Solve for payment A given PV: A=PV⋅r1−(1+r)−n.A = PV \cdot \frac{r}{1 - (1+r)^{-n}}.
  • Relationship between FV due and FV ordinary: FV<em>AD=FV</em>OA(1+r).FV<em>{AD} = FV</em>{OA} (1 + r).
  • IRR condition: IRR \ge DR implies invest; IRR is the rate that makes NPV 0.

Additional Examples (from the transcript)

  • Mortgage payment example (30-year loan, end-of-period): for a $1$-period monthly rate $r = 0.05/12$ over 360 periods, the payment is found by solving:
    A=PV⋅r1−(1+r)−n.A = PV \cdot \frac{r}{1 - (1+r)^{-n}}.
  • Beginning vs end payments: starting earlier increases FV by about one period of compounding; starting earlier yields higher FV by the factor of the annuity term.