Chapter 2 Notes: Time Value of Money Calculations

Four DCF rules

• Money has a time value.

• Quantities of money cannot be added or subtracted unless they occur at the same points in time.

• To move money forward one time unit, multiply by $1+i$ (the discount or interest rate).

• To move money backward one time unit, divide by $1+i$.

Time Value of Money (TVOM) structure in this chapter

• LO 2.1: Construct a cash flow diagram (CFD) for an investment alternative.

• LO 2.2: TVM calculations for single cash flows with annual compounding.

• LO 2.3: TVM calculations for irregular series with annual compounding.

• LO 2.4: TVM calculations for uniform series with annual compounding.

• LO 2.5: TVM calculations for gradient series with annual compounding.

• LO 2.6: TVM calculations for geometric series with annual compounding.

• LO 2.7: TVM calculations for multiple compounding periods per year.

Key concepts and notations

• Present Worth (P) and Future Worth (F) of a single cash flow: $F = P(1+i)^n, ag{F|P} \ P = \frac{F}{(1+i)^n}.$

• Present Worth factor for a uniform series: $P = A\,(P|A,i,n),\quad F = A\,(F|A,i,n).$

• Present Worth of irregular cash flows: $P = \sum<em>{t=1}^{n} A</em>t\,(1+i)^{-t}.$

• For a uniform series of cash flows, key factors are:

• Present worth factor: $(P|A,i,n) = \frac{1 - (1+i)^{-n}}{i}$

• Future worth factor: $(F|A,i,n) = \frac{(1+i)^n - 1}{i}$

• For a single payment converted to a series, the relationships include: $A = P\,(A|P,i,n), \quad P = F\,(P|F,i,n),$ where the corresponding factors relate to conversions between P, F, and A.

• Effective annual interest rate (eff) when there are m compounding periods per year: $i_{eff} = (1 + \tfrac{r}{m})^{m} - 1$

• Nominal annual rate r with m compounding periods per year: used to derive $i_{eff}$ above.

Example results and checkpoints (highlights)

• Example 2.2 (5-year compounding at 10%): amount after 5 years on a $10,000 loan = $10{,}000\cdot(1+0.10)^5 = 16{,}105.10.$

• Example 2.3 (single payment at 12% for 5 years): use future worth factor for 12% over 5 years: $F|P\,12\%,5 = 1.76234,$ so amount owed after 5 years on a $1{,}000 loan ≈ $1{,}000\cdot 1.76234 = 1{,}762.34.$ The text confirms the factor and equivalent payment value.

• Example 2.4 (Doubling money): time to double for i = 2%, 4%, 12%

• Rule of 72 approximation: $n\approx \frac{72}{i\%}$ gives n ≈ 36, 18, 6 years respectively.

• Exact/numerical: $n=\frac{\ln 2}{\ln(1+i)}$ gives: i=2% → ~35.003 years, i=4% → ~17.673 years, i=12% → ~6.116 years.

• Excel methods (NPER, GOAL SEEK, SOLVER) yield the same results as shown: 35.003; 17.673; 6.116 years.

• Example 2.5 (Saving money): accumulating $10,000 in 4 years at 5% compounded annually. Present value today: $P = F\,(P|F,5$ Excel PV function gives $P = 8{,}227.02.$

• Example 2.6 (Present worth of a series): present worth of a cash flow series with irregular payments; computed as
  $P = \sum<em>{t} A</em>t\,(1+i)^{-t}.$ The numeric result shown: $P = 597.02.$

• Example 2.9 (Uniform series present worth): deposit needed today for five $2,000 withdrawals at 5% for 5 years:
  $P = A\,(P|A,5\%,5) = 2{,}000\times 4.32948 = 8{,}658.96.$ Excel PV gives $8{,}227.02$ for a different example, illustrating the method.

• Example 2.11 (Uniform withdrawals): deposit $10{,}000, withdraw 10 equal annual payments at 8% so last withdrawal depletes funds. With first withdrawal at year 1:
  $A = P\,(A|P,8\%,10) = 10{,}000\times 0.14903 = 1{,}490.30.$ (This uses the A|P factor, the inverse of P|A.)

• Example 2.13 (Future worth of a uniform series): deposits of $1,000 for 30 years at 6% →
  $F = A\,(F|A,6\%,30) = 1{,}000\times 79.05819 = 79{,}058.19.$

• Example 2.15 (Gradient series): costs start at $3,000 and increase by $1,000 per year for 5 years; with i = 8%, compute P of the composite gradient series by converting to a uniform series and adding to the base series. (Equation (2.29)/(2.34) referenced.)

• Example 2.16 (Geometric series): maintenance costs start at $1{,}000 at year 1 and grow at j = 8% per year for n = 15 years; with i = 10%, compute P of the geometric series (A1, growth j) via the standard geometric-present-worth formula.

• Example 2.19 (Car payments with monthly compounding): P = $25{,}000; nominal annual rate 8% compounded monthly; 5-year term; payments A are found with the PMT function. Period rate = 8%/12; number of periods = 60; A = PMT(0.08/12, 60, -25000).

• Example 2.20 (Effective annual rate for different compounding frequencies with r = 12%):

• quarterly: $i_{eff} = (1+0.12/4)^4 - 1 = 0.12551\,(12.551\%)$

• monthly: $i_{eff} = (1+0.12/12)^{12} - 1 = 0.12683\,(12.683\%)$

• every minute (m = 525{,}600): use $i_{eff} = (1+0.12/525{,}600)^{525{,}600} - 1\approx e^{0.12}-1\approx 12.75\%$ (value shown conceptually; exact numeric depends on integration).

• continuous compounding: $i_{eff} = e^{0.12} - 1$ (≈ 12.75% as a limit case).

• Example 2.22 (Mismatch of cash flow and compounding frequencies): when cash flows occur more or less frequently than compounding, convert to a common per-period rate:

• i = the per-cash-flow-period rate given by $i = (1 + r/m)^{m/k} - 1$ where r is nominal annual rate, m is compounding periods per year, and k is number of cash flows per year. This aligns the per-period discount/interest rate with the cash-flow schedule.

Key takeaways on methodology

• To compare alternatives, build the CFD to identify timing and scale of inflows/outflows.

• Use the appropriate TVM formula depending on cash-flow pattern: single, irregular, uniform, gradient, geometric, and/or multiple compounding regimes.

• Convert between P, F, and A using the standard factors:

• $P = A\,(P|A,i,n),\quad F = A\,(F|A,i,n)$

• For irregular series: $P = \sum<em>{t=1}^{n} A</em>t(1+i)^{-t}$

• For gradient: $P = A\,(P|A,i,n) + G\,(P|G,i,n)$

• For geometric: $P = A<em>1\frac{1}{1+i}\cdot\frac{1 - \left(\frac{1+g}{1+i}\right)^n}{1 - \frac{1+g}{1+i}} = A</em>1\frac{1 - \left(\frac{1+g}{1+i}\right)^n}{i - g}.$

• The effective annual rate is key when comparing investments with different compounding schemes: $i_{eff} = (1 + \tfrac{r}{m})^{m} - 1.$

• If cash-flow frequency does not match compounding frequency, compute the per-cash-flow interest rate via $i = (1 + \tfrac{r}{m})^{\tfrac{m}{k}} - 1.$

Quick references to figures and terms (from the figures/tables in the transcript)

• CFD example: Fig. 2.1 (illustrates cash inflows/outflows with time axis).

• Table and figures referenced include ARP/Appendix A values for F|P/P|F factors, NPV/PMT/GOAL SEEK/ SOLVER utilizations in Excel, and the key term glossary.

• Figure 2.7 shows P = PV(5%,4,,−10000) as an Excel illustration.

Key terms to memorize

• Capital Recovery Factor

• Cash Flow Diagram (CFD)

• Compounding

• Effective Annual Interest Rate

• Future Value

• Geometric Series

• Gradient Series

• Interest Rate

• Nominal Annual Interest Rate

• Period Interest Rate

• Present Value

• Sinking Fund Factor

• Uniform Series

Notes on exam relevance

• Be able to identify the correct TVM formula for each cash-flow pattern (single, irregular, uniform, gradient, geometric) and for multiple compounding frequencies.

• Be comfortable converting between P, F, and A using the appropriate factors and understanding the meaning of each factor (P|A, F|A, etc.).

• Practice with the common examples (doubling time, saving for a future target, uniform withdrawals, gradient costs, geometric growth in costs, and loan repayments).