Time Value of Money - Chapter 4 Notes

4.1 The One-Period Case

Focus: basic time value concepts for a single period; establishes forward-looking value relationships used throughout the course.
Future Value (FV) in the one-period case:
- Formula: $FV = PV \times (1 + r)$
- PV = present value (value today); r = appropriate interest rate for the period.
- Example: Invest $10,000 at 12% for one year.
- FV: $FV = 10{,}000 \times (1 + 0.12) = 11{,}200$
- Components of FV: Interest = $10{,}000 \times 0.12 = 1{,}200$ ; Principal repayment = $10{,}000$ ; Total due = $11{,}200$ .
- Excel: $=FV(rate, nper, pmt, pv)$
Present Value (PV) in the one-period case:
- Formula (basic form): $PV = \frac{C1}{1 + r}$ where $C1$ is the cash flow at date 1; alternatively, $PV = \frac{FV}{1 + r}$ .
- Example: If you are promised $11{,}200 in one year and the rate is 12%, today’s value is:</li><li>PV:$ PV = \frac{11{,}200}{1.12} = 10{,}000 $</li><li>Excel:$ =PV(rate, nper, pmt, fv) $</li></ul></li><li>Textbook exercise reference: Q2 (page 122)<ul><li>Future value of$ 1{,}640 $compounded annually for: 10 years at 5% →$ FV = 1640(1+0.05)^{10} = 2671.39 $</li><li>20 years at 5% →$ FV = 1640(1+0.05)^{20} = 4351.41 $</li></ul></li><li>Connection to Excel tools: FV and PV functions help value future cash flows, whether for single cash flows or series.</li></ul><h3 id="fc7eb8c6-a216-48d9-b734-d6a1d169887d" data-toc-id="fc7eb8c6-a216-48d9-b734-d6a1d169887d" collapsed="false" seolevelmigrated="true">4.2 The Multiperiod Case</h3><ul><li>The transcript lists 4.2 as part of the chapter outline but does not provide explicit content for this section.</li><li>Key idea (general): extend one-period results across multiple periods, summing or discounting cash flows across several time steps; introduces multi-period FV/PV, and interactions with compounding.</li><li>Note for study: rely on the 4.1–4.4 material to handle multi-period timing, discounting, and compounding concepts in later worked examples.</li></ul><h3 id="a35f90b4-e510-4f07-b5d1-9ead233a8675" data-toc-id="a35f90b4-e510-4f07-b5d1-9ead233a8675" collapsed="false" seolevelmigrated="true">4.3 Compounding Periods</h3><ul><li>Core concepts:<ul><li>APR (Annual Percentage Rate): the nominal annual rate without adjusting for compounding periods.</li><li>m = number of compounding periods per year.</li><li>Co = initial investment (capital outlay).</li></ul></li><li>Effective Annual Rate (EAR): the actual rate earned per year after compounding.</li><li>Relationship between APR, compounding, and EAR:<ul><li>General formula for EAR given APR r and m compounding periods per year:</li><li>$ EAR = \left(1 + \frac{r}{m}\right)^m - 1 $</li></ul></li><li>Examples from the notes:<ul><li>Example 1: APR = 24% compounded monthly (m = 12)</li><li>$ EAR = \left(1 + \frac{0.24}{12}\right)^{12} - 1 \approx 0.2682 = 26.82\% $</li><li>Interpretation: EAR reflects the impact of monthly compounding.</li><li>Example 2: APR = 10% compounded quarterly (m = 4)</li><li>$ EAR = \left(1 + \frac{0.10}{4}\right)^4 - 1 \approx 0.10381 = 10.381\% $</li></ul></li><li>Excel helper: =EFFECT(rate, nper) can compute EAR given the nominal rate and compounding frequency.</li><li>Notes:<ul><li>EAR captures the true yearly return when compounding occurs more than once per year.</li><li>Increasing the compounding frequency (with the same APR) increases EAR.</li></ul></li></ul><h3 id="0c536410-1a5a-4780-a186-51fdef3f4477" data-toc-id="0c536410-1a5a-4780-a186-51fdef3f4477" collapsed="false" seolevelmigrated="true">4.4 Simplifications: Perpetuities, Annuities, and Growing Variants</h3><ul><li>Perpetuity: a constant cash flow that lasts forever.<ul><li>PV of a perpetuity paying amount C per year at rate r:$ PV = \frac{C}{r} $</li><li>Example: A perpetuity paying$ 100 per year at 8\%" (r = 0.08) has value $PV = \frac{100}{0.08} = 1250$ ; if r falls to 6%, $PV = \frac{100}{0.06} = 1666.67$ ; indicates inverse relation between rate and value.
Growing perpetuity: cash flows grow at a constant rate g forever.
- Formula (with C1 as the cash flow in year 1): $PV = \frac{C_1}{r - g}$ , with r > g.
- Timing convention: cash flows start in year 1; the doctrine is that growth applies to subsequent payments.
- Example (from EX 4.19 style): Popovich pays a dividend that will grow by g = 6% per year; current dividend C = 3.00; discount rate r = 11%.
- Next-period dividend C1 = C(1+g) = 3.00(1+0.06) = 3.18
- PV of growing perpetuity: $PV = \frac{3.18}{0.11 - 0.06} = \frac{3.18}{0.05} = 63.60$
- Stock price today would be the present value of future dividends, plus any immediate dividend if paid now; the example in the notes shows price = 3 + 63.60 = 66.60.
Annuity: a finite series of equal cash flows (C) at regular intervals for T periods.
- PV of an annuity: $PV = C \times \frac{1 - (1 + r)^{-T}}{r}$
- PVIFA(r, T) is the present value interest factor for annuities, i.e., the factor that multiplies C to give PV.
Growing annuity: annuity payments grow at a constant rate g for a fixed number of periods T.
- PV of growing annuity formula (with C1 = first year cash flow):
- $PV = C_1 \times \frac{1 - \left( \frac{1+g}{1+r} \right)^T}{r - g}$
Applications/examples from the notes:
- EX 4.20 Lottery: monthly payments of $50,000 for 20 years at 8% interest.
- PV = C [ 1 − (1 + r)^−T ] / r = $50{,}000 × 9.8181 = $490{,}907
- PVIFA(8%, 20) = 9.8181 is the factor used.
- PV of an annuity example demonstrates how to value level payments over time.
Quick summary of factors:
- PVIFA(r, T) = \frac{1 - (1 + r)^{-T}}{r}
- FVIFA(r, T) = \frac{(1 + r)^T - 1}{r}
Excel references: use PV, FV, and the PVIFA/FVIFA concepts when computing annuities.

4.5 Loan Amortization

Three loan types:
- Pure Discount Loans: borrower receives money today and repays a single lump sum later (principal + interest).
- Interest-Only Loans: pay interest each period; principal due at maturity.
- Amortized Loans: payments include both interest and principal reduction in each period.
Pure Discount Loans (example): Treasuries (T-bills) example.
- If face value to be repaid at t = 1 year is $10,000 and market rate is 7%:
- PV today: $PV = \frac{10{,}000}{1.07} = 9{,}345.79$
Interest-Only Loans:
- Example structure: 5-year loan, 7% interest, principal $10,000.
- Annual interest payments: $0.07 \times 10{,}000 = 700$ per year for years 1–4; year 5 includes interest plus principal repayment: 10,700.
- Cash flow stream resembles that of a bond with a bullet repayment at maturity.
Amortized Loans with Fixed Principal Payment:
- Example: $50,000 loan, 10 years, 8% interest; fixed principal payments of $5,000 per year plus interest on remaining balance.
- Interest declines over time as principal is repaid.
Amortized Loans with Fixed Payment (level payments):
- Solve for annual payment C using the amortization formula (or financial function):
- $C = \text{PMT}(r, n, PV)$
- Example from notes: With $50{,}000 PV, rate 8%, term 10 years, annual payment ≈ $1{,}509.60$
- Process: each payment covers interest on the outstanding balance and reduces principal.
Practical notes:
- Amortization schedules show how much of each payment goes to interest vs principal over time.
- The PMT function is a standard tool to compute fixed payments for amortized loans.

4.6 What Is a Firm Worth?

Core concept: a firm’s value is the present value of its expected future cash flows, discounted at a required return that reflects risk.
Challenge: determining the size, timing, and risk of those cash flows.
Example framework from notes:
- Cash flows expected: $CF1 = 5{,}000$ at the end of year 1; $CFt = 2{,}000$ for years 2–6; sale value (terminal value) at year 7: $T_7 = 10{,}000$ .
- Required return (discount rate): $r = 10\%$ .
- Firm value: sum of the present values of all cash flows:
- $Value = \sum{t=1}^{7} \frac{CFt}{(1 + r)^t}$ (including terminal value discounted to present).
Real-world relevance: valuation hinges on accurate cash flow forecasting, discount rate determination, and risk assessment.

Textbook Exercises and Examples (Key Takeaways)

Q36 (page 125): Monthly payments of $225$ begin at end of this month, rate 6.5% APR compounded monthly, reaching a balance of $15{,}000$ . Answer: 57 payments.
Q37 (page 125): Borrow $95{,}000<ul><li>Monthly payments of$ 1{,}850;
Term: 60 months;
Find the highest rate you can afford with monthly compounding.
Answer: approximately 6.30%.

Recurring Excel references throughout: =PMT(rate, nper, pv), =PV(rate, nper, pmt, fv), =FV(rate, nper, pmt, pv), =NPER(rate, pmt, pv, fv), =RATE(nper, pmt, pv, fv).

Additional practice items in the notes cover the use of EAR, PV, FV, and annuity/growing annuity formulas for various timing and cash-flow patterns.

Notes on practical interpretation and connections:

The FV and PV formulas underpin everything from investment planning to corporate valuation.
Effective rates (EAR) are critical for comparing investments or loans with different compounding frequencies.
Perpetuities, growing perpetuities, annuities, and growing annuities model common real-world cash-flow patterns (e.g., bonds, dividends, retirements).
Loan amortization links time value concepts to financing decisions, showing how repayment structure affects total interest and cost of capital.
The firm-worth framework emphasizes risk-adjusted discounting and the importance of forecasting cash flows realistically.

Ethical/philosophical/practical implications (brief):

Valuation depends on forecasts of uncertain future cash flows; misestimation can lead to mispriced securities or poor investment decisions.
The choice of discount rate reflects risk tolerance and market conditions; incorrect risk assessment can systematically under- or overvalue projects.
Understanding compounding and timing aids in personal finance (e.g., saving for education or retirement) and corporate financial strategy (capital budgeting, debt management).