Notes on Chapter 4 Part 2: Rate of Return and Amortization (Nominal vs Periodic Rates; Compounding)

Nominal rate, periodic rate, and compounding frequency

• Time value of money recap: future value (FV) and present value (PV) depend on the interest rate and how often it is applied (compounded).
• Nominal rate (quoted rate): a rate stated in contracts or disclosures. It does not automatically equal the rate used directly in computations unless you align it with the compounding frequency.
• APR vs nominal rate: in practice, the quoted annual rate in contracts is often called APR (annual percentage rate), but the computation uses the periodic rate after dividing by the number of compounding periods per year.
• Compounding frequency (m): the number of times per year the interest is applied.
• Periodic rate (i): the rate used in each compounding interval, given by
  $i = \frac{r}{m}$
  where $r$ is the annual nominal rate and $m$ is the number of compounding periods per year.
• Common values of $m$:
  • Monthly: $m=12$
  • Semiannual: $m=2$
  • Quarterly: $m=4$
  • Daily: $m=365$ (or 360 in some contexts)
• You may see a gap between the quoted annual rate and the rate used for calculations; you must convert to the periodic rate to compute values like FV or PV.
• Example terminology:
  • If a credit card quote says 24% APR, that is the nominal annual rate; the daily or monthly periodic rate is what actually accrues interest depending on how often they compound.
  • Daily compounding vs monthly compounding affects the effective cost of borrowing even if the quoted APR is the same.

Periodic rate, compounding, and payments

• Daily compounding example: daily periodic rate is
  $i<em>{\text{daily}} = \frac{r}{365}$ If $r = 0.10$ (10%), then $i</em>{\text{daily}} = \frac{0.10}{365} \approx 0.00027397.$
• Monthly compounding example: monthly periodic rate is
  $i_{\text{monthly}} = \frac{r}{12} = \frac{0.10}{12} \approx 0.0083333.$
• If the same nominal annual rate is applied, increasing frequency generally increases the effective annual rate (EAR).

Effective annual rate and continuous compounding

• Effective annual rate (EAR) with periodic compounding is
  $EAR = \left(1 + \frac{r}{m}\right)^{m} - 1.$
• Continuous compounding (the limit as $m \to \infty$):
  -FV with continuous compounding: $FV = PV \; e^{rt}$
  -Annual effective rate under continuous compounding: $EAR_{\text{cont}} = e^{r} - 1.$
• The continuous-compounding framework provides a theoretical upper bound on the true annual growth for a given nominal rate.

Worked concepts and common calculations

• You often compute FV or PV with a mix of cash flows (PMT) and a lump-sum PV:
  • Lump-sum FV: $FV = PV\,(1+i)^{n}$
  • With level payments (annuity): end-of-period payments (typical):
    $FV = PV\,(1+i)^{n} + PMT\;\frac{(1+i)^{n} - 1}{i}$
  • If payments occur at the beginning of each period (annuity due):
    $FV = PV\,(1+i)^{n} + PMT\;\frac{(1+i)^{n} - 1}{i}\; (1+i)$
• Sign convention in many software (Excel): cash inflows are positive, cash outflows are negative. For FV with\ PV, you typically enter PV as negative if you owe money.
• Example setup (daily compounding): a $100 loan, APR 10%, daily compounding for 30 days, no payments (PMT = 0)
  • PV = -100, $r = 0.10$, $m = 365$, $i = r/m = 0.10/365$, $n = 30$
  • FV = PV (1+i)^{n} = -100\,(1 + 0.10/365)^{30}$
  • Interest paid over the period ≈ $FV - PV$ (in absolute terms).
• Example numeric check (daily vs monthly for 30 days horizon):
  • Daily: $i{\text{daily}} = \frac{0.10}{365}$, $n = 30$ → $FV{\text{daily}} = 100\cdot (1+i_{\text{daily}})^{30} \approx 100.82$; Interest ≈ $0.82$.
  • Monthly: $i{\text{monthly}} = \frac{0.10}{12}$, $n = 1$ (one month) → $FV{\text{monthly}} = 100\cdot (1+i_{\text{monthly}})^{1} \approx 100.83$; Interest ≈ $0.83$.
  • Interpretation: for fixed horizon measured in years, increasing compounding frequency increases the effective amount; differences are small over short horizons but accumulate for longer horizons or larger balances.
• Practical note: to compare fairly, convert the horizon to years and compute using $n = m t$ with $t$ in years.

Excel usage notes (FV function and conventions)

• Excel FV function syntax: $FV(rate, nper, pmt, [pv], [type])$
  • rate: periodic rate, not the annual nominal rate. If you have an annual rate $r$ and compounding $m$ times per year, then rate = $r/m$.
  • nper: number of compounding periods (for horizon $t$ years, $nper = m t$).
  • pmt: payment per period (0 if no recurring payments).
  • pv: present value (enter as a negative number if it represents an outflow).
  • type: 0 for payments at end of period, 1 for payments at beginning.
• Examples:
  • Daily compounding scenario for 30 days with $100 loan, $r=0.10$, $m=365$, $n=30$:
  • rate = $0.10/365$, nper = 30, pv = -100, pmt = 0, type = 0
  • FV gives the amount owed after 30 days. Interest = FV - 100 (absolute value).
  • Monthly compounding for 1 month with $100 loan, $r=0.10$, $m=12$, $n=1$:
  • rate = $0.10/12$, nper = 1, pv = -100, pmt = 0, type = 0
  • FV ≈ 100.83; Interest ≈ 0.83.
• Practical guidance:
  • Always ensure rate is the periodic rate, not the annual nominal rate, when using FV or PV formulas in Excel.
  • The sign convention matters for interpreting FV and the derived interest.
  • When horizon spans multiple years, use $nper = m \cdot t$ with $t$ in years.

Real-world relevance and implications

• Why it matters: lenders advertise a nominal annual rate, but the actual cost to you depends on how often interest is compounded (EAR).
• Credit cards often advertise a high APR, but the daily or daily-equivalent periodic rate compounds many times per year, yielding a higher effective cost than the nominal rate suggests.
• For borrowers: understanding EAR helps compare loan offers with different compounding schedules.
• For savers: more frequent compounding (at a given nominal rate) increases accumulated wealth over time.
• Conceptual bridge to continuous compounding: as compounding becomes more frequent, the effective rate approaches the continuous-compounding limit; in the limit, EAR_cont = e^{r} - 1.

The big-picture takeaway

• Nominal rate is a quoted annual rate; the actual growth depends on compounding frequency m.
• Periodic rate is the rate you plug into per-period formulas: $i = \frac{r}{m}.$
• EAR captures the true annual growth after compounding effects: $EAR = \left(1 + \frac{r}{m}\right)^{m} - 1.$
• Continuous compounding provides the theoretical upper bound: $FV = PV e^{rt}, \quad EAR_{\text{cont}} = e^{r} - 1.$

Quick practice prompts

• Practice 1: A $1,000 loan carries an annual nominal rate of 8% compounded monthly. What is the FV after 2 years? (Set $PV=1000$, $r=0.08$, $m=12$, $t=2$; compute using $nper = m t$ and rate = $r/m$.)
• Practice 2: Compare the EAR for 6% nominal rate with daily compounding vs monthly compounding. Which is higher and by how much roughly? (Compute $EAR = \left(1 + \frac{0.06}{365}\right)^{365} - 1$ and $EAR = \left(1 + \frac{0.06}{12}\right)^{12} - 1$.)

Ethical and practical reflection

• Be mindful that real-world consumer debt can be expensive due to compounding frequency; understanding these concepts helps assess affordability and planning.
• Financial decisions should consider both nominal rates and the effective costs/benefits over the actual time horizon of borrowing or saving.

Summary of key formulas

• Periodic rate from nominal annual rate: $i = \frac{r}{m}$
• Effective annual rate: $EAR = \left(1 + \frac{r}{m}\right)^{m} - 1$
• Lump-sum future value: $FV = PV\,(1+i)^{n}$
• Lump-sum with payments (end of period): $FV = PV\,(1+i)^{n} + PMT\;\frac{(1+i)^{n} - 1}{i}$
• Continuous compounding FV: $FV = PV\; e^{rt}$
• Continuous compounding EAR: $EAR_{\text{cont}} = e^{r} - 1$