CHAPTER 9: Compound Interest Formula (part 2)

The Core Formula

• Definition: $A = P\left(1 + \frac{r}{n}\right)^{nt}$
  • $A$ = future value of the investment/loan
  • $P$ = principal
  • $r$ = annual interest rate (decimal)
  • $n$ = number of times interest is compounded per year
  • $t$ = time in years
• Interest earned (if you want only interest): $I = A - P = P\left(\left(1 + \tfrac{r}{n}\right)^{nt} - 1\right)$
• Quick example (monthly compounding):
  • $P = 10{,}000\,$
  • r = 0.06\
  • $n = 12$
  • $t = 20$
  • $A = 10{,}000\left(1 + \frac{0.06}{12}\right)^{12\cdot 20} = 33{,}102.04$
  • Interest earned: $I = A - P = 23{,}102.04$
• Quick takeaway: compound interest grows faster than simple interest over time.

Variations by compounding period

• General form (same as core): $A = P\left(1 + \frac{r}{n}\right)^{nt}$
• Common special cases:
  • Annual: $A = P(1 + r)^{t}$
  • Quarterly: $A = P\left(1 + \frac{r}{4}\right)^{4t}$
  • Monthly: $A = P\left(1 + \frac{r}{12}\right)^{12t}$
  • Daily: $A = P\left(1 + \frac{r}{365}\right)^{365t}$
• Inverse formulas:
  • Solve for P: $P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$
  • Solve for r: $r = n\left[\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right]\times 100\,\%$
  • Solve for t: $t = \frac{\ln\left(\frac{A}{P}\right)}{n\ln\left(1 + \frac{r}{n}\right)}$
• Note on Effective Annual Rate (EAR):
  • $EAR = \left(1 + \frac{r}{n}\right)^n - 1$
  • Nominal rate r, split across periods; EAR may differ across compounding frequencies.

How to use the compound interest formula

• Steps:
  • Identify $P, r, n, t$ from the scenario.
  • Compute $A = P\left(1 + \frac{r}{n}\right)^{nt}$.
  • If you want only interest, use $I = A - P$.
• Quick consistency check example (monthly):
  • For monthly compounding with hypothetical values, the same steps apply as in the core formula.

Monthly contributions (compound interest with deposits)

• When you make regular deposits (PMT) each period, the future value combines principal growth and the annuity part:
  • End of period deposits:
    $A = P\left(1 + \frac{r}{n}\right)^{nt} + PMT \cdot \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$
  • Beginning of period deposits (annuity due):
    $A = P\left(1 + \frac{r}{n}\right)^{nt} + PMT \cdot \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \times \left(1 + \frac{r}{n}\right)$
• Example (end-of-period deposits):
  • If $P = 5{,}000$, $PMT = 100$, $r = 0.03$, $n = 12$, $t = 10$,
  • Principal growth: $P\left(1 + \frac{r}{n}\right)^{nt} = 5{,}000\left(1 + \frac{0.03}{12}\right)^{120} \approx 6{,}746.77$
  • Series growth: $PMT \cdot \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \approx 13{,}974.14$
  • Total: $A \approx 20{,}720.91$
• Example (principal-only, and then deposits): the same structure applies; you add the annuity portion to the grown principal.

Using Excel or Google Sheets

• Basic future value (no deposits):
  • $A = P\left(1 + \frac{r}{n}\right)^{nt}$
• With monthly deposits (end of period):
  • $A = P\left(1 + \frac{r}{n}\right)^{nt} + PMT \cdot \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}$
• Related built-in function:
  • FV(rate, nper, pmt, [pv], [type]) can compute the future value directly, where
  • rate = annual rate / n
  • nper = nt
  • pmt = PMT
  • pv = -P
  • type = 0 (end of period) or 1 (beginning)

Quick reference: key notes

• The basic formula assumes a nominal annual rate r split evenly across the periods.
• For precise equivalence of effective annual rate across different compounding intervals, use EAR formula above.
• The simple vs compound comparison (example):
  • Simple interest over 10 years on $P = 10{,}000, r = 0.03$: $I_{simple} = P r t = 10{,}000 \times 0.03 \times 10 = 3{,}000$
  • Compound interest over same terms yields higher total: $A = 10{,}000\left(1 + \frac{0.03}{12}\right)^{120} \approx 13{,}493.54$, so $I_{compound} = A - P \approx 3{,}493.54$
• Period settings: same formulas apply with the appropriate n (e.g., monthly n = 12, daily n = 365).

Summary tips for quick recall

• Always identify P, r, n, t first; then apply the core formula for A.
• To include monthly deposits, add the annuity term to the principal growth term.
• Use the EAR formula to compare different compounding frequencies on an apples-to-apples basis.
• For back-solving (finding P, r, or t), use the inverse formulas provided above.