MATH 107 Comprehensive Study Notes: Budgets, Growth, Savings Plans, and Investment Returns

1. Symbol and Variable Definitions

  • PP: Starting principal, original amount deposited, invested, or borrowed.

  • AA: Accumulated balance, future value, or ending account balance after growth.

  • PMT\text{PMT}: Regular or periodic deposit made each period.

  • rr: Annual Percentage Rate (APR) expressed as a decimal (e.g., 4.8%=0.0484.8\% = 0.048).

  • nn: Number of compounding periods (and deposits) per year (e.g., monthly n=12n = 12, quarterly n=4n = 4, daily n=365n = 365).

  • YY: Time in years.

  • nYnY: Total number of compounding periods or total number of deposits.

  • rn\frac{r}{n}: Interest rate applied for each single compounding period.

  • WW: Desired annual withdrawal in retirement.

  • TA\text{TA}: Target accumulated balance needed at retirement to preserve principal.

  • APY\text{APY}: Annual Percentage Yield (effective annual rate after accounting for compounding).

2. One-Time Investments (Single Lump Sum)

Look for phrases like "deposited today", "one-time deposit", or "starts at PP" without recurring contributions.

  • Mentions "simple interest": Use A=P(1+rY)A = P(1 + rY)

  • Mentions periodic compounding (e.g., "compounded monthly", "quarterly", "daily"): Use A=P(1+rn)nYA = P\left(1 + \frac{r}{n}\right)^{nY}

  • Mentions "compounded continuously": Use A=PerYA = P e^{rY}

  • Asks "how much must be deposited today to reach a target?" (Present Value): Use P=A(1+rn)nYP = \frac{A}{\left(1 + \frac{r}{n}\right)^{nY}}

3. Savings Plans (Repeated / Recurring Deposits)

Look for phrases like "regular deposit", "monthly payment", "deposited each period", or "equal recurring deposits".

  • Gives regular deposit PMT\text{PMT} and asks for future balance AA: Use A=PMT×[(1+rn)nY−1rn]A = \text{PMT} \times \left[ \frac{\left(1 + \frac{r}{n}\right)^{nY} - 1}{\frac{r}{n}} \right]

  • Gives target future balance AA and asks for required regular depositPMT\text{PMT}: Use PMT=A×[rn(1+rn)nY−1]\text{PMT} = A \times \left[ \frac{\frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nY} - 1} \right]

  • Asks for total contributions: Use total contributions=PMT×(nY)\text{total contributions} = \text{PMT} \times (nY)

  • Asks for total interest earned in a savings plan: Use interest earned=A−total contributions\text{interest earned} = A - \text{total contributions}

4. Retirement Target (Preserved Principal)

Look for phrases like "withdraw WW dollars per year forever", "indefinitely", or "without decreasing principal".

  • Find required target balance: Use TA=Wr\text{TA} = \frac{W}{r}

5. Yield and Investment Returns

  • Asks for "effective annual rate" or "one-year effective yield": Use APY formulas:

    • Periodic compounding: APY=(1+rn)n−1\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1

    • Continuous compounding: APY=er−1\text{APY} = e^r - 1

  • Asks for "overall percentage change over entire holding period": Use total return=A−PP\text{total return} = \frac{A - P}{P}

  • Asks for "equivalent constant yearly rate": Use $$\text{annualized return} = \left(\frac{A}{P}\right)^{\frac{1}{Y}} - 1