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

p

Starting principal, original amount deposited, invested, or borrowed

A

Accumulated balance, future value, or ending account balance after growth

PMT

Regular or periodic deposit made each period

r

Annual Percentage Rate (APR) expressed as a decimal (e.g., 4.8%=0.048)

n

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

Y

Time in years

nY

Total number of compounding periods or total number of deposits

r/n

Interest rate applied for each single compounding period

W

Desired annual withdrawal in retirement

TA

Target accumulated balance needed at retirement to preserve principal

APY

Annual Percentage Yield (effective annual rate after accounting for compounding)

Simple Interest

$$A = P(1 + rY)$$

Periodic Compounding Formula (e.g., "compounded monthly", "quarterly", "daily")

$$A = P\bigg(1 + \frac{r}{n}\bigg)^{nY}$$

Continuous Compounding

$$A = P e^{rY}$$

"How much must be deposited today to reach a target?" (Present Value)

$$P = \frac{A}{\bigg(1 + \frac{r}{n}\bigg)^{nY}}$$

Asks for total contributions

$$\text{PMT}\times(nY)$$

Asks for total interest earned in a savings plan

$$\text{interest earned} = A - \text{total contributions}$$

Required Target Balance

$$\text{TA} = \frac{W}{r}$$

Periodic Compounding

$$\text{APY} = \bigg(1 + \frac{r}{n}\bigg)^n - 1$$

Continuous Compounding

$$\text{APY} = e^r - 1$$

"Overall percentage change over entire holding period"

$$\text{total return} = \frac{A - P}{P}$$

"Equivalent constant yearly rate"

$$\text{annualized return} = \bigg(\frac{A}{P}\bigg)^{\frac{1}{Y}} - 1$$

"Deposited today", "one-time deposit", or "starts at $P$" without recurring contributions.

One-Time Investments (Single Lump Sum)

"Regular deposit", "monthly payment", "deposited each period", or "equal recurring deposits"

Savings Plans (Repeated / Recurring Deposits)

"Withdraw $W$ dollars per year forever", "indefinitely", or "without decreasing principal"

Retirement Target (Preserved Principal)

Gives regular deposit $\text{PMT}$ and asks for future balance $A$

$$A = \text{PMT} \times \left[ \frac{\left(1 + \frac{r}{n}\right)^{nY} - 1}{\frac{r}{n}} \right]$$

Gives target future balance $A$ and asks for required regular deposit$\text{PMT}$

$$\text{PMT} = A \times \left[ \frac{\frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nY} - 1} \right]$$

"Effective annual rate" or "one-year effective yield"

Yield and Investment Returns

e

2.71828