Compounded Annual Interest Formula: $FV = P \times (1 + R)^t$

Future Value of an Annuity Formula: $FV_{An} = A \left[ \frac{(1+r)^n - 1}{r} \right]$

FVIF for Annuity: The term $\frac{(1+r)^n - 1}{r}$ is the future value interest factor for an annuity ($FVIF_{Arn}$).

Present Value: Calculates the worth of future money today.
Process: "Discounting" - for any positive return rate, PV will be less than future value.
Discount Rate: Interest rate used to calculate PV of future cash flow.
Inverse Relationship: PV is the mirror image of FV.
Formula: $PV = FV \times \frac{1}{(1+r)^n}$
Discounting Factor: The term $\frac{1}{(1+r)^n}$ is the discounting factor or present value interest factor ($PVIF_{r,n}$).
Common Uses for PV:
- Calculating value of pension annuity payments vs. lump sum.
- Determining if business owner's investment meets profit expectations.
- Valuing a business.

Importance: Crucial for financial planning, from asset buy/lease decisions to new equipment investments.
Financial Management Standpoint:
- Helps determine impact of debt on earnings and profits for expansion/growth companies.
- Essential for managing funds and generating profits due to future uncertainty.
- Aids financial decision-making (e.g., choosing Project 'A' with higher present value over Project 'B' with later payout).

PV of Future Cash Flows: $PV = FV \times (1+r)^{-n}$ or $FV = PV \times (1+r)^n$
- $PV$: Present Value.
- $FV$: Future Value.
- $r$: Interest rate.
- $n$: Number of periods.

Compounding frequency significantly impacts FV.
More frequent compounding (quarterly, monthly, daily) leads to higher FV.
Formula for multiple compounding periods: $FV = PV \times \left(1 + \frac{r}{t}\right)^{xt}$
- $PV$: Present Value.
- $r$: Annual interest rate.
- $x$: Number of years.
- $t$: Number of compounding periods of interest per year.

Key Concept: Used in Net Present Value (NPV), Compound Annual Growth Rate (CAGR), Internal Rate of Return (IRR).
Application: General formulas apply to any series of cash flows.
Tools: Financial calculators, spreadsheet programs (Excel functions: PV, FV, IRR, NPV).
Discount Rate: Selecting correct rate (e.g., Weighted Average Cost of Capital - WACC) is crucial; wrong decision can render process meaningless.

Definition: In investments, risk means the possibility of an "unexpected" negative or "adverse" return.
Risk vs. Risk-less: