1) Present Value (PV)

Idea: a dollar today is worth more than a dollar tomorrow; discount future cash flows to today’s dollars using an appropriate rate.

• Single cash flow: PV = Future cash / (1 + r)^n
• Ordinary annuity (equal payments, end of period): present value is the sum of discounted payments; use your preferred calculator or spreadsheet function.
• Perpetuity: constant cash each period forever; PV equals cash flow divided by discount rate.
• Growing perpetuity: cash grows at rate g; PV equals first-period cash divided by (r − g) when r > g.

2) Net Present Value (NPV)

Definition: the sum of all discounted cash inflows minus the initial outlay.

Decision rule: accept projects with NPV > 0 (they add value given the chosen discount rate); among mutually exclusive projects, prefer the one with the higher NPV.

In spreadsheets, use =NPV(rate, cash_flows_range) − initial_outlay or =XNPV(rate, dates, cash_flows) when timing is irregular.

3) Internal Rate of Return (IRR)

Definition: the discount rate that sets NPV to zero; the project’s implied average rate of return.

• Decision rule: accept if IRR > hurdle rate (cost of capital).
• Use with care: multiple sign changes can lead to multiple IRRs; for mutually exclusive projects, rely on NPV at the correct discount rate.
• Spreadsheet tip: use =IRR(range) or =XIRR(cash_flows, dates) for irregular timing.

4) Mini Examples

• Annuity: A five-year lease pays the same amount annually. Discount each payment at the contractually appropriate rate to compute PV; compare PV to buy-vs-lease alternatives.
• Perpetuity: A preferred stock paying fixed dividends is valued by dividing the dividend by the required return.
• Project selection: Two projects with different timing: compute both NPVs at the firm’s WACC; choose the larger NPV even if IRR is smaller.

5) Quick Checks

• Confirm the discount rate matches the risk and currency of the cash flows.
• Use XNPV/XIRR when dates are irregular; month-count assumptions matter.
• Sensitivity-test NPV to the discount rate and to key drivers (growth, margins, terminal values).