Time Value of Money
Money has a time value because of its earning potential over time. The fundamental factors relate present, future, and annual values:
- Future value: F = P*(1+i)^n (single payment compound amount, F/P factor)
- Present value: P = F*(1+i)^(-n) (single payment present worth, P/F factor)
- Annual from future: A = F*i/[(1+i)^n - 1] (sinking fund, A/F factor)
- Future from annual: F = A*[(1+i)^n - 1]/i (compound amount, F/A factor)
- Annual from present: A = P*i*(1+i)^n/[(1+i)^n - 1] (capital recovery, A/P factor)
- Present from annual: P = A*[(1+i)^n - 1]/[i*(1+i)^n] (present worth, P/A factor)
Project Evaluation Methods
- Net Present Value (NPV): sum all cash flows discounted to time zero. Accept if NPV > 0
- Internal Rate of Return (IRR): discount rate making NPV = 0. Accept if IRR > MARR
- Annual Worth (AW): convert all cash flows to equivalent uniform annual amount
- Benefit-Cost Ratio: B/C = PW(benefits)/PW(costs). Accept if B/C > 1
- Incremental analysis: compare mutually exclusive alternatives using incremental B/C or IRR
- MARR: Minimum Attractive Rate of Return (hurdle rate for accepting projects)
Depreciation
Depreciation allocates the cost of an asset over its useful life:
- Straight-line: D = (P - S)/n (equal amount each year; P = cost, S = salvage, n = life)
- Double declining balance: D_t = (2/n)*BV_(t-1) (book value at start of year)
- Sum-of-years digits: D_t = [(n-t+1)/SYD]*(P-S), where SYD = n(n+1)/2
- MACRS: IRS-defined percentages; salvage value is assumed zero
- Book value at end of year t: BV_t = P - accumulated depreciation through year t
Break-Even Analysis
- Break-even point: where total revenue equals total cost
- Q_be = Fixed Costs / (Price per unit - Variable cost per unit)
- Useful for deciding between alternatives with different cost structures
- Sensitivity analysis: examine how results change with key parameter variations
Effective Interest Rates
- Nominal rate (r): stated annual rate before compounding adjustment
- Effective rate: i_eff = (1 + r/m)^m - 1 (m = compounding periods per year)
- Continuous compounding: i_eff = e^r - 1
- Always match compounding period to payment period for factor calculations
Inflation
- Inflation reduces purchasing power over time
- Real (inflation-free) rate: i_r = (i - f)/(1 + f)
- Actual dollars: use market interest rate (includes inflation)
- Constant dollars: use real interest rate (inflation removed)
- Both approaches give the same present worth if applied consistently