NPV, IRR, Discounting and Amortisation Concepts for Finance
ST226 Written Answer Cheat Sheet (Plain Text)
Net Present Value (NPV) and Its Interpretation
NPV represents the present value of all future net cash flows minus the initial cost. A positive NPV indicates value creation at the chosen discount rate. A negative NPV indicates value destruction.
Why Discounting Is Required
Discounting is required because money received in the future is worth less than money now. Discounting converts future amounts into their equivalent today. The discount rate reflects the required return or opportunity cost. A higher rate reduces the weight of future cash flows and lowers NPV.
Project Selection Rules and Rate Effects
For project choice, accept all independent projects with positive NPV. For mutually exclusive projects, choose the one with the highest positive NPV. As the interest rate increases, NPV decreases because discount factors fall. NPV tends to be a smooth decreasing function of the rate.
Internal Rate of Return (IRR)
IRR is the interest rate at which NPV equals zero. It measures the compounded return of the cash-flow pattern. IRR can be misleading if cash-flow signs change multiple times or if comparing mutually exclusive projects. NPV is the preferred decision rule in such cases. If IRR exceeds the required return the project is acceptable, but NPV should guide comparison decisions.
Annuities: Immediate and Due
A level annuity has a closed form because discounted payments form a geometric progression. An annuity due has higher value because its payments are discounted for one less period compared to an annuity immediate.
Higher interest rates reduce the present value of an annuity since each payment is discounted more heavily.
Varying Interest Rates Over Time
If interest rates vary over time, each cash flow must be discounted with its specific rate. A single rate cannot represent a term structure.
Discounting with varying rates is done by multiplying one-period discount factors. The discount factor to period t is the product of all previous (1 plus rate) inverses.
Inflation, Nominal and Real Rates
Inflation indexing scales nominal cash flows by the inflation factor. Discounting nominal cash flows must use nominal interest rates. Mixing real and nominal quantities gives incorrect NPVs.
Real cash flows must be discounted at real rates and nominal cash flows at nominal rates. Using the wrong pairing misprices the present value.
Amortisation Schedules and Loan Repayments
An amortisation schedule shows how each payment splits into interest and principal. Interest equals the period rate times the outstanding balance, and principal equals payment minus interest. The balance falls period by period.
Higher interest rates increase the interest portion of each payment and slow repayment of principal.
The final payment of an amortising loan may need adjusting to bring the balance exactly to zero.
If revenue is first used to repay a loan, profits only appear once the balance is fully cleared. The timing of loan reduction therefore affects investor returns.
Plots and Visual Diagnostics
An NPV plot shows how value changes with the discount rate. The crossing point with zero is the IRR. If the plot is smooth and monotonic, only one IRR is likely.
An amortisation plot shows a declining outstanding balance. When payments are constant the balance falls faster over time because interest shrinks each period.
Revenue minus cost plots show how net cash flow evolves. Rising costs or flattening revenue will reduce net cash flow over time.
Money-Weighted Return and Time-Weighted Return
MWRR is the internal rate of return of a personal investment stream. It solves the equation equating discounted contributions with discounted final redemption. MWRR reflects investor-specific timing. It differs from the time-weighted return which removes timing effects.
Sensitivity to Timing and Subsidies
Projects with large early costs or late benefits have NPVs very sensitive to the discount rate. Subsidised loans reduce initial outflow but may increase long-term interest burden.
Numerical Root Finding and IRR
Uniroot finds roots of continuous functions. NPV as a function of interest rate is continuous and usually monotonic, which makes uniroot suitable for IRR.
Why Averages Fail for Variable Rates
Variable rates cannot be replaced with an average rate because timing matters. Early high rates discount early flows more heavily than a simple average captures.
Timing Differences and Nominal Profits
Two strategies with equal total nominal profit can have different NPVs because of timing. Earlier inflows have higher present value.
IRR Nonexistence and Multiplicity
IRR may not exist or may not be unique if the cash-flow sign changes multiple times. This produces multiple crossings of zero in the NPV curve.
Nominal Discounting Appropriateness
Nominal discounting is appropriate for nominal cash flows because both include inflation. Mixing nominal flows with real discount rates under-discounts them.
Revenue-Linked Amortising Loans
A revenue-linked amortising loan can finish early if revenue spikes, because the higher payment repays more principal quickly. Once the balance hits zero the loan terminates.
Key Concepts Recap
- Discount consistently: nominal with nominal, real with real.
- Prefer NPV for mutually exclusive project comparisons.
- Be cautious with IRR when signs change multiple times.
- Account for timing: early cash flows increase present value.