In plain English
A krona today is worth more than a krona next year, because today's krona can be put to work. Discounting is just pricing that head start.
The advanced view
Formally, a discount factor is the price today of one unit delivered at time t, so a valuation is a dot product of cash flows and discount factors. Once you see it that way, term structure (a different rate per maturity), continuous compounding, and risk-adjusted rates are all changes to the factor vector, not new mathematics.
A krona today is worth more than a krona tomorrow because it can be invested. Present value discounts a future cash flow back to today; future value compounds a current amount forward. Net present value sums the present values of all cash flows from an investment, including the initial outlay. If NPV > 0 the investment creates value. This is the decision rule that governs all of corporate finance.
PV = CF / (1 + r)^t NPV = −Investment + Σ [CF_t / (1 + r)^t] Annuity PV = CF × [(1 − (1+r)^−n) / r] Perpetuity PV = CF / r Growing perpetuity PV = CF / (r − g) Rule of 72: doubling time ≈ 72 / r%
Internal rate of return is the discount rate that makes NPV zero — it answers "what return does this earn?". Use it for ranking, not accept/reject: it misleads with non-conventional cash flows and mutually exclusive projects. Payback period measures how long until you recover the outlay; simple and intuitive but ignores time value and everything after payback. Discounted payback fixes the time-value problem and still ignores post-payback flows.
Essential vocabulary
- Discount rate
- The rate converting future cash flows to present value. Reflects opportunity cost of capital and the riskiness of the flows.
- Compounding
- Earning returns on returns. More frequent compounding raises effective yield; continuous compounding gives FV = PV × e^(rt).
- Opportunity cost
- The return foregone by choosing one investment over the next best alternative — what the discount rate represents.
- Terminal value
- Value of all cash flows beyond the forecast period, usually a growing perpetuity. Typically 60–80% of DCF value, so the growth rate matters enormously.
Intuition
Discounting is a price, not a penalty. The rate is what capital could earn elsewhere at the same risk, so dividing by (1+r)^t simply restates a future amount in today's money. Two habits make time-value questions fast: put every flow on a timeline before touching a formula, and remember that the perpetuity value CF/(r−g) is dominated by the gap r−g, not by CF.
Common pitfalls
- ×Off-by-one on timing: year-1 flows are discounted once; a valuation at year 0 never discounts year 0.
- ×Mixing nominal cash flows with a real discount rate, which double-counts inflation.
- ×Using IRR to choose between mutually exclusive projects of different size — rank on NPV.
- ×Setting perpetuity growth above the long-run growth of the economy.
Worked example — NPV of a three-year strip
Step 1 of 6
- 1Outlay 100 at t=0; flows 50, 50, 60; r =
Why it works
The formula PV = CF/(1+r)^t works because it is reversible: invest PV at r for t periods and you end with exactly CF. Discounting and compounding are the same operation read in opposite directions, so no arbitrage is possible between them. The perpetuity CF/(r−g) is the limit of that same geometric series — it converges only while g < r, which is why a growth assumption above the discount rate produces nonsense.
How it is used — Rule of 72 as a live sanity check
Step 1 of 4
- 1An interviewer says a fund compounds at 9% and asks for the value in 16 years.
Deeper
Deeper: compounding conventions and the terminal-value problem
Effective annual rate = (1 + r/m)^m − 1 for m compounding periods; continuous compounding gives e^r − 1. Quoted (nominal) rates are not comparable until you convert them. Real vs. nominal follows Fisher: (1 + nominal) = (1 + real)(1 + inflation) — discount nominal cash flows at nominal rates, real at real, never mix.
In a DCF, the terminal value usually carries 60–80% of the total. Two methods: Gordon growth TV = FCF_{n+1} ÷ (WACC − g), or an exit multiple TV = EBITDA_n × multiple. Always cross-check one against the other and back out the implied growth from the multiple.
Must know cold
- ✓EAR = (1 + r/m)^m − 1; continuous = e^r − 1.
- ✓Annuity PV = CF × [1 − (1 + r)^−n] ÷ r.
- ✓Growing perpetuity = CF₁ ÷ (r − g), and g must be below long-run nominal GDP.
- ✓Fisher: nominal ≈ real + inflation.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
A loan quotes 12% nominal, compounded monthly. What is the effective annual rate, and how long does the debt double?
Exercise 2
Year-5 FCF is 100, WACC 9%, g 2.5%. Compute terminal value and its present value. What share of a DCF whose explicit years are worth 330 does it represent?
The same 100 received later is worth progressively less: at 12% a year-five cash flow is worth about 57. Two consequences for cases: the first few years drive most of a DCF's value, and terminal value dominates only because it stands for an infinite tail.