A dollar tomorrow is worth less than a dollar today — and there is an exact rule that prices every future stream. Present value PV = Σ CFₜ/(1+r)ₜ: discount every cash flow by the horizon it lands on, sum, and you have its price now. The perpetuity C/r, the annuity, the discount rate — all one identity. Rendered, not quoted.
source Irving Fisher, The Theory of Interest (New York: Macmillan, 1930) — present value & discounting. Book-era print source; link AMBER (no versioned DOI). These are MODELS, not investment advice.
The price today of a cash flow CF that arrives t periods out, at rate r:
Discount factor (1+r)^(−t) falls monotonically in both t and r; at t=0 it is 1, so PV = CF. A level stream forever — the perpetuity — collapses to C/r. A T-period annuity is C·(1−(1+r)^(−T))/r, which → C/r as T→∞.
Tomorrow's dollar, today. Fisher (1930) made time-preference the price of the future: PV = Σ CF/(1+r)ₜ, with a level perpetuity worth exactly C/r.
This is the time-value core sitting beneath the-is-lm's investment schedule — the I(r) curve is nothing but discounted future returns re-priced as r moves. Discounting is the arithmetic; IS-LM is one place it is spent.
Live re-check. Re-runs the full selfcheck() against the engine right now — including the planted linear-discount void from window 6. Green confirms; it flips red the instant the tamper is live.
A cash flow, a rate, a horizon — and a level stream C forever.
Live engine. Discount factors, the summed value, the perpetuity, and the project's IRR — all computed from the pure functions above, no stored numbers.
| t | (1+r)^−t | PV of 100 |
|---|
Proven results, set at boot from the live engine:
Perpetuity C/r is exact; the IRR is a numeric root to 1e−6. Any forward projection of an actual rate is AMBER — a model, not advice.
wall "There is no single true discount rate. Change r and you change every price — NPV rankings flip, IRR can be multiple or undefined, and the whole result is an assumption wearing arithmetic."
True, and load-bearing. Discounting is exact given r; it is silent on which r. Term structure, risk premia and inflation all live inside that one number. IRR is unique only for a conventional sign pattern (one sign change); mixed-sign projects can have several roots or none.
"Money later is worth the same as money now — a dollar is a dollar."
→ Only at r=0. For any r>0, PV strictly falls with the horizon.
"Discounting is linear — twice as far means half as much."
→ It is geometric: (1+r)^(−t), not 1−rt. The linear form goes negative and breaks the perpetuity (see window 6).
"A positive IRR means the project is good."
→ Only if the cost of capital < IRR. IRR is the break-even r where NPV = 0, not a verdict.
Disclosed planted void. Swaps the geometric factor (1+r)^(−t) for the linear (1−r·t). The perpetuity no longer sums to C/r and far-future values go negative. The Witness (7) catches it live.