The rate you were quoted is not the rate you pay

PMT = PV · i / ( 1 − (1 + i)^−n )

i = nominal annual rate ÷ periods per year
if i = 0  →  PMT = PV ÷ n

The fixed amount paid at the end of each period so that the balance reaches exactly zero on the final instalment. Identical to Excel’s PMT.

IPMT(k) = B(k−1) · i
PPMT(k) = PMT − IPMT(k)
B(k)    = B(k−1) − PPMT(k)

On a reducing balance, interest is charged only on the balance outstanding at the start of the period. As the balance falls, the interest share of each instalment falls with it.

Total interest = PV · r · T
PMT           = ( PV + total interest ) ÷ n

A flat quote charges interest on the whole original amount for the whole term, as if no instalment had ever been paid. But you start repaying principal immediately, so on average you hold only about half the original sum. A 6% flat quote over 24 monthly instalments works out at an effective 11.9%; an 18% flat quote reaches 36.4%.

Interest-only:  PMT(k) = B · i   ,   B unchanged
Capitalised:    B(k) = B(k−1) · (1 + i)
                → B_amort = PV · (1 + i)^g

This one runs against intuition: with no fees, a grace period does not change the effective rate at all. A 20% nominal loan returns an effective 21.94% with no grace, with interest-only grace, and with capitalised grace alike — because in every case the lender charges exactly i on the outstanding balance, so the IRR stays at i. What grace changes is the amount: in that example capitalised grace lifts total repayment from 116.16m to 116.81m. Add a fee, though, and a longer horizon spreads that fee thinner and pushes the effective rate down.

Fee = PV · (pct ÷ 100) + fixed

Deducted:  net = PV − fee   , instalments based on PV
Added:     net = PV         , instalments based on PV + fee

A fee never touches the quoted rate but always moves the true one, because you either receive less or repay more. The shorter the term, the harder a fixed fee hits the effective rate — it is spread over less time.

NPV(i*) = Σ  CF(k) / (1 + i*)^k  =  0
         k=0..n

CF(0) = + net advanced
CF(k) = − instalment k

The true rate is the rate at which the present value of everything you pay equals the cash you actually received. For a conventional loan the NPV function is strictly monotonic in the rate, so the root is unique; this tool finds it by bisection over 200 iterations — no initial guess, no divergence.

Σ  CF(k) / (1 + R)^τ(d₀, d_k)  =  0

τ = year fraction under the chosen day-count convention

IRR assumes every period has equal length. XIRR uses the real distance between dates, so months of 28 to 31 days are handled honestly. Under Actual/365 Fixed this matches Excel’s XIRR exactly; under Actual/Actual it matches the basis prescribed for the European APRC.

Actual/365F   τ = days ÷ 365            ISDA §4.16(d)
Actual/360    τ = days ÷ 360            ISDA §4.16(e)
Actual/Actual τ = Σ days_y ÷ (365|366)  ISDA §4.16(b)
30/360 US     D1=31→30 ; D2=31 & D1>29→30   §4.16(f)
30E/360       D1=31→30 ; D2=31→30           §4.16(g)

Two dates do not imply one unique length of time — that depends on the convention in the contract. Actual/360 counts more year-fractions per calendar day, so the same cash flows imply a lower annual rate: 100 repaid as 110 after 365 days solves to 10.000% on Actual/365F but 9.856% on Actual/360. The choice moves XIRR and duration only; the periodic IRR, the Reg Z APR and the EAR are unaffected.

US  Reg Z    APR  = i* × m          12 CFR §1026 App. J
EU  CCD      APRC = (1 + i*)^m − 1  2008/48/EC Annex I

→  APRC ≥ APR always, equal only when m = 1

The same loan has two different legally-correct "annual percentage rates". In the United States, Regulation Z defines APR by the actuarial method as the periodic rate multiplied by the number of unit-periods per year — compounding within the year is deliberately ignored. In the European Union, Directive 2008/48/EC Annex I defines the APRC as the rate that equates present values, which is an effective rate. A 2% monthly rate is a 24.00% Reg Z APR and a 26.82% EU APRC. This tool reports both, labelled.

EAR = (1 + i*)^m − 1

EAR = (1 + APR/m)^m − 1  ≥  APR

Fully reflects compounding, which is why this tool shows it in the largest type. When comparing loans with different repayment frequencies — monthly against quarterly — it is the only sound basis. Identical in construction to the EU APRC.

D_mac = Σ [ τ_k · PV(CF_k) ] ÷ Σ PV(CF_k)
D_mod = D_mac ÷ (1 + y/m)
WAL   = Σ [ τ_k · principal_k ] ÷ Σ principal_k

Macaulay duration is the present-value-weighted average time to be repaid, discounted at the loan’s own yield; modified duration turns it into a sensitivity to a small parallel shift in rates. Weighted-average life is the same idea on principal alone, undiscounted — the securitisation convention. WAL always exceeds duration, and duration always shortens as yield rises. For a single bullet repayment, duration equals maturity exactly.

Right — one IRR from the merged flow:

  CF(t) = Σ  CF_j(t)        for every date t
           j
  Σ  CF(t) / (1 + R)^τ(t₀, t)  =  0

Wrong — a weighted average:

  R̄ = Σ (net_j × EAR_j) ÷ Σ net_j

A rate is a ratio, not a quantity, and averaging ratios is a classic error. The sound method merges every loan’s cash flow onto one shared calendar and solves a single IRR from the total, which weights each loan by money × time automatically. An expensive six-month loan gets the same weight in a naive average as an expensive five-year loan, while tying up a fraction of the money-time — an error that reaches 5.19 percentage points in a realistic pair.

Cumulative series:  S(k) = Σ CF(0..k)

If S(k) changes sign exactly once
        →  the IRR is unique ✓

A single loan is simple: one inflow, then only outflows — so the root is unique. But merge several loans with different start dates and you may receive money again mid-stream, flipping the sign more than once. By Descartes’ rule of signs, several mathematical IRRs may then exist. This tool tests Norstrøm’s condition and says so when it fails, rather than quietly reporting one root as if it were the answer.

Solar Hijri → Julian Day Number → Gregorian

Month lengths:  1–6   → 31 days
                7–11  → 30 days
                Esfand → 29 or 30 (leap)

Conversion uses the "breaks-year" algorithm, which follows the irregular Solar Hijri leap cycle exactly — unlike the naive 33-year approximation, which drifts by a day every few decades. Verified day by day against the system reference calendar across 25,567 days from 1990 to 2060. All financial mathematics runs on Gregorian dates; the Solar Hijri calendar is a display layer only, so no conversion error can enter the numbers.