What rate is this murabaha really charging?

What a stated markup works out to as an annual rate, so a cost-plus plan and an ordinary loan can be compared on one number. It converts a markup into a comparable rate, and says nothing about whether a structure is permissible, and it assumes equal instalments, no deposit, no arrangement fee and no rebate for settling early.

• The same markup over twice the term is close to half this rate, because the money stays outstanding twice as long.

100,000 paid over 4 years comes to 120,000, or 2,500 a month, which costs the same as a loan at 9.24% a year, on the markup as a total, not a rate.

A murabaha states a price and a markup: the financier buys the asset and sells it on at a stated uplift, payable in instalments. That is a different contract from a loan, and it is quoted differently — a total rather than a rate. The difficulty is that a total cannot be compared with a rate, and a reader holding one offer of each has no way to line them up.

So this converts. It takes the total the plan comes to, divides it into equal monthly instalments over the term, and solves for the annual rate an ordinary equal-instalment loan would have to charge to produce exactly those instalments on the same price. That rate is found by bisection, halving a bracket that runs from −99% to 1,000% a year until the two sides agree to nine decimal places; where no rate inside that bracket solves it, the figure is the em dash rather than a ceiling dressed up as an answer.

Whether a structure is permissible is not a question arithmetic can answer, and this page does not answer it. It states no ruling and implies none. What it does is put two offers in the same units, which is a comparison, not a judgement — and a reader who wants the first question answered wants a scholar, not a calculator.

The lesson the conversion exposes is in the term rather than in the markup. The same 20% over eight years is close to half the rate it is over four, because the money is outstanding for twice as long; and 10% over two years is very nearly the same rate as 20% over four. A markup quoted without its term is not a price anyone can compare.

price = monthly × [1 − (1 + r ÷ 12)^(−12n)] ÷ (r ÷ 12), solved for r

r is the annual rate an equal-instalment loan would have to charge to cost the same, n is the term in years, and monthly is the total divided over the months.

A worked example

A cash price of 100,000 with a 20% markup over four years comes to 120,000, or 2,500 a month across 48 instalments. An equal-instalment loan would have to charge 9.24% a year to cost the same. Leave the markup at 20% and stretch the term to eight years and the equivalent rate is 4.66%.

What it assumes

• The markup is the total uplift over the whole term, not a rate per year. This is the field most often filled with the wrong number, because a rate and a markup are both written as a percentage and only one of them compounds.
• Instalments are equal and monthly, and the term is taken as a whole number of months. A plan with a balloon payment at the end, or one that steps up, is a different set of cash flows and produces a different rate.
• There is no deposit and no arrangement fee. Both change the amount actually financed, and a fee taken at the start raises the equivalent rate by more than its size suggests.
• Nothing is settled early. Where a financier grants a rebate for early settlement — and it is discretionary in most contracts rather than owed — the rate a reader actually paid is lower than this one.