Islamic Finance Calculations: Murabaha, Ijarah and Profit-Sharing

Islamic banking has grown rapidly in Pakistan and across the world, and students of finance, banking and Islamic economics are now expected to handle its calculations as confidently as conventional interest problems. The contracts are different in substance, since Islamic finance prohibits riba (interest) and requires trade, leasing or genuine partnership, but the mathematics of time value still applies. This article works through Murabaha, Ijarah, Mudarabah and Musharakah with numbers you can check.

The Core Principle

In a conventional loan, the bank lends money and charges interest on it. In Islamic finance, the bank's return must come from a real economic activity:

• Murabaha: the bank buys an asset and sells it to the client at cost plus a disclosed profit, payable later.
• Ijarah: the bank owns an asset and rents it to the client.
• Mudarabah: one party provides capital, the other provides management; profit is shared, and financial loss falls on the capital provider.
• Musharakah: all partners contribute capital; profit is shared by agreement and loss by capital ratio.

Banks often use a benchmark such as KIBOR to price these contracts. Scholars permit this as a pricing reference, but the legal and Shariah structure of the contract is what distinguishes it from a loan.

Murabaha (Cost-Plus Sale)

The client knows the cost and the profit margin. Once the Murabaha sale is concluded, the sale price is fixed: it cannot be increased if payment is late. Late-payment amounts, where agreed, go to charity rather than to the bank's income.

Worked Example 1: Murabaha for Machinery

A Gujranwala manufacturer needs machinery costing PKR 3,000,000. The Islamic bank buys it, takes possession, and sells it to the manufacturer at a total markup of 20%, payable in 12 equal quarterly instalments over 3 years.

Step 1: Sale price.

$$ \text{Sale price} = 3{,}000{,}000 \times 1.20 = \text{PKR } 3{,}600{,}000 $$

Step 2: Quarterly instalment.

$$ \frac{3{,}600{,}000}{12} = \text{PKR } 300{,}000 $$

Step 3: What return is the bank effectively earning? The flat 20% over three years looks like 6.67% a year, but the client pays down the price steadily, so the bank's money is outstanding for less time on average. The implied quarterly rate \( q \) solves

$$ 3{,}000{,}000 = 300{,}000 \times \frac{1 - (1 + q)^{-12}}{q}. $$

Solving numerically (by trial or with Excel's RATE(12, -300000, 3000000)) gives \( q \approx 2.923\% \) per quarter.

Step 4: Annualise.

$$ \text{Nominal annual rate} = 4 \times 2.923\% \approx 11.69\%, \qquad \text{Effective} = (1.02923)^4 - 1 \approx 12.21\% $$

Sanity check: at \( q = 0.02923 \), the annuity factor \( (1 - 1.02923^{-12})/0.02923 \approx 10.0 \), and \( 300{,}000 \times 10.0 = 3{,}000{,}000 \). The flat 20% markup corresponds to an effective rate of roughly 12.2% a year, which is the figure a finance student should use to compare it with other offers.

Ijarah (Leasing)

In Ijarah the bank remains the owner throughout the lease. It therefore bears ownership risks, such as major damage or destruction not caused by the lessee, and is usually responsible for takaful (Islamic insurance) of the asset. The client pays rentals for the use of the asset. At the end, ownership may pass to the client through a separate sale or gift promise, often at a pre-agreed residual price.

Rentals are calculated so that their present value, plus the present value of the residual payment, equals the asset's cost:

$$ \text{Cost} = R \times \frac{1 - (1 + i)^{-n}}{i} + \frac{RV}{(1 + i)^n} $$

Worked Example 2: Car Ijarah

A bank purchases a car for PKR 5,000,000 and leases it to a customer for 5 years with monthly rentals. The bank targets a 15% annual return (1.25% per month) and the customer agrees to buy the car at the end for a residual value of 10%, i.e. PKR 500,000.

Step 1: Parameters. \( i = 0.0125 \), \( n = 60 \), \( RV = 500{,}000 \).

Step 2: Present value of the residual.

$$ \frac{500{,}000}{(1.0125)^{60}} = \frac{500{,}000}{2.1072} \approx 237{,}284 $$

Step 3: Amount to recover through rentals.

$$ 5{,}000{,}000 - 237{,}284 = 4{,}762{,}716 $$

Step 4: Monthly rental.

$$ R = 4{,}762{,}716 \times \frac{0.0125}{1 - (1.0125)^{-60}} = 4{,}762{,}716 \times 0.023790 \approx \text{PKR } 113{,}305 $$

Step 5: Totals. Rentals add to about \( 113{,}305 \times 60 \approx 6{,}798{,}281 \), plus the PKR 500,000 residual payment.

Sanity check: without a residual, the rental would be \( 5{,}000{,}000 \times 0.023790 \approx 118{,}950 \). A residual should lower the rental, and it does.

Rentals in Pakistani Ijarah contracts are often linked to KIBOR and reset periodically; the same formula is applied on each reset date.

Profit-Sharing: Mudarabah and Musharakah

Worked Example 3: Mudarabah

An investor (rabb-ul-maal) gives PKR 2,000,000 to an entrepreneur (mudarib) to run a trading business, with a profit-sharing ratio of 60:40 (investor : mudarib).

Case A: profit of PKR 400,000.

• Investor: \( 0.60 \times 400{,}000 = 240{,}000 \)
• Mudarib: \( 0.40 \times 400{,}000 = 160{,}000 \)

Case B: loss of PKR 100,000 (with no negligence by the mudarib). The investor bears the entire financial loss, and capital falls to PKR 1,900,000. The mudarib loses the value of the time and effort spent.

Note that profit must be a ratio of actual profit, never a fixed amount or a fixed percentage of capital. A clause like "investor gets PKR 30,000 a month" would invalidate the Mudarabah.

Worked Example 4: Musharakah

Partner A contributes PKR 3,000,000 and partner B contributes PKR 1,000,000, so capital is in the ratio 75:25. B manages the business, so they agree to share profit 60:40.

Profit of PKR 800,000:

• A: \( 0.60 \times 800{,}000 = 480{,}000 \)
• B: \( 0.40 \times 800{,}000 = 320{,}000 \)

Loss of PKR 200,000: losses must follow capital, 75:25.

• A: \( 0.75 \times 200{,}000 = 150{,}000 \)
• B: \( 0.25 \times 200{,}000 = 50{,}000 \)

B can receive a profit share above its 25% capital share because it works in the business. Under the view widely followed in Pakistan, a purely sleeping partner's profit share should not exceed its capital ratio, which is why A (contributing 75%) takes 60%, not more.

Diminishing Musharakah

Home financing in Pakistan commonly uses diminishing Musharakah. Bank and customer co-own the house; the customer pays rent on the bank's share and gradually buys the bank's units. As the bank's share shrinks, the rent falls. Each month's calculation combines the Ijarah rental idea with a unit purchase, and the schedule looks similar to a reducing-balance mortgage.

Did You Know?

• Mit Ghamr in Egypt, founded in 1963, is often cited as the first modern Islamic savings bank.
• Pakistan's Federal Shariat Court ruled in 2022 that the banking system should be made riba-free by 2027.
• In Islamic deposit pools, banks assign "weightages" to different deposit types and tenors, so a longer-term deposit receives a larger share of the pool's profit.

Key Takeaways

• Murabaha sale price is fixed at signing; convert the flat markup into an effective rate to compare offers.
• Ijarah rentals are found with the annuity formula, adjusting for any residual value.
• In Mudarabah, profit is shared by ratio and financial loss falls on the capital provider.
• In Musharakah, profit follows the agreed ratio but loss always follows capital contribution.
• Benchmarks such as KIBOR may price a contract, but the contract structure is what makes it Shariah-compliant.

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