Metric Formulas & Calculations

This chapter provides the mathematical formulas, algorithms, and computational rules used across BiiTrust’s analytics, reports, and real-time dashboard gauges.

1. Credit Risk & Delinquency Metrics

Portfolio at Risk (PAR)

Measures the proportion of the gross loan book affected by installments overdue beyond a specified threshold:

$$\text{PAR}x = \frac{\sum{i \in L_x} \text{Outstanding Principal}_i}{\text{Total Outstanding Gross Loan Portfolio}} \times 100%$$

Where:

  • $L_x$ = Set of all active loans where $\text{DPD} > x$.
  • $x \in {1, 30, 60, 90}$.

Dashboard Benchmark Status:

  • $\text{PAR } 30 \le 3.0%$: Optimal / Green
  • $3.0% < \text{PAR } 30 \le 5.0%$: Acceptable / Amber
  • $\text{PAR } 30 > 5.0%$: Critical / Red (Triggers executive intervention)

Days Past Due (DPD) Calculation

For any active loan with unpaid due installments:

$$\text{DPD} = \max\left(0, \lfloor \frac{\text{Current Date} - \text{Earliest Overdue Installment Due Date}}{86,400,000 \text{ ms}} \rfloor\right)$$

  • Weekend & Holiday Rule: If an installment due date lands on an institutional non-working day, the DPD counter only begins incrementing after the grace period expires following the next valid banking day.

Non-Performing Loan (NPL) Ratio

$$\text{NPL Ratio} = \frac{\sum \text{Principal of Loans with DPD} \ge 90}{\text{Total Gross Loan Portfolio}} \times 100%$$

2. Collection & Repayment Metrics

Collection Efficiency (Daily / Period)

Evaluates field collection performance against scheduled demand:

$$\text{Collection Efficiency} = \frac{\text{Total Cash & M-Pesa Collected in Period}}{\text{Total Scheduled Installment Demand Falling Due in Period}} \times 100%$$

  • On-Target: $\ge 98.0%$
  • Warning: $90.0% - 97.9%$
  • Deficient: $< 90.0%$

Cumulative Repayment Rate

$$\text{Repayment Rate} = \frac{\text{Total Principal Repaid to Date}}{\text{Total Principal Disbursed + Accrued Contractual Interest}} \times 100%$$

3. Amortization & Installment Math

Flat Interest Calculation

$$\text{Total Interest} = P \times \left(\frac{\text{Annual Rate %}}{100}\right) \times \left(\frac{\text{Tenor Months}}{12}\right)$$

$$\text{Monthly Installment} = \frac{P + \text{Total Interest}}{\text{Tenor Months}}$$

$$\text{Principal per Installment} = \frac{P}{\text{Tenor Months}}, \quad \text{Interest per Installment} = \frac{\text{Total Interest}}{\text{Tenor Months}}$$

Reducing Balance (Equal Installment PMT)

$$\text{Periodic Installment (PMT)} = P \times \frac{r \cdot (1 + r)^n}{(1 + r)^n - 1}$$

Where:

  • $P$ = Disbursed Principal.
  • $r$ = Periodic interest rate ($r = \frac{\text{Annual Rate}}{m}$, where $m=12$ for monthly, $m=52$ for weekly).
  • $n$ = Total number of repayment installments.

Schedule Breakdown for Period $k$: $$\text{Interest Portion}k = \text{Remaining Principal}{k-1} \times r$$ $$\text{Principal Portion}_k = \text{PMT} - \text{Interest Portion}_k$$ $$\text{Remaining Principal}k = \text{Remaining Principal}{k-1} - \text{Principal Portion}_k$$

4. Late Payment Penalty Algorithm

When a borrower exceeds the contractual grace period (default: 3 days), BiiTrust applies daily compounding or simple penalty accruals:

$$\text{Daily Penalty} = \text{Overdue Installment Amount} \times \left(\frac{\text{Penalty Rate % per Annum}}{365}\right)$$

Alternatively (Fixed Daily Fee): A fixed fee (e.g., KES 50 per day overdue) accrues daily at midnight via the background worker until the overdue installment is cleared.

5. Collateral Security & LTV

Loan-to-Value (LTV) Ratio

$$\text{LTV} = \frac{\text{Requested or Outstanding Loan Principal}}{\text{Forced Sale Value (FSV) of Pledged Collateral}} \times 100%$$

$$\text{FSV} = \text{Open Market Value (OMV)} \times (1 - \text{Haircut Rate})$$

Example:

  • Asset: Toyota Probox (Motor Vehicle).
  • Appraised OMV: KES 800,000.
  • Mandatory Vehicle Haircut: $35%$.
  • Calculated FSV: $\text{KES } 800,000 \times (1 - 0.35) = \text{KES } 520,000$.
  • Institutional LTV Ceiling ($75%$ of FSV): $\text{KES } 520,000 \times 0.75 = \text{KES } 390,000$ (Maximum safe borrowing limit).

6. Financial Solvency & Profitability

Net Interest Margin (NIM)

$$\text{NIM} = \frac{\text{Total Loan Interest Income} - \text{Total Cost of Borrowed Funds}}{\text{Average Total Earning Assets}} \times 100%$$

Loan Loss Reserve Coverage Ratio

$$\text{Coverage Ratio} = \frac{\text{Total Allowance for Loan Impairment Reserve}}{\text{Total Non-Performing Loans (DPD } \ge 90\text{)}} \times 100%$$

  • Prudential Guideline: Should remain at $\ge 100%$ to ensure the institution can absorb full default write-offs without endangering capital adequacy.