Flat Versus Reducing Balance, With a Very Wide Default Interval
An SME client, a packaging unit, asks for PKR 3,000,000 over 36 months. Your product sheet offers two structures: (A) a flat rate of 14% per annum, with markup calculated on the original amount and spread equally over 36 instalments; or (B) 19% per annum on the reducing balance, equal monthly instalments. The client's accountant has already told him option A is 'obviously cheaper because 14 is less than 19'. Your credit team keeps a file of 50 comparable packaging-sector loans written over the last three years; 6 of them went into default, and recovery on defaulted SME exposure averages 45% of outstanding. Bank policy: approve only if expected loss is at most 30% of expected net spread, otherwise require collateral or decline. Funding cost of the loan is 11% per annum on the outstanding balance. You must reply to the client and to the credit committee using the same file.
1. Compute the monthly instalment and total repayment under both structures, estimate the effective annual cost of each, and explain the accountant's error.
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Option A - flat 14%. Markup = 3,000,000 x 14% x 3 = PKR 1,260,000. Total repayable PKR 4,260,000; instalment = 4,260,000 / 36 = PKR 118,333. The markup is charged on the full principal for all 36 months even though the balance falls every month.
Option B - 19% reducing balance. Monthly rate = 19% / 12 = 1.5833%. Instalment = 3,000,000 x 0.015833 / (1 - 1.015833^-36). The annuity factor is about 27.28, so instalment = about PKR 109,970; total repayable about PKR 3,958,900; total markup about PKR 958,900.
Effective cost. Option A: solve 3,000,000 = 118,333 x annuity factor over 36 months. The factor 25.35 gives a monthly rate of about 2.03%, a nominal APR near 24.4% and an effective annual rate near 27%. Option B: the contract rate is the effective rate - 19% nominal, about 20.7% effective.
The error. The accountant compared quoted rates rather than the balance each rate is charged on. The 'cheap' 14% option costs about five percentage points a year more than the 'expensive' 19% option and roughly PKR 300,000 more in cash. Show the client the two total-repayment figures; they are much harder to argue with than a rate comparison, and put the illustration in writing so the sale is documented.
2. Using the 50-loan file, compute a 95% confidence interval for the default rate and test the application against the 30% policy limit. What do you recommend?
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Default rate. p-hat = 6/50 = 12%. Standard error = sqrt(0.12 x 0.88 / 50) = sqrt(0.002112) = 0.046. 95% interval = 12% +/- 1.96 x 4.6 points = 3.0% to 21.0%. The sample is small, so the interval is wide - the width is information, not noise to be discarded.
Expected loss. LGD = 1 - 45% recovery = 55%. Average outstanding on an amortising 36-month loan is about 54% of principal, roughly PKR 1.62 million, so this is the exposure base. At the point estimate: EL = 0.12 x 0.55 x 1,620,000 = about PKR 107,000.
Net spread. Interest income under option B is about PKR 959,000 over three years; funding cost = 11% x 1,620,000 x 3 = about PKR 535,000; net spread = about PKR 424,000.
Test. EL / spread = 107 / 424 = 25%, inside the 30% limit, so approve on the point estimate. But at the upper confidence bound: 0.21 x 0.55 x 1,620,000 = about PKR 187,000, which is 44% of spread - a fail.
Recommendation. The application passes only because the estimate is precise enough to look comfortable; the sample cannot rule out a portfolio default rate of one in five. Condition the approval: take collateral covering at least half the exposure (stock and receivables lien plus personal guarantee), price at the top of the band, review at twelve months, and start collecting default-timing data, because exposure at default matters more than the default rate alone.
3. The credit committee wants the sector default rate measured to within plus or minus 5 percentage points at 95% confidence. How many loans are needed, and what caveats must accompany the number?
Read the worked answer
Calculation. n = z^2 p(1-p) / e^2. Using the observed p-hat = 0.12: n = 3.8416 x 0.12 x 0.88 / 0.0025 = 0.4057 / 0.0025 = 162.3, so 163 loans.
Caveats to put in the memo:
- Using p = 0.5, the safe assumption when the rate is uncertain, gives n = 3.8416 x 0.25 / 0.0025 = 384. If the true rate is above 12%, 163 loans will undershoot the precision target.
- The 50 loans are not a random sample of the portfolio - they are three years of packaging loans from essentially one segment and probably a handful of branches. Clustering means the effective sample size is smaller than the count, so 163 loans from one branch does not buy plus or minus 5 points for the bank as a whole.
- If the population is finite - say 400 comparable loans - apply the finite population correction of about 0.77, which cuts the requirement to roughly 97 loans.
- Record default timing, not just occurrence. A default in month 6 costs far more than one in month 30, and the expected-loss calculation is only as good as the exposure-at-default assumption behind it.
So the answer to the committee is 163 under the point estimate, 384 if they want the assumption-proof number, and in either case a sampling frame that reaches beyond the current file.