Motor Own-Damage Rate Indication: On-Level, Trend and the Fixed-Expense Trap
An Indian general insurer's motor own-damage book earned premium of Rs 480 crore in the year just ended, at rates that applied throughout that year. A 6% rate increase filed during the year applied to all policies issued from 1 October; written premium for the year was therefore Rs 500 crore, while the Appointed Actuary computes earned premium at current rate level as Rs 512 crore. Earned losses, including claim handling costs, were Rs 378 crore, of which Rs 24 crore is IBNR. Variable expenses (commission and premium-related costs) run at 18% of premium. Fixed underwriting overhead was Rs 60 crore and is not expected to change with the rate level. The board requires a 5% underwriting profit margin. Claim frequency has been falling 2% a year and average severity rising 7% a year. The average accident date of the expiring period is 1 July of last year, and the new rates will apply to a period centred on 1 July two years later.
1. Calculate the indicated rate change. Show the on-level adjustment, the trend adjustment and the treatment of fixed expenses, and state clearly what the final percentage applies to.
Read the worked answer
Step 1 - On-level the premium. The Rs 480 crore earned was written at a mixture of old and new rates. At current rate level the same exposure would have produced Rs 512 crore, so the on-level factor is 512/480 = 1.0667. The loss ratio at current rate level is 378/512 = 73.8%, not 378/480 = 78.75%.
Step 2 - Trend to the prospective period. The average accident date of the data is 1 July last year; the new rates will be in force for a period centred on 1 July two years later. That is 2 years of trend. Frequency factor = 0.98^2 = 0.9604. Severity factor = 1.07^2 = 1.1449. Combined trend = 0.9604 x 1.1449 = 1.0996. Trended loss and LAE = 378 x 1.0996 = Rs 415.6 crore.
Step 3 - Load fixed expenses as rupees, not as a percentage. Required premium = (losses and LAE + fixed expenses) / (1 - variable expense ratio - profit margin) = (415.6 + 60) / (1 - 0.18 - 0.05) = 475.6 / 0.77 = Rs 617.7 crore.
Step 4 - Indicated change. 617.7 / 512 - 1 = +20.6%, applied to current rate levels.
Judgement point. A shortcut that converts the Rs 60 crore into 11.7% of on-level premium and uses a permissible loss ratio of 1 - 18 - 11.7 - 5 = 65.3% gives 81.2/65.3 = +24.3%. That is wrong if the overhead really is fixed in rupees, because it forces the fixed cost to grow with premium. The two methods differ by about 4 points here, and the choice must be justified, not assumed.
2. The two-wheeler segment of the book earned Rs 45 crore at current rate level and produced 320 claims in the year. Explain how much weight the segment's own experience should carry against the book-wide indication, and what you would actually file for the segment.
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Use the standard for full credibility on claim counts: n_full = (z/k)^2, with z = 1.645 for 90% confidence and k = 0.05 for 5% precision, giving 1,082 claims. The segment has 320 claims, so the credibility factor is Z = sqrt(320/1,082) = 0.54.
Only about 54% of any movement in the segment's own loss ratio is signal; the rest is noise. The credible indication is therefore Z x (segment indication) + (1 - Z) x (complement). The complement should be the book-wide indication from Task 1, or better, a class-driven expected loss ratio for two-wheelers, not simply the segment's prior rate level.
Two cautions before filing. First, credibility is claim-count based and assumes the claim size distribution is not dominated by a few large losses; if the segment had one or two total-theft or third-party death claims above Rs 20 lakh, severity credibility is lower than frequency credibility and a further reduction is needed. Second, 320 claims in one year is thin; pooling three accident years before testing credibility is preferable, provided the rate level and product mix are on-levelled.
Practical outcome: do not move the two-wheeler rates by the full segment indication. File a weighted movement, and state in the file note the credibility standard, the complement used and the reason. If the segment indication is close to the book indication the weight hardly matters - the weight only bites when the segment is genuinely diverging, which is exactly when a wrong full-weight filing does the most damage.
3. The on-level adjustment in Task 1 was omitted by a colleague who computed the indication as 378/480 trended and compared against the unadjusted earned premium. Quantify the error and explain the direction in which it would mislead the board.
Read the worked answer
The colleague's loss ratio is 378/480 = 78.75% instead of 73.8%, an overstatement of about 4.9 points. The trended loss amount is unaffected (378 x 1.0996 = 415.6), so the error arises entirely in the denominator: required premium of 617.7 is compared with 480 rather than with 512.
Indicated change = 617.7/480 - 1 = +28.7%, against the correct +20.6%. The board would be asked for roughly 8 points more rate than the experience supports - about 40% more rate action than needed on a relative basis.
The direction is not accidental. Whenever a rate increase took effect during the experience period, earned premium is a blend of old and new rates and therefore understates the premium the current rate level would produce. Dividing losses by that depressed premium inflates the loss ratio, and the insurer overshoots on rate. The reverse happens after a mid-year rate cut: the unadjusted premium is too high, the loss ratio is understated, and the insurer under-files - a far more dangerous error, because it erodes margin quietly.
The control is procedural: the rate file should reconcile earned premium, written premium and on-level earned premium, and the on-level factor should be disclosed with the exposure and rate-change history behind it. A single reconciliation line would have caught this.