What a Short-Rate Cancellation Changes
When a policy is cancelled before its scheduled expiration, the insurer returns some of the premium for the coverage that will no longer be provided. Under a pro rata cancellation, the premium retained generally reflects the proportion of the policy term that has elapsed. A short-rate cancellation retains more than that pro rata amount, according to a factor or schedule specified in the policy or applicable rating rules. The additional retained amount reduces the return premium.
The short-rate factor is not universal. Its value depends on the contract’s schedule and the cancellation date or elapsed policy period. In every calculation, first determine what the factor represents: it may specify the percentage of premium retained, or, less commonly, the percentage returned. The examples below explicitly use hypothetical schedules and state their conventions.
These formulas determine the return premium credit, not necessarily the cash amount the policyholder receives at cancellation. If some premium remains unpaid, the return credit first reduces the amount owed. If the credit exceeds the unpaid balance, the difference is payable to the policyholder, subject to the contract’s terms.
Calculate from the Contractual Factor
A short-rate table or schedule supplies the factor; the calculation applies it to the premium to which the cancellation clause applies. It is not valid to substitute a remembered table, assume one standard factor for all policies, or infer a factor from the word “short-rate” alone. For a hypothetical example, a policy might state that 40% of its term premium is retained when cancellation occurs during a specified elapsed-time band. That 40% is a contract-specific input, not a general industry value.
For comparison, if \(t\) days have elapsed in a policy term of \(T\) days, straight pro rata retention is \(P(t/T)\), and pro rata return is \(P(1-t/T)\). This comparison assumes premium is earned uniformly over the term. A short-rate schedule can be stated by days, months, or another period; use the period and day-count convention the contract specifies. Do not replace a table lookup with a day ratio unless the question asks for a pro rata calculation.
Why the Return Formula Works
The return premium is the part of the applicable premium that is not retained. Once the factor \(r(t)\) has identified the retained fraction, the remaining fraction is \(1-r(t)\), since the retained and returned fractions sum to one. Multiplying that remaining fraction by the premium gives the return.
For a short-rate provision that retains more than pro rata, \(r(t)>t/T\). Subtracting each retained amount from the same premium reverses the comparison:
Thus the short-rate provision retains more premium and returns less than pro rata, for the same premium and elapsed period. The inequality describes the usual short-rate effect; the actual calculation must still follow the particular contract’s schedule.
Worked Calculations
Worked Example: Applying an Annual Short-Rate Factor
A hypothetical policy has a 365-day term and a $1,200 premium subject to cancellation. The policyholder cancels after 120 days of coverage. The hypothetical short-rate schedule specifies that 40% of the premium is retained at that point. Find the short-rate return and compare it with pro rata. There is no separate fee or unpaid balance. This is a deterministic premium calculation, so no interest rate, mortality assumption, probability distribution, or distribution parameters apply.
Method. Use the schedule’s retained-premium factor for the short-rate calculation, then calculate a separate pro rata benchmark using elapsed days.
Short-rate retained premium. The factor is \(r=0.40\), so the amount retained is \(1{,}200(0.40)=\$480.00\).
Short-rate return premium. Subtract the retained premium from the applicable premium: \(\$1{,}200.00-\$480.00=\$720.00\). Equivalently, the return fraction is \(1-0.40=0.60\), and \(1{,}200(0.60)=\$720.00\).
Pro rata comparison. The elapsed fraction is \(120/365=0.328767\ldots\). Pro rata retention is \(\$1{,}200(120/365)=\$394.5205\ldots\), or \(\$394.52\) rounded to cents. Pro rata return is \(\$1{,}200.00-\$394.52=\$805.48\).
Check. The short-rate retained amount exceeds the pro rata retained amount by \(\$480.00-\$394.52=\$85.48\); accordingly, the short-rate return is \(\$85.48\) less than the pro rata return: \(\$805.48-\$720.00=\$85.48\). The returned and retained short-rate amounts sum to \(\$1,200.00\).
Common mistake avoided: Using \(120/365\) as the short-rate factor would calculate pro rata retention, not the contractual short-rate amount. The day ratio is useful here only for the requested comparison.
Worked Example: A Shorter Policy Term
A hypothetical 180-day policy has a $900 term premium. It is cancelled after 45 days. Its hypothetical schedule specifies that 35% of the term premium is retained at this elapsed period. Calculate the short-rate return and the pro rata return. No fees or unpaid premium are involved. Interest, mortality, and a probability distribution are not relevant to this deterministic calculation.
Method. Apply the schedule factor to the full $900 term premium. For the benchmark, use the actual 180-day term, not a 365-day year.
Short-rate retention. The retained fraction is 0.35, so retained premium is \(\$900(0.35)=\$315.00\).
Short-rate return. The return fraction is \(1-0.35=0.65\). Therefore, return premium is \(\$900(0.65)=\$585.00\). Check: \(\$315.00+\$585.00=\$900.00\).
Pro rata benchmark. The elapsed fraction is \(45/180=0.25\). Pro rata retention is \(\$900(0.25)=\$225.00\); pro rata return is \(\$900-\$225=\$675.00\).
Comparison and check. Short-rate retention exceeds pro rata retention by \(\$315-\$225=\$90\), and short-rate return is \(\$90\) less: \(\$675-\$585=\$90\). The comparison uses the policy’s 180-day term consistently.
Common mistake avoided: Dividing 45 days by 365 would use the wrong term for the pro rata comparison. The factor-based short-rate result is unaffected by that mistake, but the comparison would be wrong.
Worked Example: Return Credit Versus Cash Refund
A hypothetical 365-day policy has a $1,500 premium subject to cancellation. After 150 days, the applicable schedule specifies that 48% is retained. The policyholder has paid $900, with $600 of the billed premium still unpaid. Find the return premium credit and the cash amount payable to the policyholder after the unpaid balance is accounted for. There are no separate fees. Because this is a deterministic cancellation calculation, interest, mortality, and a probability distribution do not apply.
Method. Calculate the contractual return premium first. Then offset the unpaid balance against that return credit to find the settlement.
Retained premium. The schedule retains \(0.48\) of the $1,500 premium: \(\$1{,}500(0.48)=\$720.00\).
Return premium credit. The return fraction is \(1-0.48=0.52\), so the return credit is \(\$1{,}500(0.52)=\$780.00\). Check: \(\$720+\$780=\$1,500\).
Cash settlement. The $780 return credit offsets the $600 unpaid balance. The amount remaining is \(\$780-\$600=\$180.00\), payable to the policyholder. A second check reaches the same result: the policyholder paid $900 and the insurer retains $720, leaving \(\$900-\$720=\$180.00\) to return.
Pro rata comparison. The elapsed fraction is \(150/365=0.4109589\ldots\). Pro rata retention is \(\$1{,}500(150/365)=\$616.4383\ldots\), or \(\$616.44\) to cents. Pro rata return is \(\$1,500-\$616.44=\$883.56\), which is $103.56 more than the short-rate return.
Common mistake avoided: The $780 return premium is a credit, not the cash refund. Paying out $780 without offsetting the $600 unpaid balance would ignore the policy’s remaining account balance.
Reading the Schedule and Avoiding Errors
The arithmetic is simple once the schedule convention and premium base are clear. Most mistakes arise before the multiplication: using the wrong factor, applying it to the wrong amount, or treating a return credit as a cash payment. A reliable calculation follows these steps.
Use the premium amount the cancellation provision applies to, and note whether the problem specifies any excluded charge or adjustment.
Use the cancellation date, policy-term boundaries, and the schedule’s stated units or counting convention. Do not silently substitute a different day-count basis.
Determine whether its entry is a retained percentage, a return percentage, or another stated quantity. For a return percentage \(q\), return premium is \(Pq\); do not treat \(q\) as a retained factor.
For a retained factor \(r\), calculate \(Pr\) and \(P(1-r)\). Confirm that the two amounts sum to \(P\), then reconcile any return credit with amounts already paid or still due.
A common exam trap is reading a table heading too quickly. If a hypothetical schedule says “52% returned,” then the return is 52% of the applicable premium and the retained fraction is 48%. If it says “52% retained,” those results reverse. Another trap is applying a short-rate factor to only the paid-to-date amount. Unless the contract or problem says otherwise, the factor applies to the premium subject to cancellation; payment history is used separately to determine the settlement.
The pro rata comparison is also a useful reasonableness check. For an ordinary short-rate factor, the retained fraction should generally be greater than the elapsed-time fraction, so the short-rate return should be smaller. If the result points the other way, verify the factor convention, cancellation period, and premium base before concluding that the result is unusual.
Check Your Understanding
For each problem, use only the hypothetical schedule convention stated in the problem. Round money to cents where needed.
- A hypothetical annual policy has a $2,000 premium. Its schedule states that 32% is retained at the cancellation date. Find the retained premium and return premium. (Answer: retained $640.00; return $1,360.00)
- A hypothetical 240-day policy with a $1,200 premium is cancelled after 90 days. The schedule says 45% is retained. Find the short-rate return and the pro rata return. (Answer: short-rate return $660.00; pro rata return $750.00)
- A $1,750 premium applies to a hypothetical 365-day policy cancelled after 210 days. The schedule specifies 70% retained. Calculate the return premium and compare it with pro rata; give the difference between the two returns. (Answer: short-rate return $525.00; pro rata return $743.15; short-rate return is $218.15 less)
- A hypothetical policy has a $2,400 premium and is cancelled after 120 days of a 365-day term. The schedule specifies 40% retained. The policyholder has paid $1,700 and owes $700. Find the return credit and the cash settlement. (Answer: return credit $1,440.00; $740.00 payable to the policyholder)
- A hypothetical schedule states that 52% of premium is returned, not retained. The applicable premium is $1,000. Find the retained amount and return premium, and explain which factor should be used as the retained factor. (Answer: retained $480.00; return $520.00; retained factor 48%)