Excess (Deficiency) Ratio

Study guide: Exam 6C

The Excess (Deficiency) Ratio measures how a year-end estimate of claim liabilities has run off: the excess (positive: the liabilities proved more than enough) or deficiency (negative) that emerged over one or more calendar years, as a percentage of the liabilities being tested. When the liabilities are discounted, the CIA's runoff educational note credits them with the investment income earned while they were held, so a liability that was exactly right runs off to zero.

\text{Excess (Deficiency)}_t = L_{t-1} + I_t - P_t - L_t

I_t = y_t \times \frac{L_{t-1} + L_t}{2}

\text{Ratio} = \frac{\sum_t \text{Excess (Deficiency)}_t}{L_{\text{start}}}

• L_t is the discounted claim liability for the accident year(s) tested at the end of calendar year t, P_t the amount paid during t (incremental, not cumulative — using cumulative paid is an error several examiners' reports list), y_t the annual yield for t, and L_{\text{start}} the liability at the start of the period measured. For a cumulative ratio that is the liability at the end of the accident year itself, which is how the note's Table 6 and every published cumulative answer divide; for a one-year ratio it is the prior year-end liability.
• Two ways to allow for the time value of money (note §1.2): discount the payments and the closing liability back to t-1, or add the investment income earned during t on the assets supporting the liabilities. They should agree; the note uses the second because it is simpler to calculate and present. Examiners marked down answers that said only "discount": the answer has to say that the time-t amounts are discounted back to t-1.
• Undiscounted basis (§1.1): the emergence is the ultimate estimated at t-1 less the ultimate estimated at t, or equivalently L_{t-1} - P_t - L_t on undiscounted liabilities, with no investment income; the ratio divides by the undiscounted unpaid amount at the start. On the discounted basis the same shortcut works with actuarial present value ultimates (cumulative paid plus discounted unpaid): excess = APV ultimate at the start - APV ultimate now + investment income. Pre-IFRS 17 questions build L_t as case reserves plus IBNR plus the "effect of discounting and PfAD" — the provision for adverse deviations the 2011 note refers to.
• Which yield (§3): the investment income attributable to policy liabilities, consistent with any allocation of assets. With no formal allocation, the default yield follows exhibit 10.60 of the return, applied to the average of the opening and closing net unpaid claims + net unearned premium − gross DPAC + premium deficiency provisions + unearned commissions − agents', brokers' and policyholders' receivables − instalment premiums (see Deferred Policy Acquisition Expenses). The result is capped at total investment income, and a negative overall yield is used as it is, so the runoff is penalized.
• Where it appears. Before IFRS 17 the return carried the margin (deficiency) for unpaid claims at the prior year-end on page 60.30, and the ratio on P&C-1 page 60.41. The 2023 annual return reports, on page 60.45, the prior-year development excess or (deficiency) on net, undiscounted ultimates (column 09 less column 10), with the effects of discounting and of the risk adjustment shown separately below; page 10.60 reports claims development as a percentage of adjusted equity. The OSFI Memorandum asks the Appointed Actuary to explain differences between page 60.45 and the actual-versus-expected comparison.
• Reading it. Always say whether the result is an excess or a deficiency. The note says it is useful for the Appointed Actuary's Report to identify the runoff's components — the undiscounted liabilities, changes in the discount rate and changes in the provision for adverse deviations (§1.2). A deficiency repeated across accident years and calendar years says the insurer keeps understating its initial liabilities, and the published 2019 answer says OSFI may intervene (see Reserve Adequacy, Runoff).

Cumulative Discounted Ratio for One Accident Year For accident year 2023 ($000s): discounted claim liabilities are 50{,}000 at 31 December 2023, 28{,}000 at 31 December 2024 and 15{,}000 at 31 December 2025. Payments were 20{,}500 during 2024 and 12{,}000 during 2025. The annual yield was 4.0\% in 2024 and 3.5\% in 2025.

Calculate the cumulative discounted excess (deficiency) ratio for accident year 2023 as at 31 December 2025, and the one-year ratio for 2025.

Answer Investment income on the average liability in each year:

\begin{align*}
I_{2024} &= 0.040 \times \frac{50{,}000 + 28{,}000}{2} \\
&= 1{,}560 \\
I_{2025} &= 0.035 \times \frac{28{,}000 + 15{,}000}{2} \\
&= 752.5
\end{align*}

Excess in each calendar year:

\begin{align*}
E_{2024} &= 50{,}000 + 1{,}560 - 20{,}500 - 28{,}000 \\
&= 3{,}060 \\
E_{2025} &= 28{,}000 + 752.5 - 12{,}000 - 15{,}000 \\
&= 1{,}752.5
\end{align*}

Both ratios:

\begin{align*}
\text{Cumulative ratio} &= \frac{3{,}060 + 1{,}752.5}{50{,}000} \\
&= 9.6\% \\
\text{One-year ratio (2025)} &= \frac{1{,}752.5}{28{,}000} \\
&= 6.3\%
\end{align*}

Both are excesses: the liability set at the end of 2023 has so far proved 9.6\% more than enough, and the cumulative ratio divides by that original 50{,}000, not by an average.

Undiscounted Versus Discounted From Ultimates For accident year 2024 ($000s), as at 31 December 2025:

	12 months	24 months
Cumulative paid	30{,}000	52{,}000
Undiscounted ultimate	95{,}000	97{,}000
Actuarial present value ultimate	92{,}000	94{,}600

The annual yield for 2025 was 3\%. Calculate the undiscounted and the discounted excess (deficiency) ratios for accident year 2024 as at 31 December 2025, and explain the difference.

Answer Undiscounted. The unpaid amount at 12 months is 95{,}000 - 30{,}000 = 65{,}000.

\begin{align*}
\text{Ratio} &= \frac{95{,}000 - 97{,}000}{65{,}000} \\
&= -3.1\%
\end{align*}

Discounted. The discounted liabilities are 92{,}000 - 30{,}000 = 62{,}000 and 94{,}600 - 52{,}000 = 42{,}600.

\begin{align*}
I_{2025} &= 0.03 \times \frac{62{,}000 + 42{,}600}{2} \\
&= 1{,}569 \\
\text{Excess} &= 92{,}000 - 94{,}600 + 1{,}569 \\
&= -1{,}031 \\
\text{Ratio} &= \frac{-1{,}031}{62{,}000} \\
&= -1.7\%
\end{align*}

Both bases show a deficiency, and the pieces reconcile: the undiscounted ultimate rose 2{,}000; the effect of discounting and PfAD moved from -3{,}000 to -2{,}400, adding a further 600 to the discounted liability; and the 1{,}569 of investment income earned on the liability offsets part of both, since -2{,}000 - 600 + 1{,}569 = -1{,}031. Separating undiscounted development from the discount and margin effects is the kind of analysis the CIA note says is useful in the Appointed Actuary's report.