Decomposing Changes in the Indicated Rate

Why Decompose an Indicated Rate Change?

When two rate indications differ, the difference alone does not explain what changed. The new indication might reflect higher projected losses, changed fixed expenses, a revised variable expense provision, a different profit target, or several of these together. Decomposition assigns portions of the overall dollar change to those inputs, making the calculation easier to audit and interpret.

An allocation is a reconciliation, not a claim that each input caused a uniquely determined amount. When the rate formula is nonlinear, changing the inputs in different orders can assign different amounts to individual components even though the total change remains the same. This tutorial uses a specified hierarchical midpoint allocation, so the method and its allocation choices must be stated.

Definition: Let $L$ be projected loss cost per exposure, $F$ fixed expense per exposure, $v$ the variable expense provision as a proportion of premium, and $p$ the profit and contingency provision as a proportion of premium. Write $N=L+F$. The indicated rate per exposure is $$ R=\frac{N}{1-v-p}=\frac{L+F}{1-v-p}, $$ provided $v+p<1$. The denominator is the share of premium available to fund losses and fixed expenses after the variable provisions.

The formula assumes the provisions are expressed on the same premium basis and that $L$, $F$, $v$, and $p$ are defined consistently in both indications. It does not apply in this form if a provision is a fixed dollar amount but has been included in $v$ or $p$, or if the denominator is zero or negative. A fixed dollar provision belongs in $N$; a percentage-of-premium provision belongs in the denominator.

The Hierarchical Midpoint Allocation

Use subscripts 0 and 1 for the old and new assumptions. Thus $N_0,v_0,p_0$ produce $R_0$, and $N_1,v_1,p_1$ produce $R_1$. Define $g(v,p)=1/(1-v-p)$. Let $g_{00}=g(v_0,p_0)$, $g_{10}=g(v_1,p_0)$, $g_{01}=g(v_0,p_1)$, and $g_{11}=g(v_1,p_1)$. First allocate the total rate change between $N$ and the combined factor $g$; then split the factor contribution between $v$ and $p$.

\Delta R=R_1-R_0 =\underbrace{\Delta N\frac{g_{00}+g_{11}}{2}}_{C_N} +\underbrace{\frac{N_0+N_1}{2}(g_{11}-g_{00})}_{C_g}, \qquad \Delta N=N_1-N_0.

The factor contribution is then allocated between the two percentage provisions by averaging the two possible orders for changing $v$ and $p$:

C_v=\frac{N_0+N_1}{2}\cdot \frac{(g_{10}-g_{00})+(g_{11}-g_{01})}{2}, \qquad C_p=\frac{N_0+N_1}{2}\cdot \frac{(g_{01}-g_{00})+(g_{11}-g_{10})}{2}.

The term $C_v$ is the variable-expense contribution and $C_p$ is the profit-and-contingency contribution. If $L$ and $F$ are both separated within $N$, their contributions are $\Delta L(g_{00}+g_{11})/2$ and $\Delta F(g_{00}+g_{11})/2$. These add to $C_N$.

Why the Allocation Reconciles

The first identity follows by expanding its right-hand side. Since $R_j=N_jg_{jj}$, the right-hand side is

\frac{(N_1-N_0)(g_{00}+g_{11})+(N_0+N_1)(g_{11}-g_{00})}{2} =N_1g_{11}-N_0g_{00}.

That equals $R_1-R_0$, so the allocation between $N$ and $g$ is exact. For the second stage, add the two bracketed midpoint terms. Each cross-term cancels, leaving $g_{11}-g_{00}$. Consequently $C_v+C_p=C_g$, and therefore $C_N+C_v+C_p=R_1-R_0$.

This is a hierarchical midpoint allocation: it first treats $N$ and the combined factor $g$ as two groups, then splits the factor group between $v$ and $p$. It is not an average over all six possible orders of changing $N,v,p$. A full permutation-average allocation is another possible convention, but it can give different component amounts. State which convention is being used rather than describing this hierarchy as an average over all input orders.

Worked Example: Separating Loss, Fixed Expense, and Percentage Provisions

A hypothetical coverage has projected loss cost per exposure increasing from $L_0=\$600.00$ to $L_1=\$660.00$, and fixed expense increasing from $F_0=\$100.00$ to $F_1=\$110.00$. The variable expense provision changes from $v_0=0.20$ to $v_1=0.22$, while the profit and contingency provision changes from $p_0=0.05$ to $p_1=0.06$. Use the hierarchical midpoint method to attribute the indicated rate change.

Method and numerator. First combine loss and fixed expense: $N_0=600+100=700$ and $N_1=660+110=770$. The initial and final factors are $g_{00}=1/(1-0.20-0.05)=4/3$ and $g_{11}=1/(1-0.22-0.06)=25/18$. The crossed factors are $g_{10}=1/(1-0.22-0.05)=100/73$ and $g_{01}=1/(1-0.20-0.06)=50/37$.

Indications and total change. The initial rate is $R_0=700(4/3)=\$933.33$, and the final rate is $R_1=770(25/18)=\$1,069.44$, rounded to cents. Thus $\Delta R=\$136.11$ to cents.

Numerator contribution. The midpoint factor is $(4/3+25/18)/2=49/36$. Hence $C_N=(770-700)(49/36)=\$95.28$. Separating the numerator change gives a loss contribution of $(660-600)(49/36)=\$81.67$, and a fixed-expense contribution of $(110-100)(49/36)=\$13.61$. Together they equal $\$95.28$.

Split the factor contribution. The factor change is $25/18-4/3=1/18$, and the midpoint numerator is $(700+770)/2=735$. Thus $C_g=735(1/18)=\$40.83$. The factor portions are

\Delta g_v=\frac{(100/73-4/3)+(25/18-50/37)}{2} \approx 0.03703361,

and

\Delta g_p=\frac{(50/37-4/3)+(25/18-100/73)}{2} \approx 0.01852200.

Multiplying each by $735$ gives a variable-expense contribution of $\$27.22$ and a profit-and-contingency contribution of $\$13.61$, to cents. The four displayed component amounts sum to $\$136.11$: $\$81.67+\$13.61+\$27.22+\$13.61=\$136.11$. This check confirms the reconciliation. The midpoint allocation avoids assigning the entire interaction between changing inputs to whichever one happens to be changed last.

Worked Example: Allocating a Smaller Change in Percentage Provisions

A hypothetical product has $L_0=\$500.00$, $F_0=\$80.00$, $v_0=0.18$, and $p_0=0.04$. The revised assumptions are $L_1=\$540.00$, $F_1=\$90.00$, $v_1=0.20$, and $p_1=0.05$. Find the change and its hierarchical midpoint contributions.

Method and factors. The numerators are $N_0=500+80=580$ and $N_1=540+90=630$. The factors are $g_{00}=1/0.78=50/39$, $g_{10}=1/0.76=25/19$, $g_{01}=1/0.77=100/77$, and $g_{11}=1/0.75=4/3$.

Check the indicated rates. The initial indication is $R_0=580(50/39)=\$743.59$, and the revised indication is $R_1=630(4/3)=\$840.00$, rounded to cents. The total increase is $\$96.41$.

Numerator effect. The midpoint factor is $(50/39+4/3)/2=17/13$. The numerator contribution is $50(17/13)=\$65.38$. Its loss portion is $40(17/13)=\$52.31$, and its fixed-expense portion is $10(17/13)=\$13.08$; using unrounded amounts, these sum to $\$65.38$.

Percentage-provision effects. The midpoint numerator is $(580+630)/2=605$. The variable-expense factor portion is \[ \frac{(25/19-50/39)+(4/3-100/77)}{2}\approx 0.03418511, \] so its rate contribution is $605(0.03418511)\approx \$20.68$. The profit-and-contingency factor portion is \[ \frac{(100/77-50/39)+(4/3-25/19)}{2}\approx 0.01709694, \] so its rate contribution is approximately $\$10.34$. Before rounding, the two factor contributions sum to $605(4/3-50/39)=\$31.03$. Together with the numerator contribution, the unrounded total is $\$96.41$, as required. The small apparent cent difference that can arise when adding individually rounded components is only rounding; retain full precision until the final presentation.

Worked Example: Avoiding an Order-Dependent Attribution

A hypothetical rate indication changes from $N_0=\$900.00$, $v_0=0.10$, and $p_0=0.10$ to $N_1=\$990.00$, $v_1=0.12$, and $p_1=0.13$. Compare a sequential calculation with the hierarchical midpoint allocation.

Method and endpoint check. The initial factor is $g_{00}=1/0.80=1.25$; the final factor is $g_{11}=1/0.75=4/3$. Therefore $R_0=900(1.25)=\$1,125.00$, $R_1=990(4/3)=\$1,320.00$, and $\Delta R=\$195.00$.

Hierarchical midpoint result. The numerator contribution is $90(1.25+4/3)/2=90(31/24)=\$116.25$. The factor contribution is $945(4/3-1.25)=945(1/12)=\$78.75$. The crossed factors are $g_{10}=1/0.78=50/39$ and $g_{01}=1/0.77=100/77$. The variable-expense factor portion is \[ \frac{(50/39-5/4)+(4/3-100/77)}{2} =\frac{267}{8008}\approx 0.03334166, \] giving $945(267/8008)\approx \$31.51$. The profit-and-contingency portion is the remainder of the factor change, $1/12-267/8008\approx0.04999168$, giving approximately $\$47.24$. Those amounts sum to the $\$78.75$ factor contribution, and all contributions sum to $\$195.00$.

Why the order matters. If $N$ is changed first at the old factor, its sequential contribution is $90(1.25)=\$112.50$, followed by a factor contribution of $990(1/12)=\$82.50$. If the factor is changed first, the factor contribution is $900(1/12)=\$75.00$, followed by a numerator contribution of $90(4/3)=\$120.00$. Both orders sum to $\$195.00$, but neither gives the same component amounts as the hierarchical midpoint method. The method used must therefore be explicit when comparing attributions.

Practical Interpretation and Common Traps

A decomposition is most useful when the two indications are genuinely comparable: the exposure basis, premium basis, coverage definition, and treatment of expenses should be consistent. Before interpreting a component, verify that its input changed in the stated direction and that the rate formula represents the selected provisions. If a combined assumption such as $p$ contains both profit and contingencies, this calculation attributes their combined effect; it cannot distinguish them without separate inputs and a further allocation rule.

A common mistake is to call an allocation “symmetric” or “order-averaged” without specifying what orders were averaged. The hierarchical method averages $N$ against the combined factor, and then averages the two orders for $v$ and $p$ within that factor. It does not average every ordering of all three inputs. Another trap is to add percentage-point changes directly to the indicated rate. A change in $v$ or $p$ affects the denominator, so its rate effect depends on the other assumptions and must be calculated through $g(v,p)$.

Key takeaway: Calculate the endpoint indications first, use a stated allocation convention, retain full precision during the calculations, and verify that all component contributions add to $R_1-R_0$. Treat the component allocation as a transparent reconciliation rather than a uniquely determined causal result.

Check Your Understanding

For each hypothetical indication, use the definitions and hierarchical midpoint method above. Round money to cents only at the end.

A rate changes from $N_0=\$480.00,\ v_0=0.15,\ p_0=0.05$ to $N_1=\$528.00,\ v_1=0.17,\ p_1=0.06$. Find the total indicated rate change and the numerator contribution. (Answer: total change $\$85.71$; numerator contribution $\$61.17$.)
A hypothetical risk has $L_0=\$300.00,\ F_0=\$60.00,\ v_0=0.08,\ p_0=0.07$. Only loss cost changes, to $L_1=\$330.00$. Find the revised rate and the rate change. (Answer: revised rate $\$458.82$; increase $\$35.29$.)
For $N_0=\$700.00,\ v_0=0.12,\ p_0=0.08$, the revised assumptions are $N_1=\$735.00,\ v_1=0.13,\ p_1=0.10$. Find the total change and the numerator contribution. (Answer: total change $\$79.55$; numerator contribution $\$44.60$.)
A hypothetical indication changes from $N_0=\$800.00,\ v_0=0.10,\ p_0=0.05$ to $N_1=\$840.00,\ v_1=0.11,\ p_1=0.06$. Calculate the initial rate, revised rate, and total change; then state why a sequential attribution could differ by order. (Answer: initial rate $\$941.18$; revised rate $\$1,012.05$; increase $\$70.88$; sequential component amounts can differ because each changed input is evaluated using the level of the other inputs at that step.)

What You'll Learn