6  General Pricing

Published

February 6, 2026

posts/060-general-pricing.qmd

Compiled: 2026-02-28 09:50:25.620916300

6.1 Pricing from the Top Down

posts/060-files/top-down.qmd

not the main focus but want to say something about how the quantum to be allocated is determined.

  • Diagram(s) of flow - see One Note sketches

We start with a basic way of taking a required return on capital, splitting it into insurance and non-insurance costs, and arriving at an insurance operating profit. At this point, we assume you start with a set of simulation outputs of loss by accident year, and the only other information is the emergence runoff of that is the supporting asset number requirement and the profit target.

from bit.qmd

Bühlmann (1985) lays out a general philosophy for pricing from the top down.

  1. We have first considered the total of all risks from a given portfolio (or from a whole insurance company). For this total we have
    1. formulated a stability criterion (e.g., probability of ruin criterion),
    2. imposed certain conditions regarding yield of invested capital.
  2. The goals set out in (1) have then led us to find a total premium to be charged for the whole portfolio.
  3. In a final step we have then argued how this total should be split in a fair way among all the individual risks.

We now apply the top down pricing philosophy to investigate pricing for a steady-state portfolio. Usual set up with portfolio loss outcomes \(X\), \(X = \sum_i \Delta_iX\) and \(\tilde X\) some calendar year decoupling of \(X\). The one-year risk SRM is \(g\), meaning \(g\) is a distortion with associated pricing functional \[ X\mapsto g(X) := \int_0^\infty gS(x)\,dx. \] Initially assume that the outcome \(X\) is incurred over multiple years but is paid in full at time \(T\).

Define \(\Pi^b_{m}(X)\) to be the profit signature of \(X\) under basis \(b\) and metric \(m\) either operating result or distributable dividends. The former model assumes the risk is part of a portfolio large enough that all operating income is distributable because losses and negative cash flows are offset elsewhere.

Step 1 of Bühlmann’s top down approach is to determine the amount of capital \(K\) needed to support writing \(X\). Assume it is invested at an expected return \(r_K\) (CASH!). The one-year risk margin for the steady-state portfolio is \(RM = g(\tilde X)\)

Notation

Thorny issue!

Symbol Interpretation Reference
\(A\)

6.2 The DMC Framework

posts/060-files/dmc.qmd

This section gives describes the DMC framework (Decoupled Marginal Cost) in detail. The core idea is to price the risk that is transferred to capital providers on a one-year basis using a SRM measure consistent with macro market observables.

6.2.1 Design Parameters

QUOTED FROM ROBBIN: sec-robbin-taxonomy.

  1. Cash or accounting value: cash flow is largely independent of accounting, but cash values do not reflect distributable amounts, and are not in an accounting language used to communicate with investors. Bias toward using a accounting-based metric.

  2. Margin or return: investors, the audience, use return, whereas underwriters, regulators and customers use margin. However, it is easy to translate between the two, though margin depends crucially on accounting—are losses discounted losses or not?. Bias toward return as the metric.

  3. Time frame: DCF and standard capital budgeting for projects are inter-temporal, and follow a policy from inception through claims payment. An intra-temporal (CY) view is easier to align with the current portfolio, and analysis must take place within the context of the whole portfolio. Investors also care about intra-temporal reporting. There is no need to tie to a historic CY view—other than from regulators’ desire for verification. Pure DCF methods are hard to apply to risky, quick paying lines. Bias toward intra-temporal.

  4. Policyholder funds: historical or payout pattern pro forma based. Robbin’s intra-temporal methods use CY reported, but could equally use pro forma adjusted portfolios to determine or adjust reserves, assets for expected business to address Robbin’s concern that calendar year data are backward looking and may be out of date. Bias toward payout pattern.

  5. Anticipated asset yields: the same considerations apply as for policyholder funds. Bias toward prospective yield.

  6. Target: each unit must be analyzed with the context of the total portfolio risk. Methods should avoid unjustified exogenous variables. Bias toward allocation of required top down target.

  7. Capital allocation: capital allocations are meaningless (PCA) and hard to interpret when the cost of capital is not constant (PIR, CMM). However, a capital allocation of some kind is needed for to allocate the tax burden on capital investment earnings. Use natural capital allocation to allocate tax burden.

EXPAND

How these design parameters are incorporated into DMC is explained in @ REF. ] yield the METHOD we propose. SOMEWHER should be a written description of the algo! NAME. Target derived from pre-tax cost of equity capital plus cost of debt minus pre-tax investment income on non-policyholder funds. It is conceptually aligned with the CY ROE method, but differs in top down determination of overall target, and openness to prospective estimates of key parameters (reserve levels, invested assets, investment yields, tax rates). FLESH OUT.

6.2.2 Detailed Framework Description

Decision Variables

  1. Accounting framework

Inputs

  1. Prospective plan, expected losses
  2. Simulation output
  3. Starting balance sheet
  4. Asset variables
  5. Macroeconomic variables

To ensure prices are appropriate for prospective pricing and not dependent on the firm’s current financial state, the framework should be applied to an adjusted steady state portfolio.

Calculation Steps

6.3 Example

posts/060-files/example.qmd

../

Bühlmann, Hans. 1985. “Premium Calculation from Top Down.” ASTIN Bulletin 15 (2): 89–101. https://doi.org/10.2143/ast.15.2.2015021.