index.qmd

Compiled: 2026-02-28 09:45:53.327183300

Preface

Pricing Multi-period Insurance Risk (PMIR) extends Pricing Insurance Risk (written jointly with John Major and hereinafter PIR) by allowing for multi-period loss payouts, reserve uncertainty, and time value of money. PIR was published at the tail-end of a 20 year period of historically very low interest rates when it was almost permissible to ignore time value—but not reserves. Such as it was, that bye ended with the recent normalization of rates, and hence this work is needed.

PMIR is part manifesto and part summary of current practice. It is a statement of what I believe practice should be that is based on a 25 year career working with senior executives on pricing and risk problems and listening to what they want to do, blended with an appreciation for the underlying theories. There are many loose ends and solutions hinted at; I hope these are incorporated into the future research agenda.

Two words do not appear in the main text. The first is fair, as in fair premium or fair return. Fair is oft used in the context of regulated rates, and appears numerous times in PIR. The second lacuna is allocation, as in capital allocation or premium allocation. The phrase natural allocation does appear, following PIR, but is a synonym for marginal cost by Delbaen’s theorem. The phrases fair premium and allocated capital imply a value judgment: fair to whom? allocated how? Both of these wooly concepts are replaced by marginal cost which has a clear and objective definition. Marginal cost delivers facts that the user can confidently and robustly incorporate into their decision-making. How they choose to do so involves the same considerations as determining whether a premium is fair—questions beyond the scope of this monograph. We regard this choice as analogous to presenting the chef with raw ingredients rather than a selection of processed ingredients, and believe that it introduces greater clarity to the problem of pricing. Ultra processed ingredients should be avoided in both foods and actuarial work!

The monograph supports hands-on learning, offering many reproducible examples. All examples are created using Python, and the monograph’s Quarto (RMarkdown) files are available on GitHub. The reader is encouraged to download, reproduce - and tinker.

Remarks provide optional, additional background. They are not relied upon elsewhere and can be skipped.

Thanks to A, B and C.

Notation

This section sets out various conventions used throughout the monograph. Standard terminology is laid out in Table tbl-index-terminology, definitions in Table tbl-index-definitions, and notation in Table tbl-index-notation.

Capital Roman letters denote monetary quantities at the accounting or reporting level: loss $L$, premium $P$, margin $M$, equity $Q$, capital $K$, debt $D$, and assets $a$. These represent recognized or expected amounts unless randomness is explicit.

All monetary amounts are nominal and expressed in a single currency. Each value refers to an amount at the relevant time within context. Premiums are paid and received at the start of each period and losses at the end. Discounting to present value is always explicit. There are no inflation adjustments.

When stochastic treatment is required, the same symbols are viewed as random variables on $(\Omega,\mathcal{F},\mathsf P)$, and their expectations are written $\mathsf PL$ and so on, where $\mathsf P$ serves as both probability measure and expectation operator following Pollard’s convention.

Bars, hats, and tildes may appear locally, e.g. $\bar L$, $\hat L$, or $\tilde L$, when both realized and expected versions are needed in the same equation; they mark the random quantity. Context determines meaning: probabilistic in probability sections, deterministic in accounting or valuation discussions..

Increasing and decreasing are used in a non-strict sense. We say that $f$ is increasing if it is non-decreasing. This convention means that many statements are clearer, because we do not have to mentally process the negative.

We use the (infamous to Python programmers) Walrus operator: $f(x):=x^2$ defines the function $f$. This is a very handy shortcut. We use it like a programmer.

Table 1: Common terminology.

Term	Context	Meaning
Estimate	Profit targets	compute profit targets by unit
Determine	Profit targets	same
Communicate	Profit targets	describe process, amount, and implementation
Implement	Premiums	compute premium to achieve target, usually for rates
Calculate	Premiums	same, usually for an individual risk
Evaluate	Premiums	compute profit target embedded in a premium
Validate	Models	check that the model is working as intended, compare different types of model

Table 2: Common definitions.

Term	Definition
Actuarial reserve	Expected discounted value of future cash flows, no risk adjustment.
Actuarial value	Objective expected value with no risk margin. Discounted if appropriate. Undiscounted actuarial value for clarity.
Duration	contract duration, short-d and long-d
Emergence
Incurred	Always means ultimate incurred loss and never case incurred.
Individual risk (IR)
Market value	Risk adjusted expected value. Discounted if appropriate.
Payout pattern	long-tail (as distinct from heavy tail)
Statutory reserve	Management best estimate liability, no discount, no (explicit) risk adjustment
Surplus	Old terminology for capital
Unit

Table 3: List of symbols.

Symbol	Meaning	Reference
$X$	Random variable
$\mathscr F$	Sigma algebra

--- part: front-matter --- ::: {.content-hidden when-meta="production-mode" .source-path} `index.qmd` ::: ```{python} #| label: index-datetime from datetime import datetime import time now = datetime.now() ns = time.time_ns() % 1_000_000_000 print(f"Compiled: {now.strftime('%Y-%m-%d %H:%M:%S')}.{ns:09d}") ``` # Preface {.unnumbered} Pricing Multi-period Insurance Risk (PMIR) extends *Pricing Insurance Risk* (written jointly with John Major and hereinafter PIR) by allowing for multi-period loss payouts, reserve uncertainty, and time value of money. PIR was published at the tail-end of a 20 year period of historically very low interest rates when it was almost permissible to ignore time value---but not reserves. Such as it was, that bye ended with the recent normalization of rates, and hence this work is needed. PMIR is part manifesto and part summary of current practice. It is a statement of what I believe practice should be that is based on a 25 year career working with senior executives on pricing and risk problems and listening to what they want to do, blended with an appreciation for the underlying theories. There are many loose ends and solutions hinted at; I hope these are incorporated into the future research agenda. Two words do not appear in the main text. The first is *fair*, as in fair premium or fair return. Fair is oft used in the context of regulated rates, and appears numerous times in PIR. The second lacuna is *allocation*, as in capital allocation or premium allocation. The phrase *natural allocation* does appear, following PIR, but is a synonym for marginal cost by Delbaen’s theorem. The phrases *fair premium* and *allocated capital* imply a value judgment: fair to whom? allocated how? Both of these wooly concepts are replaced by *marginal cost* which has a clear and objective definition. Marginal cost delivers facts that the user can confidently and robustly incorporate into their decision-making. How they choose to do so involves the same considerations as determining whether a premium is fair—questions beyond the scope of this monograph. We regard this choice as analogous to presenting the chef with raw ingredients rather than a selection of processed ingredients, and believe that it introduces greater clarity to the problem of pricing. Ultra processed ingredients should be avoided in both foods and actuarial work! The monograph supports hands-on learning, offering many reproducible examples. All examples are created using Python, and the monograph's Quarto (RMarkdown) files are available on GitHub. The reader is encouraged to download, reproduce - and tinker. Remarks provide optional, additional background. They are not relied upon elsewhere and can be skipped. Thanks to A, B and C. ## Notation {.unnumbered} This section sets out various conventions used throughout the monograph. Standard terminology is laid out in @tbl-index-terminology, definitions in @tbl-index-definitions, and notation in @tbl-index-notation. Capital Roman letters denote monetary quantities at the accounting or reporting level: loss $L$, premium $P$, margin $M$, equity $Q$, capital $K$, debt $D$, and assets $a$. These represent recognized or expected amounts unless randomness is explicit. All monetary amounts are nominal and expressed in a single currency. Each value refers to an amount at the relevant time within context. Premiums are paid and received at the start of each period and losses at the end. Discounting to present value is always explicit. There are no inflation adjustments. When stochastic treatment is required, the same symbols are viewed as random variables on $(\Omega,\mathcal{F},\mathsf P)$, and their expectations are written $\mathsf PL$ and so on, where $\mathsf P$ serves as both probability measure and expectation operator following Pollard’s convention. Bars, hats, and tildes may appear locally, e.g. $\bar L$, $\hat L$, or $\tilde L$, when both realized and expected versions are needed in the same equation; they mark the random quantity. Context determines meaning: probabilistic in probability sections, deterministic in accounting or valuation discussions.. **Increasing** and **decreasing** are used in a non-strict sense. We say that $f$ is increasing if it is non-decreasing. This convention means that many statements are clearer, because we do not have to mentally process the negative. We use the (infamous to Python programmers) **Walrus** operator: $f(x):=x^2$ *defines* the function $f$. This is a very handy shortcut. We use it like a programmer. ```{python} #| label: tbl-index-terminology #| tbl-cap: "Common terminology." from pmir.format import qd qd(""" | Term | Context | Meaning | |:------------|:---------------|:------------------------------------------------------------------------------| | Estimate | Profit targets | compute profit targets by unit | | Determine | Profit targets | same | | Communicate | Profit targets | describe process, amount, and implementation | | Implement | Premiums | compute premium to achieve target, usually for rates | | Calculate | Premiums | same, usually for an individual risk | | Evaluate | Premiums | compute profit target embedded in a premium | | Validate | Models | check that the model is working as intended, compare different types of model | """) ``` ```{python} #| label: tbl-index-definitions #| tbl-cap: "Common definitions." qd(""" | Term | Definition | |:---------------------|:-------------------------------------------------------------------------------------------------------------------| | Actuarial reserve | Expected discounted value of future cash flows, no risk adjustment. | | Actuarial value | Objective expected value with no risk margin. Discounted if appropriate. Undiscounted actuarial value for clarity. | | Duration | contract duration, short-d and long-d | | Emergence | | | Incurred | Always means ultimate incurred loss and never case incurred. | | Individual risk (IR) | | | Market value | Risk adjusted expected value. Discounted if appropriate. | | Payout pattern | long-tail (as distinct from heavy tail) | | Statutory reserve | Management best estimate liability, no discount, no (explicit) risk adjustment | | Surplus | Old terminology for capital | | Unit | | """) ``` ```{python} #| label: tbl-index-notation #| tbl-cap: "List of symbols." qd(""" | Symbol | Meaning | Reference | |:--------------|:----------------|:----------| | $X$ | Random variable | | | $\\mathscr F$ | Sigma algebra | | """) ```