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Annex F. Derivation of the simplification formula for the counterparty default adjustment F.1. Starting point of the derivation is a simple deterministic model for the recoverables as follows: CFt BE Re c = , t ( 1 + r ) where BERec CFt ∑ t ≥ 1 = Best estimate of recoverables taking not account of expected loss due to default of the counterparty = Expected cash-flow underlying the recoverable at the end of year t r = Risk-free rate (a flat curve is assumed) F.2. Let PD be the probability that the counterparty will default during the next year and let this probability be constant over time. Let further RR be the recovery rate of the counterparty. The expected loss can be approximated as follows Adj CD ≈ − ∑ t ≥1 = −( 1 − RR) ≈ −( 1 − RR) ( 1 − ( 1 − PD) ) ⋅ ( 1 − RR) ⋅ CF ⋅ BE ⋅ BE Re c Re c + ( 1 − RR) + ( 1 − RR) t ( 1 + r ) ⋅ ⋅ t ∑ t ≥1 ∑ t ≥1 ⎛1 − PD ⎞ ⎜ ⎟ ⋅ CFt ⎝ 1 + r ⎠ ⎛ 1 ⎞ ⎜ ⎟ ⎝1 + s ⎠ 110/112 t t t ⋅ CF PD where s = r + . (The last approximation is based on the assumption 1 − PD that is r/(1-PD) ≈ r, because PD is small.) F.3. This shows that AdjCD ≈ −( 1 − RR) ⋅ ( BERe c − BERe ′ c ) where BE’Rec is the best estimates of recoverables as defined above, but discounted with interest rate s instead of r. F.4. BE’Rec can be approximated by means of the duration approach as follows: BE ′ Re c = BE Re c ≈ BE Re c − Dur − Dur mod mod PD ⋅ ⋅ BE 1 − PD ⋅ ( s − r ) ⋅ BE Re c Re c , t © CEIOPS 2010
where Durmod denotes the modified duration of the recoverables, defined as 1 1 t ⋅CFt Dur mod = . t BE 1+ r ( 1+ r) Re c ∑ t≥ 1 F.5. Replacing BE’Rec with this approximation in the formula in paragraph F.3 yields the following result: PD AdjCD ≈ −( 1 − RR) ⋅ Durmod ⋅ ⋅ BERe c 1 − PD 111/112 © CEIOPS 2010
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CEIOPS e.V. - Westhafenplatz 1 - 60
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1. Introduction 1.1. In its letter
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2.2.2. The following recitals provi
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“[…]Where, in specific circumst
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3.6 It is noted that in the recital
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have in place internal processes an
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not be appropriate to reduce this v
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3.36 Hence where an (re)insurance u
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Scale 3.46 Assigning a scale introd
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“scale” that will apply in all
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circumstances of its omission or mi
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3.81 Under these circumstances, it
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• the level of certainty i.e. the
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3.105 All in all, at least the foll
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the chosen valuation method. If it
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development, so that it is likely t
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• As part of supervisory guidance
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case the threshold is not exceeded.
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if the threshold is not exceeded th
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principles. In following paragraphs
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sign ( x) = −1 if x < 0 . 3.179 E
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3.193 The time value of the investm
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distribution to policyholders as th
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• assume independency between ass
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company or new line of business, al
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easonable be supposed proportional
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incurred when the method is applied
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3.261 The general rule for calculat
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• the loss absorbing capacity of
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Page 91 and 92: 3.5. Quarterly calculations 3.444 A
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