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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