The impact of counterparty risk and collateralization on longevity - - PowerPoint PPT Presentation

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

The impact of counterparty risk and collateralization on longevity swaps

Enrico Biffis Imperial College London & Pensions Institute David Blake Cass Business School & Pensions Institute Lorenzo Pitotti Imperial College London & Algorithmics Ariel Sun Risk Management Solutions Paris - February 3, 2011

1/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

2/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

3/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Recent transactions

Date Hedger Size Term (yrs) Type Interm./supplier Jan 08 Lucida Not disclosed 10 indexed JPM ILS funds Jul 2008 Canada Life GBP 500m 40 indemnity JPM ILS funds Feb 2009 Abbey Life GBP 1.5bn run-off indemnity DB ILS funds Partner Re Mar 2009 Aviva GBP 475m 10 indemnity RBS Jun 2009 Babcock GBP 750m 50 indemnity Credit Suisse International Pacific Life Re Jul 2009 RSA GBP 1.9bn run-off indemnity GS (Rothesay Life) Dec 2009 Berkshire GBP 750m run-off indemnity Swiss Re Council Feb 2010 BMW UK GBP 3bn run-off indemnity DB Paternoster Dec 2010 Swiss Re US 50m 8 indexed ILS funds (Kortis bond) Feb 2011 Pall GBP 70m 10 indexed JPM Pension Fund

4/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Motivation

Hedgers’ concerns

• longevity swaps introduce credit risk to a counterparty
• Global Financial Crisis and tighter regulation of OTC transactions

Other markets: interest-rate swaps (IRSs) for example

• almost every swap bilaterally collateralized
• cash collateral in over 90% of the cases
• collateral thresholds based on credit ratings, CDS spreads, etc.

Questions

• insurance (indemnity) vs. capital markets (value) paradigm
• to what extent does collateralization reduce pooling/diversification

benefits traditionally used as substitute for credit enhancement?

• how does impact of collateral costs compare with interest-rate swaps?
• longevity swap pricing with bilateral default risk and collateral

5/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Indemnity-based longevity swaps

Stylized example: single payment at time T

• notional n, fixed payment p
• variable payment ST (realized survival rate)

n × p n × ST Party A (hedger) Party B (hedge supplier)

6/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Indemnity-based longevity swaps

Stylized example: single payment at time T

• notional n, fixed payment p
• variable payment ST (realized survival rate)

n × p n × ST Party A (hedger) Party B (hedge supplier) Swap value (hedger’s viewpoint) V0 = nEQ

• exp
• −

T rtdt

• (ST − p)
• 6/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Indemnity-based longevity swaps

Stylized example: single payment at time T

• notional n, fixed payment p
• variable payment ST (realized survival rate)

n × p n × ST Party A (hedger) Party B (hedge supplier) Longevity swap rate p = EQ [ST ] + CovQ exp

• −

T

0 rtdt

• , ST
• EQ
• exp
• −

T

0 rtdt

• 6/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Indemnity-based longevity swaps

Stylized example: single payment at time T

• notional n, fixed payment p
• variable payment ST (realized survival rate)

n × p n × ST Party A (hedger) Party B (hedge supplier) Longevity swap rate p = EQ [ST ] + 0

6/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Indemnity-based longevity swaps

Stylized example: single payment at time T

• notional n, fixed payment p
• variable payment ST (realized survival rate)

n × p n × ST Party A (hedger) Party B (hedge supplier) Longevity swap rate p = EQ [ST ] + 0 We focus on baseline swap rate p = EP[ST ].

6/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

7/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Case study

UK-based hedger

• 10,000 individuals (England & Wales) aged 65 in 1980
• indemnity-based solution over 1980 − 2005
• interest risk hedged away

Realized cashflows

• population evolves as in Human Mortality Database (HMD)
• cashflow hedge in operation:
• realized rate
• −
• swap rate
• Marking to market (MTM)
• assume swap rates coincide with Lee-Carter forecasts based on most

recent HMD information available

• allow for mortality experience/counterparty risk

8/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Cashflows and marking-to-market

1980 1985 1990 1995 2000 2005 200 400 600 800 1000 1200 year GBP CFs MTM

9/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Longevity swap rates

1980 1985 1990 1995 2000 2005 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 year LC forecast 10/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Hedge supplier’s credit deterioration

1980 1985 1990 1995 2000 2005 −2500 −2000 −1500 −1000 −500 500 1000 year GBP CFs MTM MTM + 50bps MTM + 25bps 11/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

12/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Bilateral default risk

Counterparty risk is bilateral

• if the hedger is a pension plan...

⋆ sponsor’s covenant (“ability and willingness of the sponsor to pay sufficient advance contributions to ensure that the scheme’s benefits can be paid as they fall due”) in the UK ⋆ use/extrapolate spreads in credit/CDS market ⋆ analyze funding level/strategy of the scheme

Need to allow for credit enhancement tools

• termination rights (credit puts, break clauses, etc.)
• collateralization (posting of cash/securities)

Need to allow for MTM and costs accruing from MTM procedure

13/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Valuation

• hedger’s viewpoint, unitary notional (n = 1)
• default intensities (λA

t )t≥0, (λB t )t≥0

• ψA, ψB ∈ [0, 1] recovery rates upon default of the counterparty
• swap’s market value (e.g., Duffie/Huang, 1996)

V0 = EQ

• exp
• −

T (rt + Λt)dt ST − pd Λt := 1{Vt<0}(1 − ψA)λA

t + 1{Vt≥0}(1 − ψB)λB t .

14/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Valuation

• hedger’s viewpoint, unitary notional (n = 1)
• default intensities (λA

t )t≥0, (λB t )t≥0

• ψA, ψB ∈ [0, 1] recovery rates upon default of the counterparty
• swap’s market value (e.g., Duffie/Huang, 1996)

V0 = EQ

• exp
• −

T (rt + Λt)dt ST − pd Λt := 1{Vt<0}(1 − ψA)λA

t + 1{Vt≥0}(1 − ψB)λB t .

• swap rate

pd = EP[ST ] + CovQ exp

• −

T

0 (rt + Λt)dt

• , ST
• EQ
• exp
• −

T

0 (rt + Λt)dt

• ≤ EP[ST ]

14/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Valuation

• hedger’s viewpoint, unitary notional (n = 1)
• full recovery: ψA = ψB = 1
• swap’s market value

V0 = EQ

• exp
• −

T rtdt P − pd

• swap rate

pd = EP[ST ] + CovQ exp

• −

T

0 rtdt

• , ST
• EQ
• exp
• −

T

0 rtdt

• ≤ EP[ST ]

15/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Valuation

• hedger’s viewpoint, unitary notional (n = 1)
• full recovery: ψA = ψB = 1
• swap’s market value

V0 = EQ

• exp
• −

T rtdt P − pd

• swap rate

pd = EP[ST ] + CovQ exp

• −

T

0 rtdt

• , ST
• EQ
• exp
• −

T

0 rtdt

• ≤ EP[ST ]

Where is the flaw?

• credit enhancement can make ψA, ψB close to 1, but can’t be costless!
• focus on costs accruing from MTM procedure

(e.g., Johannes/Sundaresan, 2007)

15/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

16/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Collateral

Collateral strategy (Ct)t≥0, yield on collateral (δt)t≥0

• hedger’s viewpoint
• Ct > 0 (Ct < 0) represents cash to be held (posted) at time t in response

to changes in market conditions Ct =

• cA

t 1{Vt−<0} + cB t 1{Vt−≥0}

• Vt−
• ci, i ∈ {A, B}, fraction of swap’s market value posted by party i
• holding (posting) collateral of amount Ct yields (costs) δtCt after rebate

17/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Collateral

Collateral strategy (Ct)t≥0, yield on collateral (δt)t≥0

• hedger’s viewpoint
• Ct > 0 (Ct < 0) represents cash to be held (posted) at time t in response

to changes in market conditions Ct =

• cA

t 1{Vt−<0} + cB t 1{Vt−≥0}

• Vt−
• ci, i ∈ {A, B}, fraction of swap’s market value posted by party i
• holding (posting) collateral of amount Ct yields (costs) δtCt after rebate

Examples

• cA = cB = 1 (full collateralization)
• cA = 1{St−≤β(t)}, cB

t = 1{St−≥α(t)}∪{λB

t−≥γ} (with α > β, γ ≥ 0) 17/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Longevity swap rates with costly collateral

Full collateralization

• swap market value

V0 = EQ

• exp
• −

T (rt + Γt)dt

• (ST − pc)
• Γt :=
• λA

t (1 − cA t )−δtcA t

• 1{Vt<0} +
• λB

t (1 − cB t )−δtcB t

• 1{Vt≥0}
• longevity swap rate with full collateralization (cA = cB = 1)

pc = EP[ST ] + CovQ exp

• −

T

0 (rt−δt)dt

• , ST
• EQ
• exp
• −

T

0 (rt−δt)dt

• ≥ EP[ST ]

18/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Longevity swap rates with costly collateral

Full collateralization

• swap market value

V0 = EQ

• exp
• −

T (rt + Γt)dt

• (ST − pc)
• Γt :=
• λA

t (1 − cA t )−δtcA t

• 1{Vt<0} +
• λB

t (1 − cB t )−δtcB t

• 1{Vt≥0}
• longevity swap rate with full collateralization (cA = cB = 1)

pc = EP[ST ] + CovQ exp

• −

T

0 (rt−δt)dt

• , ST
• EQ
• exp
• −

T

0 (rt−δt)dt

• ≥ EP[ST ]

Intuition

• A receives collateral when ST is high, liability more capital intensive
• A posts collateral when ST is low, liability less capital intensive

18/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Longevity swap rates with costly collateral

Full collateralization

• swap market value

V0 = EQ

• exp
• −

T (rt + Γt)dt

• (ST − pc)
• Γt :=
• λA

t (1 − cA t )−δtcA t

• 1{Vt<0} +
• λB

t (1 − cB t )−δtcB t

• 1{Vt≥0}
• longevity swap rate with full collateralization (cA = cB = 1)

pc = EP[ST ] + CovQ exp

• −

T

0 (rt−δt)dt

• , ST
• EQ
• exp
• −

T

0 (rt−δt)dt

• ≥ EP[ST ]

Intuition

• B receives collateral when ST is low, liability less capital intensive
• B posts collateral when ST is high, liability more capital intensive

18/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Computing the swap rate

Markov setting, state variable vector X. Least-Squares Monte Carlo procedure:

• 1. Fix pc

i, generate M paths of X under Q along time grid T .

• 2. Compute recursively swap value as V m,i

tj

= β∗

j · e(Xm tj ), where

β∗

j = arg min βj∈RH M

• m=1
• V i,m

tj+1 + f i,m tj+1 − βj · e(Xm tj )

• .
• 3. Iterate over different pc

i’s: determine pc i∗ making initial price 1 M

M

m=1 V m,i∗ t0

close enough to zero.

19/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Opportunity cost of collateral

Q-dynamics of δ = X(i)

• a. Simulate X(−i) along time grid ˆ

T under pricing measure Q.

• b. On simulated paths, compute capital needed at each t ∈ ˆ

T to support representative liability St+T .

⋆ T proxies average liability duration in longevity swap market. ⋆ Solvency II framework, 99.5% VaR of the net assets over one year.

• c. Compute gains/costs to update capital at each t ∈ ˆ

T along each simulated path.

⋆ Funding costs incurred at realized LIBOR rate plus spread reflecting cost of capital. Spread values of 6% and 12%. ⋆ Risk-free rate rebated, netting of collateral with liability.

• d. Estimate parameters of X(i)’s Q-dynamics along each simulated

path, average across simulations.

20/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

21/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Example: collateral thresholds

2008 2013 2018 2023 2028 2033 2038 0.7 0.75 0.8 0.85 0.9 0.95 1

projection year St(ω) α(25) β(25)

22/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Example: constant r, λ, δ

(0,100) (5,95) (10,90) (20,80) (30,70) (40,60) (50,50) 5 10 15 20 25 −0.02 0.02 0.04 0.06 0.08 collateral rule (εα,εβ) payment date T swap margin (%)

23/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Fully fledged calibration

Swap spreads (basis points), pc

T − EP[ST ]; different collateralization rules.

(cA, cB) maturity (0, 0) (0, 0.5) (0.5, 0) (0.5, 0.5) (0, 1) (1, 0) (1, 1) 15.0 3.1 174.0

• 158.0

9.5 233.8

• 218.1

10.3 20.0 3.3 296.5

• 254.3

38.7 436.9

• 253.7

45.9

Maturities of interest rate swaps (IRS) giving the same spreads as fully collateralized longevity-linked payments with maturity 15 and 20.

(cA, cB) IRS implied maturity (1, 1) maturity 15.0 10.3 4.5 20.0 45.9 13.1

24/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Longevity swap spreads

5 10 15 20 25 −10 −8 −6 −4 −2 2 4 6 8 10 payment date swap margin and percentiles (%) 95 75 25 5

pc EP[ST ] − 1 against Lee-Carter mortality improvements quantiles

25/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Agenda

1 Motivation 2 Counterparty risk 3 Bilateral default risk 4 Collateralization 5 Examples 6 Conclusion

26/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

Conclusion

Longevity swap valuation under counterparty risk

• bilateral default risk
• MTM process and costly collateral
• endogenous longevity swap rate (Least-Square Monte Carlo)
• positive margin in the presence of symmetric default risk / collateral rules

Implications

• robust framework to compare (credit enhanced) indemnity-based solutions

with securitization and indexed solutions

• collateralization costs comparable with those arising in IRSs market
• fixed-income market may provide the right framework for standardized

MTM and collateral rules

27/28

Motivation Counterparty risk Bilateral default risk Collateralization Examples Conclusion

THANK YOU

28/28