Securitization of Mortality Risks in Life Annuities

Yijia Lin

Outline• Introduce mortality-based securities• Swiss Re mortality bond (December 2003)• Other side of the “mortality tail”--

longevity risk.• Mortality risk bond• Uncertainty in mortality forecasts.• Potential expansion of the annuity market.

Insurance Securitizations• Jack Gibson predicts a flurry of life

based deals in 2004.• Really asset securitizations.• Mortality securitizations – pure

hedge of mortality risk.

Securitization: Insurance linked bond

Bt + Dt =1,000C

Bond Coupons and Insurance Benefits

Mortality Bond Coupon

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5 6 7 8 9 10 11

Insurance

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11

Floating for fixed swaps• Floating cash flow swapped for fixed.• Swap dealer replaces SPC

Equivalent swap structure

Bt + Dt = x + y

Swiss Re’s Mortality Bond• Issued December 2003, matures

January 1, 2007• No coupons at risk• Priced to sell at par with coupon

LIBOR + 1.35%• Principal at risk during four calendar

years

Swiss Re’s Mortality Bond• Weighted average mortality rate for

US, UK, France, Italy Switzerland

q = max(q1,q2,q3,q4 ) where qi = 2002 + i Index

q0 = 2002 Index

Maturity value =

400,000,000 if q ≤ q0

400,000,0001.5q0 − q

0.2q0

if q0 < q ≤1.5q0

0 if q >1.5q0

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Wang Transform PricingMortality distribution implied by annuity

pricesF(t) = Pr(T ≤ t)

Φ(u) is the standard normal cdf

F *(t) = Φ[Φ−1(F(t)) − λ]

λ is the market price of risk

Market Price of Mortality Risk

• Realistic distribution 1996 IAM• Observed male age 65 market pricesF(t)=t q65,m

F *(t) = Φ[Φ−1(t q65,m ) − λ]

λ = λ65,m

Transform Pricing• Annuity market: solve for MPR

• Bond market price

Annuity price (1− e) = ax(12)

=1

12 i /12 px*d(0,i /12)

i=1

∞

∑

t px* =1− Φ Φ−1

t qx( )− λx[ ]

V = Fd(0,T) + E*[Dt ]d(0,t)t=1

T

∑

Female, age 65

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

65 70 75 80 85 90 95 100 105 110

Age

lx

1995 US Buck Experience

1995 Market Mortality based on the WangTransform

Investor Coupon

Dt

1,000=

C if l x+t ≤ Xt

C + Xt − l x+ t if Xt < l x+t ≤ C + Xt

0 if l x+t > C + Xt

⎧

⎨ ⎪

⎩ ⎪

= Max(C + Xt − l x+ t ,0) − Max(Xt − l x+ t ,0)

Coupon Market Value

E* Dt[ ]d(0,t) =1,000E* Max C + Xt − l x+ t ,0( )[ ]d(0,t)

−1,000E* Max Xt − l x+ t ,0( )[ ]d(0,t)

l x, tpx* Binomial distribution with parameters

Insurance Benefit

Bt =1,000C − Dt

=1,000

0 if l x+ t ≤ Xt

l x − Xt if Xt < l x+ t ≤ C + Xt

C if l x+ t > C + Xt

⎧

⎨ ⎪

⎩ ⎪

E* Bt[ ]=1,000C − E* Dt[ ]

Strike levels

1996 IAM is the reference table

Mortality projections• Trend is improvement (lower q)• Optimistic: Life expectancy will

increase to 150 or 200 years.• Pessimistic: Projecting trends is

dangerous.• Data situation in the US: inadequate,

but improving.

Demand for longevity hedge• Longevity risk in corporate benefit

plans shifting to individuals• Proposed reforms in social security

shift longevity risk to individuals• Life insurers have not responded to

the opportunity

Final comments• Securitization - a tool for managing

longevity risks through mortality linked bonds or swaps.

• A market for mortality-based securities - could develop.

• Needed - Better collection and dissemination of data and research in mortality dynamics.