Finance for Actuaries

Interest Rate Sensitive Insurance Products

2000 Investment Conference

Jeroen van Bezooyen

Shyam Mehta

Finance for Actuaries 2

Agenda

• Product Description

• Valuation Methodology

• Interest Rate Model

• Example

• Hedging Strategies

• Model Extensions

Finance for Actuaries 3

Product Description• Based on French interest rate products

• Single premium

• Guaranteed minimum annual bonus

• Discretionary additional bonus based on portfolio yield

• Maturity: 8 years with option to extend

• Early surrender option (subject to surrender and/or tax penalties)

Finance for Actuaries 4

Product Description

• Bonus policy is not hard-coded

• Awarded bonus credits driven by– Guarantee– Competitor bonuses– Portfolio yield

• Modelling approach: specify functional form that captures the above effects

Finance for Actuaries 5

Valuation Methodology

Traditional actuarial approach

• ‘Single scenario’

• Specify discount rate and bonus credit assumptions

• Specify lapse rates

• Project cash flows

• Value product as present value

Finance for Actuaries 6

Valuation Methodology

Drawback of actuarial approach

• No explicit allowance for optionality (actuarial judgement?)

• No risk management/hedging policy (what to do when interest rates go to 2%?)

Finance for Actuaries 7

Valuation Methodology

Financial economic approach

• Specify stochastic interest rate model

• Calibrate model to current market conditions

• Specify bonus rate function

• Specify lapse rates

• Solve model ‘backwards’ (numerically)

Finance for Actuaries 8

Valuation Methodology

Advantages of financial economic

approach

• Allows for optionality

• Consistency with market prices

• Specifies risk management/hedging policy

• Possible to ‘hedge’ model assumptions

Finance for Actuaries 9

Interest Rate Model• Hull-White (1-factor) interest rate model, i.e.

model short rate as– Mean-reverting and– Normal process

• Parameters– Volatility– Mean reversion rate– Mean reversion level (time-dependent for calibration)

• Implement as trinomial tree

Finance for Actuaries 10

Trinomial Tree

Interest rate (t = 1, up)

Interest rate (t = 0) Interest rate (t = 1, middle)

Interest rate (t = 1, down)

tttt dBdtrdr

Finance for Actuaries 11

Trinomial Tree

Backward solution method

• Set value (N,j) at final time layer equal to 1

• Product value at node (i,j) is product of– exp [ -interest rate (i,j) ]*– [ 1 + bonus(i,j) ]*– Probability up-move * up value (i+1) +

Probability middle-move * middle value (i+1) + Probability down move * down value (i+1)

Finance for Actuaries 12

Example

Bonus

• 3.5% guarantee, plus fraction of difference (if positive) between– weighted average of 8-t year spot rate and

fixed 5% rate, and– 3.5% guarantee

Finance for Actuaries 13

Example

• Early surrender– Sliding scale of surrender penalties– Fraction of investors withdraws rationally– Withdrawal behaviour of rest is driven by a

lapse rate function, where surrender rate depends on difference between yield and bonus

• Option to extend at maturity

Finance for Actuaries 14

Example - Lapse Rate Function

0%

10%

20%

30%

40%

-4% 0% 4% 8% 12%

(Yield - Bonus)

Lap

se r

ate

(pa)

Finance for Actuaries 15

Example - Valuation Results

• Euro term structure as at 31st May 2000

• Lump sum investment of €100, 5% initial charge. Product value to investor:– Bonus only: €97.26 / 92.40– Including early surrender: €100.92 / 95.88– Including extension option €102.31 / 97.20

Finance for Actuaries 16

ExampleProduct Value as Function of Yield Level

80%

100%

120%

140%

2% 3% 4% 5% 6% 7% 8% 9%

Yield

Val

ue

Product

8-year Bond

Finance for Actuaries 17

ExampleProduct Value as Function of Yield Curve Tilt

80%

100%

120%

140%

160%

180%

-0.3% -0.2% -0.1% 0.0% 0.1% 0.2% 0.3% 0.4% 0.5%

Slope

Val

ue

Product

8-year Bond

Finance for Actuaries 18

ExampleProduct Value as Function of Volatility

102%

104%

106%

108%

110%

112%

50% 75% 100% 125% 150% 175% 200%

Volatility (% original level)

Val

ue

Product 8-year Bond

Finance for Actuaries 19

Hedging

• Valuation method is based on replicating portfolio

• Consequently, outcome is product value as well as hedging strategy

• Hedging strategy is dynamic, i.e. depends on interest rate level

Finance for Actuaries 20

Hedging - Example

• Consider two assets– 8-year bond with 6% coupon– Cash

• 31st May 2000 hedge strategy is to invest– 84% in bond, and– 16% in cash

Finance for Actuaries 21

Hedging - Dynamic StrategyHedge Ratio as Function of Yield Level

0%

20%

40%

60%

80%

100%

120%

2% 4% 6% 8%

Yield Level

Hed

ge R

atio

Finance for Actuaries 22

Hedging - Model Assumptions• Hedging strategies based on model

parameters• Parameters can be calibrated against

market prices (term structure, options, etc.)• However, parameters can change over

time! (e.g. volatility, yield curve slope)• These modelling assumptions can also be

hedged

Finance for Actuaries 23

Hedging - Model Assumptions

• Volatility parameter can be hedged with an option position

• Introduce third asset: 8-year, 3.5% floor contract

• Hedge portfolio is to invest– 55% in 8-year bond– 43% in cash– 2% in floor (total notional € 198)

Finance for Actuaries 24

Model Extensions

• Additional behavioural analysis– Bonus declarations– Lapse rate function– Extension rationality

• Two-factor Hull-White model

• Calibration to other instruments

Finance for Actuaries 25

Conclusions

• Advantages of using a market value model– Allows explicitly for optionality– Calibration to market prices– Delivers value as well hedge strategy

• This should be in every Actuary’s toolkit!

• Actuarial judgement on specification of lapse rate and bonus credit functions

Finance for Actuaries

Interest Rate Sensitive Insurance Products

2000 Investment Conference

Jeroen van Bezooyen

Shyam Mehta