Advanced methods of insurance

Lecture 1

Example of insurance product I

• Assume a product that pays – A sum L if the owner dies by time T – A payoff max(SP(T)/SP(0), 1 + k) • Pricing factors – Risk free discount factor v(t,T) – Survival function S(t,T) – Level of the underlying asset SP(t) – Volatility of SP(t)

Example of insurance product II

• Assume a product that pays – A sum L if the owner dies by time T – A payoff max(min(S i (T)/S(0)), 1 + k) • Pricing factors – Risk free discount factor v(t,T) – Survival function S(t,T) – Level of the underlying asset SP i – Volatility of SP i (t) – Correlation of the asset SP i (t).

(t)

How do you pay for the product?

• You may pay for the product in a unique payment. • Alternatively, you may pay on a running basis, with several payments until maturity, if you survive to maurity. • In case one dies, the payments would stop and a fraction of the amount paid is given to the heirs.

Financial and insurance products

• Financial products allow to trnasfer consumption from the current to future periods. • Insurance products introduce actuarial risks such as the risk of death for an individual underwriting a life insurance policy or the risk of catastrophic loss for a product that is indexed non-life insurance risks. • In this course we review the main instruments that could be used to transfer consumption and risk from the present to the future.

Financial and insurance products

• Fixed income. Bonds. Pay-off is defined independently from the project funded.

• Variable income. Equity. Pay-off is a function of the proceedings from the project.

• Derivatives.

Contingent claims

. Products whose value is defined as a function of other risky assets • Managed funds: funds aggregated and managed on behalf of customers • Insurance policies: life, death and mixed.

Financial structures: ingredients

• Schedule: {

t

0 ,

t

1 , …,

t n

} – Calendar conventions – Day-count conventions • Coupon plan: {

c

0 ,

c

1 , …,

c n

} – Deterministic – Indexed (interest rates, inflation, equity, credit, commodities, longevity, …) • Repayment plan {

k

0 ,

k

1 , …,

k n

} – Deterministic – Stochastic (callable, putable, exchangeable, convertible)

Working in finance or insurance

• Structurer: design products, identifying possible customers and including possible clauses. • Pricer: evaluated the product, “marking-to market” the elementary elements of the product • Risk manager: evaluate the exposures to risk factors, and both expected and unexpected risks, as well as their dependence.

• NB. All these operations are based on the decomposition of the product in elementary units.

Arbitrage principle

• We say that there exists an arbitrage opportunity (

free lunch

) if in the economy it is possible to build a position that has negative or zero value today and positive value at a future date (positive meaning non-negative in one state and positive in at least one)

Replicating portfolio

• A replicating portfolio or a replicating strategy of a financial product is a set of postions whose value at some future date is equal to that of the financial product with probability one. • If it is possible to build a replicating portfolio or strategy of a financial product for a price different from that of the product, one could exploit infinite arbitrage profits selling what is more expensive and buying what is cheaper.

Replicating portfolio for valuation and hedging

• Saying that no arbitrage profits are possible means to require that the value of each financial product is equal to the value of its replicating portfolio and strategy (

pricing

) • Buying the financial product and selling the replicating portfolio enables to immunize the position (

hedging

).

Zero-coupon-bond

• Define P(t,t k ,x k ) the value at time t of a

zero-coupon bond

(ZCB). It is a security that does not pay coupons before maturity and that gives right to receive a quantity x k futurre date t k at a • Define

v

(t,t k ) the discount funtion, that is the value at time t of a unit of cash available in t k • Assuming infinite divisibility of each bond, down to the bond paying one unit at maturity, we obtain that

P(t,t k ,x k ) = x k v(t,t k )

Coupon bond evaluation

Let us define P(t,T;c) the price of a bond paying coupon c on a schedule {t 1 , t 2 , …,t m =T}, with trepayment of capital in one sum at maturity T. The cash flows of this bond can be replicated by a basket of ZCB with nominal value equal to c corresponding to maturities t i for i = 1, 2, …, m – 1 and a ZCB with a nominal value 1 + c iat maturity T. The arbitrage operation consisting in the purchase/sale of coupons of principal is called

coupon stripping

.

P

(

t

,

T

;

c

) 

k m

  1

cv

(

t

,

t k

) 

v

(

t

,

t m

)

Bond prices and discount factors

• Based on zero-coupon bond prices and the prices of coupon bonds observed on the market it is possible to retrieve the discount function.

• The technique to retrieve the discount factor is based on the no-arbitrage principle and is called

bootstrapping

• The discount function establishes a financial equivalence relationship between a unit amount of cash available at a future date t k and an amount v(t,t k ) available in t. • Notice that the equivalence holds for each issuer.

Bootstrapping procedure

Assume that at time

t

the market is structured on

m

periods with maturities

t k = t + k, k=1....m,

and assume to observe zero-coupon-bond P(t,t k ) prices or coupon bond prices P(t,t k ;c k ). The

bootstrapping

procedure enables to recover discount factors of each maturity from the previous ones.

v

 

k



P



t

,

t k

;

c k





c k i

 1

k

  1

v

 

i

1 

c k

The term structure of interest rates

The term structure is a way to represent the discount function.

It may be represented in terms of discrete compounding

v

(

t

,

t k

)  1  1 

i

(

t

,

t k

)  

t k



t



i

(

t

,

t k

)  

v

(

t

,

t k

)   1 / 

t k



t

  1

The term structure of interest rates

The term structure is a way to represent the discount function.

It may be represented in terms of continuous compounding

v

(

t

,

t k

)  exp  

i

 

t k k



t

 

i

(

t

,

t k

)   ln 

v

(

t

,

t k t k



t

) 

The term structure of interest rates

The term structure is a way to represent the discount function.

It may be represented in terms of discrete compounding

v

(

t

,

t k

)  1  

t k

 1

t



i

(

t

,

t k

)

i

(

t

,

t k

) 

t k

1 

t

  1

v

(

t

,

t k

)   1 

Term (forward) contracts

• A forward contract is the exchange of an amount

v

(t,  ,T) fixed at time t and paid at time  ≥ t in exchange for one unit of cash available at T.

• • A spot contract is a specific instance in which  = t, so that

v

(t,  ,T) =

v

(t,T).

v

(t,  ,T) is defined as the (

forward price

) established in t of an investment starting at  ≥ t and giving back a unit of cash in T.

Spot and forward prices

• • • 1.

Consider the following strategies Buy a nominal amount at time 

v

(t,  ,T) availlable at , giving a unit of cash available on T  on the spot market and buy a forward contract for settlement 2.

Issue debt on the spot market for repayment of a unit of cash at time T.

It is easy to see that this strategy yields a zero pay-off at time both at time  and at time T.

If the value of the strategy at time t is different from zero, there exists an arbitrage opportunity for one of the two parties.

Arbitrage example

–

v

(t,  )

v

(t,  ,T)

v

(t,  ,T) – –

v

(t,  ,T)

v

(t, T) – – 1 – 1 Total

v

(t, T) –

v

(t,  )

v

(t,  ,T) 0 0

Spot and forward prices

• Spot and forward prices are then linked by a relationship that rules out the arbitrage opportunity described above

v

(t,T)=

v

(t,  )

v

(t,  ,T) • All the information on forward contracts is then completely contained in the spot discount factor curve.

• Caveat. This is textbook paradigm that is under question today. Can you guess why?

The forward term structure

Forward term structure is a way of representing the forward discount function.

It may be represented with discrete compounding.

f

(

t

,



,

T

)   

v

(

t

,



,

T

)   1 / 

T

 

v

(

t

,



v

(

t

,

T

) )   1 / 

T

       1  1

The forward term structure

Forward term structure is a way of representing the forward discount function.

It may be represented with continuous compounding.

f

(

t

,  ,

T

)    ln ln 

i

(

t

, 

v

(

t

,  ,

T



v

(

t T

,   )  

T

)(

T T

 

t

)  

T

)   

v

(

t

,

i

(

t

, 

T

) )(   

t

)

The forward term structure

Forward term structure is a way of representing the forward discount function.

It may be represented with linear compounding.

f

(

t

,  ,

T

) 

T

1   

T

1     1

v

(

t

,  ,

T

)   1   

v

(

t

, 

v

(

t

,

T

) )   1 

Indexed (floating) coupons

• An indexed coupon is determined based on a reference index, typically an interest rates, observed at time  , called the

reset

date

.

• The typical case (known as

natural time lag

) is a coupon with – reference period from  –

reset

date  to T and payment date T – reference interest rate for determination of the coupon

i

(  ,

T

) (

T –

 ) = 1/

v

(  ,

T

) – 1

Replicating portfolio

• What is the replicating portfolio of an floating coupon, indexed to a linear compounded interest rate for one unit of nominal? • Notice that at the reset date  the value of the coupon, determined at time  and paid at time T, will be given by

v

(  ,

T

)

i

(  ,

T

) (

T –

 ) = 1 –

v

(  ,

T

) • The replicating portfolio is then given by – A long position (investment) of one unit of nominal available at time  – A short position (financing) for one unit of nominal available at time T

Cash flows of a floating coupon

• Notice that a floating coupon on a nominal amount C corresponds to a position of debt (

leverage

) C t  T C

No arbitrage price: indexed coupons

• The replicating portfolio enables to evaluate the coupon at time t as: indexed coupons =

v

(t,  ) –

v

(t,T) At time  we know that the value of the position is: 1 –

v

(  ,T) =

v

(  ,T) [1/

v

(  ,T) – 1] =

v

(  ,T)

i

(  ,T)(T –  ) = discount factor X indexed coupon • At time t the coupon value can be written

v

(t,  ) –

v

(t,T) = =

v v

(t,T)[ (t,T)

f v

(t, (t,   ) /

v

(t,T) – 1] ,T)(T –  ) = discount factor X

forward

rate

Indexed coupons: some caveat

• It is wrong to state that expected future coupons are represented by

forward

rates, or that forward rates are unbiased forecasts of future forward rates • The evaluation of expected coupons by forward rates is NOT linked to any future scenario of interest rates, but only to the current interest rate curve.

• The

forward

term structure changes with the

spot

term structure, and so both expected coupons and the discount factor change at the same time (in opposite directions)

Indexed cash flows

• Let us consider the time schedule  t,t 1 ,t 2 ,…t m  where t i , i = 1,2,…,m – 1 are coupon

reset

and each of them is paid at t i+1 .

times, t is the valuation date. • It is easy to verify that the value the series of flows corresponds to – A long position (investment) for one unit of nominal at the reset date of the first coupon (t 1 ) – A short position (financing) for one unit of nominal at the payment date of the last coupon (t m )

Floater

• A floater is a bond characterized by a schedule  t,t 1 ,t 2 ,…t m  – at t 1 the current coupon c is paid (value cv(t,t 1 )) – t i , i = 1,2,…,m – 1 are the

reset

dates of the floating coupons are paid at time t i+1 (value v(t,t 1 ) – – principal is repaid in one sum t m .

• Value of coupons: cv(t,t 1 ) + v(t,t 1 ) – • Value of principal: v(t,t m ) • Value of the bond v(t,t m )) v(t,t m ) Value of bond = Value of Coupons + Value of Principal = [cv(t,t 1 ) + v(t,t 1 ) – v(t,t m )] + v(t,t m ) =(1 + c) v(t,t 1 ) • A floater is financially equivalent to a short term note.

Forward rate agreement

(FRA)

• A FRA is the exchange, decided in t, between a floating coupon and a fixed rate coupon

k

, for an investment period from  to T. • Assuming that coupons are determined at time equal to interest rate

i

(  ,T), and paid, at time T,  , and set FRA(t) = v(t,  ) – v(t,T) – v(t,T)k = v(t,T) [v(t,  )/ v(t,T) –1 – k] = v(t,T) [

f

(t,  ,T) – k] • At origination we have FRA(0) = 0, giving k =

f

(t,  ,T) • Notice that market practice is that payment occurs at time  (

in arrears

) instead of T (

in advance

)

Natural lag

• In this analysis we have assumed (natural lag) – Coupon reset at the beginning of the coupon period – Payment of the coupon at the end of the period – Indexation rate is referred to a tenor of the same length as the coupon period (example, semiannual coupon indexed to six-month rate) • A more general representation Expected coupon = forward rate + convexity adjustment + timing adjustment • It may be proved that only in the “ natural lag” case convexity adjustment + timing adjustment = 0

Esercise Reverse floater

• A reverse floater is characterized by a time schedule  t,t 1 ,t 2 ,…t j , …t m  – From a reset date t j formula coupons are determined on the r Max –  i(t i ,t i+1 ) where  is a

leverage

parameter.

– Principal is repaid in a single sum at maturity

Swap contracts

• The standard tool for transferring risk is the swap contract: two parties exchange cash flows in a contract • Each one of the two flows is called

leg

• Examples of

swap

– Fixed-floating plus spread (plain vanilla swap) – Cash-flows in different currencies (

currency swap

) – Floating cash flows indexed to yields of different countries (

quanto swap

) –

Asset swap

,

total return swap

,

credit default swap

…

Swap

: parameters to be determined

• The value of a swap contract can be expressed as: – – Fixed rate coupon (

swap

rate): the value of fixed rate payment such that the fixed leg be equal to the floating leg –

Net-present-value

(NPV); the difference between the present value of flows

Spread

: the value of a periodic fixed payment that added to to a flow of floating payments equals the fixed leg of the contract.

Plain vanilla swap (fixed-floating)

• In a fixed-floating swap – the long party pays a flow of fixed sums equal to a percentage c, defined on a year basis – the short party pays a flow of floating payments indexed to a market rate • Value of fixed leg:

c i m

  1



t i



t i

 1

  

i

• Value of floating leg: 1 

v

 

m



i m

  1

v

 

i t i



t i

 1  

t

,

t i

 1 ,

t i



Swap rate

• In a fixed-floating swap at origin Value fixed leg = Value floating leg swap rate m  i  1



t i



t i

 1

v t t i

 1 

v

swap rate  m  i  1



t i

1  

v t t m t i

 1

  

i t t m

Swap rate

• Representing a floating cash flow in terms of

forward

rates, a swap rate can be seen as a weghted average of forward rates swap rate m  i  1 

t i



t i

 1

v t t i

 m  i  1

v t t i

swap rate  m  i  1

v t t i

m  i  1 

t i t i



t i

 1  

t

,

t i

 1 ,

t i

 

t i

 1   

i t i



t i

 1  

t

,

t i

 1 ,

t i



Swap rate

• If we assume ot add the repayment of principal to both legs we have that swap rate is the so called par yield (i.e. the coupon rate of a fixed coupon bond trading at par) swap rate m



i  1



t i



t i

 1

v t t i

 1 

v t t m

swap rate m



i  1



t i



t i

 1

    

i m

 1

Bootstrapping procedure

Assume that at time

t

the market is structured on

m

periods with maturities

tk = t + k, k=1....m,

and assume to observe swap rates on such maturities. The

bootstrapping

procedure enables to recover discount factors of each maturity from the previous ones.

v

 

k

 1  swap rate 1 

   

k

swap rate

k

 1 

i

 1

 

k i

Forward swap

rate

• In a

forward start swap

the exchange of flows determined at

t

begins at

t j

. Value fixed leg = Value floating leg forward forward swap rate m  i  j



t i



t i

 1

v t

  swap rate  i  m  j  1



t i j

 

v v t i

 1

t t t i t m



  

i v t t j



v t t m

Swap rate: summary

1.

2.

3.

The swap rate can be defined as: A fixed rate payment, on a running basis, financially equivalent to a flow of indexed payments A weighted average of

forward rates

with weights given by the discount factors The internal rate of return, or the coupon, of a fixed rate bond quoting at par (par yield curve)

Asset Swap

(

ASW

)

• L’

asset swap

is a package of – A bond – A

swap

contract • The two parties pay – The cash flows of a bond and the difference between par and the market value of the bond, if positive – A

spread

over the floating rate and the difference between the market value of the bond and par, if positive

Asset Swap

(

ASW

)

• Asset Swap on bond

DP

(

t

,

T

;

c

) • Value of the fixed leg: max  1 

DP



t

,

T

;

c

 , 0  

c i m



 1 

t i



t i

 1 ,

t i

• Value of the floating leg: max 

DP



t

,

T

;

c

  1 , 0   1 

v

 

m



spread i m

  1

v

 

i t i



t i

 1 

Asset Swap

(

ASW

)

Spread

• The

spread

is obtained equating the value of the two legs

spread



c



tasso swap

 1 

DP



t

,

T

;

c



i m

  1



t i



t i

 1

  

i

Structuring choices

• Natural lag: – Reference period of payment is equal to the tenor of the reference rate – Reset date at the beginning of the period (in advance) • “

In arrears”

: – Coupons reset and paid at the same date • CBM/CMS: coupon indexed to long term interest rates and swap rates.