1. Advanced methods of insurance Lecture 1

2. 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)

3. 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 (t) –Volatility of SP i (t) –Correlation of the asset SP i (t).

4. 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.

5. 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.

6. 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.

7. 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)

8. 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.

9. 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)

10. 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.

11. 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).

12. 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 at a futurre date t k 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 )

13. 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.

14. 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.

15. 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.

16. 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

17. 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

18. 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

19. 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.

20. Spot and forward prices Consider the following strategies 1.Buy a nominal amount v(t, ,T) availlable at  on the spot market and buy a forward contract for settlement at time , giving a unit of cash available on T 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.

21. Arbitrage example – v(t,  ) v(t, ,T) v(t, ,T) – – – v(t, ,T) 1 v(t, T)–– 1 Total v(t, T) – v(t,  ) v(t, ,T) 00

22. 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?

23. The forward term structure Forward term structure is a way of representing the forward discount function. It may be represented with discrete compounding.

24. The forward term structure Forward term structure is a way of representing the forward discount function. It may be represented with continuous compounding.

25. The forward term structure Forward term structure is a way of representing the forward discount function. It may be represented with linear compounding.

26. 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  to T – reset date  and payment date T – reference interest rate for determination of the coupon i( ,T) (T –  ) = 1/v ( ,T) – 1

27. 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

28. Cash flows of a floating coupon Notice that a floating coupon on a nominal amount C corresponds to a position of debt (leverage)

29. 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(t,T)[v(t,  ) / v(t,T) – 1] = v(t,T) f(t, ,T)(T –  ) = discount factor X forward rate

30. 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)

31. 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 times, and each of them is paid at t i+1. 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 )

32. 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 ) – v(t,t m )) –principal is repaid in one sum t m. Value of coupons: cv(t,t 1 ) + v(t,t 1 ) – v(t,t m ) Value of principal: v(t,t m ) Value of the bond 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.

33. 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 , and set equal to interest rate i( ,T), and paid, at time T, 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)

34. 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

35. 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 coupons are determined on the formula r Max –  i(t i,t i+1 ) where  is a leverage parameter. –Principal is repaid in a single sum at maturity

36. 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 …

37. Swap: parameters to be determined The value of a swap contract can be expressed as: – Net-present-value (NPV); the difference between the present value of flows –Fixed rate coupon (swap rate): the value of fixed rate payment such that the fixed leg be equal to the floating leg –Spread: the value of a periodic fixed payment that added to to a flow of floating payments equals the fixed leg of the contract.

38. 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: Value of floating leg:

39. Swap rate In a fixed-floating swap at origin Value fixed leg = Value floating leg

40. 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

41. 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)

42. 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.

43. 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

44. Swap rate: summary The swap rate can be defined as: 1.A fixed rate payment, on a running basis, financially equivalent to a flow of indexed payments 2.A weighted average of forward rates with weights given by the discount factors 3.The internal rate of return, or the coupon, of a fixed rate bond quoting at par (par yield curve)

45. 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

46. Asset Swap (ASW) Asset Swap on bond DP(t,T;c) Value of the fixed leg: Value of the floating leg:

47. Asset Swap (ASW) Spread The spread is obtained equating the value of the two legs

49. 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.