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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(Si(T)/S(0)), 1 + k) • Pricing factors – – – – – Risk free discount factor v(t,T) Survival function S(t,T) Level of the underlying asset SPi (t) Volatility of SPi(t) Correlation of the asset SPi(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: {t0, t1, …,tn} – Calendar conventions – Day-count conventions • Coupon plan: {c0, c1, …,cn} – Deterministic – Indexed (interest rates, inflation, equity, credit, commodities, longevity, …) • Repayment plan {k0, k1, …,kn} – 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-tomarket” 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,tk,xk) 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 xk at a futurre date tk • Define v(t,tk) the discount funtion, that is the value at time t of a unit of cash available in tk • Assuming infinite divisibility of each bond, down to the bond paying one unit at maturity, we obtain that P(t,tk,xk) = xk v(t,tk) Coupon bond evaluation Let us define P(t,T;c) the price of a bond paying coupon c on a schedule {t1, t2, …,tm=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 ti 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. m P(t , T ; c) cv(t , t k ) v(t , t m ) k 1 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 tk and an amount v(t,tk) 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 tk = t + k, k=1....m, and assume to observe zero-coupon-bond P(t,tk) prices or coupon bond prices P(t,tk;ck). The bootstrapping procedure enables to recover discount factors of each maturity from the previous ones. k 1 vt , t k Pt , t k ; ck ck vt , ti i 1 1 ck 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 1 v(t , t k ) t k t 1 i(t , tk ) 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 , t k t k t ln v(t , t k ) i (t , t k ) tk 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 1 v(t , t k ) 1 t k t i (t , t k ) 1 i (t , t k ) tk t 1 1 v(t , t k ) 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 • • • 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. 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) 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 ) v (t , ) v (t , T ) 1 / T 1 1 / T 1 The forward term structure Forward term structure is a way of representing the forward discount function. It may be represented with continuous compounding. ln v(t , , T ) f (t , , T ) T ln v(t , ) ln v(t , T ) T i (t , T )(T 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. 1 f (t , , T ) T 1 T 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 to T – reset date 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(t,T)[v(t,) / v(t,T) – 1] = v(t,T) f(t,,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,t1,t2,…tm where ti, i = 1,2,…,m – 1 are coupon reset times, and each of them is paid at ti+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 (t1) – A short position (financing) for one unit of nominal at the payment date of the last coupon (tm) Floater • A floater is a bond characterized by a schedule t,t1,t2,…tm – at t1 the current coupon c is paid (value cv(t,t1)) – ti, i = 1,2,…,m – 1 are the reset dates of the floating coupons are paid at time ti+1 (value v(t,t1) – v(t,tm)) – principal is repaid in one sum tm. • Value of coupons: cv(t,t1) + v(t,t1) – v(t,tm) • Value of principal: v(t,tm) • Value of the bond Value of bond = Value of Coupons + Value of Principal = [cv(t,t1) + v(t,t1) – v(t,tm)] + v(t,tm) =(1 + c) v(t,t1) • 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 , 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) 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,t1,t2,…tj, …tm – From a reset date tj coupons are determined on the formula rMax – i(ti,ti+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: – 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. 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: m c ti ti 1 vt , ti i 1 • Value of floating leg: m 1 vt , t m vt , ti ti ti 1 f t , ti 1 , ti i 1 Swap rate • In a fixed-floating swap at origin Value fixed leg = Value floating leg m swap rate ti ti 1 vt , ti 1 vt , t m i 1 swap rate 1 vt , t m m t i 1 i ti 1 vt , ti 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 m m i 1 i 1 swap rate ti ti 1 vt , ti vt , ti ti ti 1 f t , ti 1 , ti m swap rate vt , t t i i 1 m t i 1 i i ti 1 f t , ti 1 , ti ti 1 vt , ti 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) m swap rate ti ti 1 vt , ti 1 vt , t m i 1 m swap rate ti ti 1 vt , ti vt , t m 1 i 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. k 1 vt , t k 1 swap rate t,tk vt , ti i 1 1 swap rate t,tk Forward swap rate • In a forward start swap the exchange of flows determined at t begins at tj. Value fixed leg = Value floating leg forward swap rate ti ti 1 vt , ti vt , t j vt , t m m i j forward swap rate vt , t j vt , t m m t i j1 i ti 1 vt , ti 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) 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: m max 1 DP t , T ; c ,0 c ti ti 1 vt , ti i 1 • Value of the floating leg: m max DP t , T ; c 1,0 1 vt , t m spread vt , ti ti ti 1 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 m t i 1 i ti 1 vt , ti 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.