1. Financial Products & Markets Lecture 1

2. Markets and financial products Financial products allow to tranfer consumption from the current period to the future In this course we show the tools that can be used (and how they can be used) to achieve that goal One can decide to carry consumption to the future by simply investing in the risk-free rate and then collecting the amount of consumption at expiration. One can decide to carry consumption to the future by taking some risk in exchange for a higher consumption. One can decide to consume in the future, contingent on the state of nature. Consume more if some particular event takes place.

3. Financial products Fixed income, i.e. bonds. The pay-off is not contingent on the project that is funded by the loan, apart for the case in which the project goes bust (credit risk). Variable income, i.e. equity. The pay-off is defined from the cash flow of the project, as the residual after all other claims have been satisfied (residual claim) Derivatives or contingent claims. Products whose pay-off depends on the state of nature (but also prioduts that include debt, that is leverage). Structure finance products. Include both standard equity or bond and a derivative contract.

4. Risk transfer tools Derivatives are used to transfer risk from one institution to the other, without any unwinding of the portfolio. Risk transfer can be designed both for single positions and for portfolios. Risk transfer can be achieved either using regulated markets or by re-insurance, that is on the so-called Over-the-Counter (OTC) market (Metalgesellschaft case).

5. Financial actors and strategies Non financial firms may raise funding by –Issuing equity or relying on debt –Debt may be through loans or bonds –Equity and bonds may be listed or not in a market –Bonds may be designed for the retail market or institutional investors –New sources: P2P loans, micro-credit, crowdfunding Investors may provide funds by: –Individual investment on the market –Collective investment in funds and other intermediaries

6. Financial intermediaries Banking system: regulated, issue deposits and bonds and provide loans, or invest in the market. They also provide services to manage risk by designing derivative deals for clients (dicussion about the dual capacity, Volcker- Vickers-Liitkanen rule) Insurance market: invest in bonds and stock to provide long term consumption (life insurance) and provide re- insurance on risk to the financial system (cases: CDS in AIG and the surety bonds in the Enron case). Shadow banking system: hedge funds, SIV, SPV and in general all un-regulated intermediaries. Shadow banking system is good or bad? Is it here to stay, or it is a residual from the crisis

7. Regulators ECB (and NCBs) supervise the major banks (about 130) of the Euro Area, with the task of ensuring financial stability EBA (for banks) and EIOPA (for insurance) are the European agencies for enforcement of the regulation rules Transparency regulation (ESMA and local regulators, such as CONSOB, Basfin, and so on) Accounting regulators (IASB): define the rules for the transparent evaluation of assets and liabilities.

8. Markets Organized markets: open-outcry vs screen based, quote- driven vs order-driven, anonymous vs non-anonymous. OTC markets: collateral-based or with counter party risk Microstructure theory studies organization of exchanges and the effectiveness in processing the information flows (price discovery) Exchange venues and markets are places where agents exchange information. Information is then backed out from liquid and transparent markets in order to evaluate products not traded in the market.

9. Brand new problems Negative interest rates: carrying consumption to the future now is costly, and this increases the value of 1 euro tomorrow above the value of it today Insurance companies getting negative returns must take more risk to be able to face future payments (case of German insurance companies in the stress test) Negative interest rates, due to loss aversion (prospect theory), push investors towards taking more risk. QE blues: under the Quantitative Easing structure negative yields are excluded from the assets that can be bought from ECB, so leaving only long term maturities as possible “ammunition” for the QE weapon.

10. A general template for financial products Time schedule: {t 0, t 1, …,t n } –Market holidays and day-count and calendar conventions (following, preceding, mod following or preceding) –Day-count conventions Coupon/dividend plans: {c 0, c 1, …,c n } –Deterministic –Indexed (interest rate, inflation, equity, credit, commodities, longevity) Repayment plan {k 0, k 1, …,k n } –Deterministic –Stochastic (callable, putable, exchangeable, convertible)

11. The arbitrage principle No arbitrage, or no free lunch means Build a position whose value is zero at time t and in the future took non-negative value for sure, with possibly a positive value in some of them Build a position whose value be negative at time t and in future get non-negative value for sure.

12. Arbitrage and replicating portfolio The replicating portfolio or a replicating strategy of a financial product is a set of positions whose value at some future time is equal to that of the financial product in all possible states of nature If it is possible to build a replicating portfolio of a financial product for a price different from that of the financial product, one can exploit infinite profits selling the portfolio (if it is cheaper than the financial product to be replicated) and buying the financial product, followed by repurchase of the portfolio and sale of the bond when they have the same value.

13. Replicating portfolio: pricing and hedging Assuming that no arbitrage profits are possible means requiring that the value of each financial product be equal to that of its replicating portfolio (pricing) Buying a financial product and selling the corresponding replicating portfolio means building an immunized position or hedging the position.

14. Structured finance building block approach Structured finance products are built by aggregating financial products and derivatives. Duty of the structurer is to put these products together so that they could be useful to a set of customers Duty of the pricer is to disentangle the financial product in the elementary components in order to evaluate them on prices consistent with the market. Duty of the risk manager is to evaluate the risk of the different components and to decide what and how to hedge.

15. 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 )

16. 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.

17. 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.

18. 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.

19. 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

20. 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

21. 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

22. 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.

23. 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.

24. 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

25. 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?

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

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

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

29. 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

30. 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

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

32. 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

33. 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)

34. 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 )

35. 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.

36. 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)

37. 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

38. 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

39. 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 …

40. 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.

41. 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:

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

43. 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

44. 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)

45. 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.

46. 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

47. 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)

48. 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.

49. The relevance of tenor: basis swaps From what we have seen, the value of a stream of indexed payments is the same both if the reference rate is on 3m, 6m, or other. After the crisis this is no more true, and payments based on different tenors have different values. Basis swaps is the exchange of indexed payments with respect to interest rates of different tenors (i.e. 3m vs 6m). Basis swaps emerged during the crisis mainly because of the appearance of credit risk in the banking market. In fact, the interest rates on monthly tenors were much higher than the swap rate on the daily tenor (EONIA).

50. Forward contracts The long party in a forward contract defines at time t the price F at which a unit of the security S will be purchased for delivery at time T At time T the value of the contract for the long party will be S(T) - F

51. Contratti forward: ingredienti Date of the deal 16/03/2005 Spot price ENEL 7,269 Discount factor 16/05/2005: 99,66 Enel forward price: 7,269/0,9966 = 7,293799 ≈ 7,2938 Long position (purchase) in a forward for 10000 Enel forward for delivery on May 16 2005 for price 7,2938. Value of the forward contract at expiration date 16/05/2005 10000 ENEL(15/09/2005) – 72938

52. Derivatives and leverage Derivative contracts imply leverage Alternative 1 Forward 10000 ENEL at 7,2938 €, 2 months 2 m. later: Value 10000 ENEL – 72938 Alternative 2 Long 10000 ENEL spot with debt 72938 for repayment in 2 months. 2 m. later: Value 10000 ENEL – 72938

53. Syntetic forward A long/short position in a linear contract (forward) is equivalent to a position of the same sign and same amount and a debt/credit position for an amount equal to the forward price In our case we have that, at the origin of the deal, 16/03/2005, the value of the forward contract CF(t) is CF(t) = 10000 x 7,269 – 0,9966 x 72938 ≈ 0 Notice tha at the origin of the contract the forward contract is worth zero, and the price is set at the forward price.

54. Futures Asssume that a forward contract is closed and settled every day (mark-to-market) You have obtained a futures market –Margin (buyers and sellers post a deposit to guarantee their performance on the contract –Prices are marked-to-market every day and profits and losses are settled on the margin (margin call) –Products traded are standardized, and in some cases are adjusted for “grade”. The seller has a “delivery option” (if the contract is for “physical delivery”)