Advanced Risk Management I

Lecture 3

Market risk transfer – Hedging

Choice of funding

• Assume you want to fund an investment. Then, one first has to decide the funding. What would you recommand?

• What are the alternatives? – Fixed rate funding– Floating rate fundign– Structured funding (with derivatives)

Fixed rate funding

• Pros: future cash flows are certain

• Cons: future market value of debt certain

• Fixed rate funding risks– In case of buy-back lower interest rates would

imply higher cost– If the investment cash flows are positively

correlated with interest rates, when rates go down the value of the asset side decreases and the value of liabilities decreases.

Floating rate funding

• Pros: stable market value of debt

• Cons: future cash flows are uncertain

• Floating rate funding risk: – An increase of the interest rates can induce a

liquidity crisis– If the investment cash flows are negatively

correlated with interest rates, when rates go up the value of the asset side decreases and the value of liabilities increases.

Intermediate funding choices

• Plain fixed and floating funding presents extreme risks of opposite kind: swing of mark-to-market value vs swing of the future cash-flows.

• Are there intermediate choices? – Issuing part of debt fixed and part of it floating– Using derivatives: automatic tools to switch

from fixed to floating funding or vice versa.

Why floating coupons stabilize the value of debt?

• Intuitively, if coupons are fixed, the increase in interest rates reduces the present value of future cash flows

• Il coupons are designed to increase with interest rates, then the effect of an interest rate upward shock on the present value of future cash flows is mitigated by the increase in future coupons

• If coupons are designe to decrease with interest rates, then the effect of an interest rate upward shock on the present value of future cash flows is reinforced by the decrease in future coupons (reverse floater)

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)

t T

C

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 schedulet,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)

Floaters preserve the value of debt

• A floater is a bond characterized by a schedulet,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.

EserciseReverse 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

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 representationExpected coupon = forward rate

+ convexity adjustment + timing adjustment• It may be proved that only in the “ natural lag” case

convexity adjustment + timing adjustment = 0

Managing interest rate risk

• Interest rate derivatives: FRA and swaps.

• If one wants to change the cash flow structure, one alternative is to sell the asset (or buy-back debt) and buy (issue) the desired one.

• Another alternative is to enter a derivative contract in which the unwanted payoff is exchanged for the desired one.

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)

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:

• Value of floating leg:

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Swap rate

• In a fixed-floating swap at origin

Value fixed leg = Value floating leg

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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

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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)

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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.

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Forward swap rate

• In a forward start swap the exchange of flows determined at t begins at tj.

Value fixed leg = Value floating leg

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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)

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

Derivatives and leverage

• Derivative contracts imply leverage• Alternative 1

Forward 10 000 ENEL at 7,2938 €, 2 months

2 m. later: Value 10000 ENEL – 72938• Alternative 2

Long 10 000 ENEL spot with debt 72938 for repayment in 2 months.

2 m. later: Value 10000 ENEL – 72938

Hedging and speculation

• Assume you have 10000 ENEL stocks in your portfolio, and say that:

• You go short a forward contract on 10 000 ENEL for 72 938. Then, using the replicating portfolio, we have that the 10 000 ENEL are virtually removed from the portfolio, and the portfolio is worth the present discounted value of 72 938.

• You go long a forward contract on 10 000 ENEL for 72 938. Then, using the replicating portfolio you are exposed to 20 000 ENEL with a debt of 72 398 euros. This is the “leverage” effect.

Non linear contracts: options

• Call (put) European: gives at time t the right, but not the obligation, to buy (sell) at time T (exercise time) a unit of S at price K (strike or exercise price).

• Payoff of a call at T: max(S(T) - K, 0)

• Payoff of a put at T: max(K - S(T), 0)

Black & Scholes model

• Black & Scholes model is based on the assumption of normal distribution of returns. The model is in continuous time. Recalling the forward price F(Y,t) = Y(t)/v(t,T)

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Put-Call Parity

• Portfolio A: call option + v(t,T)Strike• Portfolio B: put option + underlying• Call exercize date: T• Strike call = Strike put• At time T:

Value A = Value B = max(underlying,strike)…and no arbitrage implies that portfolios A and B

must be the same at all t < T, implyingCall + v(t,T) Strike = Put + Undelrying

Put options

• Using the put-call parity we getPut = Call – Y(t) + v(t,T)K

and from the replicating portfolio of the callPut = ( – 1)Y(t) + v(t,T)(K + W)

• The result is that the delta of a put option varies between zero and – 1 and the position in the risk free asset varies between zero and K.

Structuring principles

• Questions:

• Which contracts are embedded in the financial or insurance products?

• If the contract is an option, who has the option?

Who has the option?

• Assume the option is with the investor, or the party that receives payment.

• Then, the payoff is:

Max(Y(T), K)

that can be decomposed as

Y(T) + Max(K – Y(T), 0) or

K + Max(Y(T) – K, 0)

Who has the option?

• Assume the option is with the issuer, or the party that makes the payment.

• Then, the payoff is:

Min(Y(T), K)

that can be decomposed as

K – Max(K – Y(T), 0) or

Y(T) – Max(Y(T) – K, 0)

Convertible

• Assume the investor can choose to receive the principal in terms of cash or n stocks of asset S

• max(100, nS(T)) =

100 + n max(S(T) – 100/n, 0)

• The contract includes n call options on the underlying asset with strike 100/n.

Reverse convertible

• Assume the issuer can choose to receive the principal in terms of cash or n stocks of asset S

• min(100, nS(T)) = 100 – n max(100/n – S(T),

0)• The contract includes a short position of n

put options on the underlying asset with strike 100/n.

Interest rate derivatives

• Interest rate options are used to set a limit above (cap) or below (floor) to the value of a floating coupons.

• A cap/floor is a portfolio of call/put options on interest rates, defined on the floating coupon schedule

• Each option is called caplet/floorlet Libor – max(Libor – Strike, 0) Libor + max(Strike – Libor, 0)

Call – Put = v(t,)(F – Strike)

• Reminding the put-call parity applied to cap/floor we have

Caplet(strike) – Floorlet(strike)

=v(t,)[expected coupon – strike]

=v(t,)[f(t,,T) – strike] • This suggests that the underlying of caplet and

floorlet are forward rates, instead of spot rates.

Cap/Floor: Black formula

• Using Black formula, we have

Caplet = (v(t,tj) – v(t,tj+1))N(d1) – v(t,tj+1) KN(d2) Floorlet =

(v(t,tj+1) – v(t,tj))N(– d1) + v(t,tj+1) KN(– d2) • The formula immediately suggests a replicating

strategy or a hedging strategy, based on long (short) positions on maturity tj and short (long) on maturity tj+i for caplets (floorlets)

Swaption

• Swaptions are options to enter a payer or receiver swap, for a swap rate at a given strike, at a future date.

• A payer-swaption provides the right, but not the obligation, to enter a payer swap, and corresponds to call option, while the receiver swaption gives the right, but not the obligation, to enter a receiver swap, and corresponds to a put option.

Swaption• A swaption gives the right, but not the

obligation, to enter a swap contract at a future date tn for swap rate Rs.

• Reset dates {tn , tn+1,……tN} for the swap, with payments due at dates {tn +1 , tn +2,……tN + 1}

• Define i = ti +1– ti the daycount factors

N

niii ttvNntA 1,,;

The pay-off of a swaption…

• A swaption with strike Rs has payoff

i max[R(tn;n,N) - Rs ,0]

where R(tn;n,N) is the swap rate that will be observed at time tn with present value,

A(tn;n,N) max[R(tn;n,N) - Rs ,0]

…and valuation

• The value of a swaption is computed using

Swaption = A(t;n,N) EA{max[R(tn;n,N) - Rs ,0]}

• Assuming the swap rate to be log-normally distributed (Swap Market Model), we have Black formula

Swaption = A(t;n,N) Black[S(t;n,N),K,tn,(n,N)]

or explicitly:

Swaption = (v(t,tj) – v(t,tN))N(d1) – iv(t,ti) KN(d2)