Transcript Advanced Methods of Risk Management I

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)
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)
Floaters preserve the value of debt
• 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.
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
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
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:
m
c ti  ti 1 vt , ti 
i 1
m
1  vt , tm    vt , 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 vt , ti   1  vt , t m 
i 1
swap rate
1  vt , t m 
m
 t
i 1
i
 ti 1 vt , 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 vt , ti    vt , ti ti  ti 1  f t , ti 1 , ti 
m
swap rate
 vt , t t
i
i 1
m
 t
i 1
i
i
 ti 1  f t , ti 1 , ti 
 ti 1 vt , 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 vt , ti   1  vt , t m 
i 1
m
swap rate ti  ti 1 vt , ti   vt , 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
vt , t k  
1  swap ratet,tk  vt , 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
forwardswap rate ti  ti 1 vt , ti   vt , t j   vt , tm 
m
i j
forwardswap rate
vt , t j   vt , t m 
m
 t
i  j1
i
 ti 1 vt , 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)
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)
callY , t ; K , T   Y t N d1   v t , T KN d 2 
lnF Y , t  / K   1 / 2 T  t 
d1 
 T t
2
d 2  d1   T  t
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, implying
Call + v(t,T) Strike = Put + Undelrying
Put options
• Using the put-call parity we get
Put = Call – Y(t) + v(t,T)K
and from the replicating portfolio of the call
Put = ( – 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
At; n, N     i vt , ti 1 
i n
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)