1. Advanced Risk Management I
Lecture 3
Market risk transfer – Hedging

2. Funding strategies

3. 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 funding
Structured funding (with derivatives)

4. 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.

5. 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.

6. 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.

7. 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)

8. 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.

9. 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.

10. 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)

11. 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…

12. 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.

13. 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:

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

15. 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

16. 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)

17. 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.

18. Value fixed leg = Value floating leg
Forward swap rate
In a forward start swap the exchange of flows determined at t begins at tj.
Value fixed leg = Value floating leg

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

20. Caps/floors and swaptions

21. 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
min(i(ti,ti+1), rcap) = i(ti,ti+1) – max(i(ti,ti+1) – rcap, 0)
max(i(ti,ti+1), rfloor) = i(ti,ti+1) + max(rfloor – i(ti,ti+1), 0)

22. 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.
This is for beginners. Graduate students know that forward rates are unbiased predictor of the future interest rates under the forward martingale measure (FMM)

23. 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)

25. 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.

26. Payer swaption
Assume one is paying floating and wants to have protection against an increase in interest rate, but does not want to buy a cap

27. Receiver swaption
Assume one is receiving floating and wants to have protection against a decrease in interest rate, but does not want to buy a floor

28. Swaption
A swaption gives the right, but not the obligation, to enter a swap contract at a future date tn for swap rate k.
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 and the unit discount factor

29. Swaption = A(t;n,N) Black[S(t;n,N),k,tn,(n,N),ω]
Swaption valuation
The value of a swaption is computed using
Swaption = A(t;n,N) EA{max[R(tn;n,N) - k ,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),ω]
with ω = 1,–1 for call (payer) and put (receiver), or explicitly:
Swaption = ω[(v(t,tn) – v(t,tN))N(d1) – iv(t,ti) kN(d2)]

31. CVA, DVA, …

32. Counterparty risk in swap contracts
In a swap cotnract both the legs are exposed to counterparty risk.
In the event of default of one of the two parties the other takes a loss equal to the marked to market value of the contract, equal to the net value of the cash-flows.
Remember that the net value of the swap contract is positive for the long end of the contract if the swap rate on the day of default of the contract is greater than the rate on the origin of the contract.

33. CVA and DVA
In recent years new risk factors have been added to the replicating portfolio, taking into account several valuation adjustment.
CVA and DVA refer to adjustment for credit risk
The price of a swap is then
Swap – CVA + DVA

34. Swap counterparty risk exposure
Assume the set of dates at which swap payments are made be {t1, t2,…, tN} and default of the counterparty that receives fixed payments (B) took place between time tn-1 and tn. In this case, the loss for the party paying fixed is given by
where R(tn, tN) is the swap rate at time tn and k is the swap rate at the origin of the contract. Notice that this is the payoff of a payer swaption (a call option on a swap).

35. Swap counterparty risk exposure
Assume the set of dates at which swap payments are made be {t1, t2,…, tn} and default of the counterparty that pays fixed payments (A) took place between time tj-1 and tj. In this case, the loss for the party receiving fixed is given by
where R(tn, tN) is the swap rate at time tn and k is the swap rate at the origin of the contract. Notice that this is the payoff of a receiver swaption (a put option on a swap).

36. Credit risk: long party

37. Swap – CVA + DVA