6 Summary: the ideal features of an ITR system
Tải bản đầy đủ - 811trang
94
Risk Management and Shareholders’ Value in Banking
(10) Lastly, it’s best to assign speciﬁc ITRs to each of the existing individual cash ﬂows
originating from an asset or liability. That is, every individual deposit/loan transaction must be ‘broken down’ into a number of zero-coupon transactions corresponding
to the cash ﬂows deriving from it (using a logic similar, to some extent, to coupon
stripping); a speciﬁc transfer rate should then be applied to each cash ﬂow.
SELECTED QUESTIONS AND EXERCISES
1. Branch A of a bank only
100 million euros; branch
a maturity of three years.
currently are 5 % and 4 %.
has ﬁxed-rate deposits, with a maturity of one year, for
B has only ﬁxed-rate loans, for the same amount, with
Market rates, on the 1 and 3 year maturity respectively,
Consider the following statements:
I. One cannot use market rates as ITRs, as they would lead to a negative margin for
the Treasury Unit;
II. 3- and 4-years ITRs must be set exactly at 5 % and 4 %;
III. The Treasury Unit can both cover interest rate risk and meanwhile have a positive
net income;
IV. The Treasury Unit can cover interest rate risk, but doing so it brings its net income
down to zero.
Which one(s) is (are) correct?
(A)
(B)
(C)
(D)
All four.
II and IV.
I and III
II and III.
2. Consider the following statements: “internal transfer rate systems based on ﬂat rates
(uniform rates for all maturities). . .:
(i) . . .are wrong because internal trades take place at non-market rates”;
(ii) . . .are correct because internal trades involve no credit risk, so there is no need
for maturity-dependent risk-premiums”;
(iii) . . .are wrong because they are equivalent to a system where only net balances are
transferred between the branches and the Treasury department”;
(iv) . . .are wrong because part of the interest rate risk remains with the branches”.
Which of them would you agree with?
(A)
(B)
(C)
(D)
WWW.
Only (ii);
Both (i) and (iv);
All but (ii);
Both (iii) and (iv).
3. A bank has two branches, A and B. Branch A has 100 million euros of one-year term
deposits, at a ﬁxed rate of 1.5 %, and 40 million euros of three-year ﬁxed-rate loans at
5 %. Branch B has 100 million euros of three-year ﬁxed rate loans at 5 %, and 6-month
deposits for 80 million euros, at 1 %. Market yields for 6-month, 1-year and 3-years
funds are 2 %, 3 %, 4 % respectively. The overnight market rate is 1 %.
Internal Transfer Rates
95
Compute the (maturity adjusted) one-year repricing gap and the expected annual proﬁts
of both branches (assuming that short term items can be rolled over at the same rate),
under the following alternative assumptions:
(a) each branch funds (or invests) its net balance (between assets and liabilities with
customers) on the market for overnight funds;
(b) each branch funds (or invests) its net balance (between assets and liabilities)
through virtual one-year deals with the Treasury;
(c) the bank has a system of ITRs based on gross cash ﬂows and market rates.
Finally, suppose you are the manager of branch B and that your salary depends on the
proﬁts and losses experienced by your branch. Which solution, among (a), (b) and (c)
would you like best if you were expecting rates to stay stable? How could your choice
be criticised?
4. A branch issues a 10-year ﬂoating-rate loan at Libor + 1 %. The borrower may convert
it to a ﬁxed rate loan after ﬁve years; also, he may payback the entire debt after eight
years. If the bank is using a complete and correct system of internal transfer rates the
branch should
(A) buy from the treasury a ﬁve-year swaption and an eight-year call option on the
residual debt;
(B) buy from the Treasury a ﬁve-year swaption and sell the Treasury an eight-year
call option on the residual debt;
(C) sell the Treasury a ﬁve-year swaption and an eight-year put option on the residual
debt;
(D) buy from the Treasury a ﬁve-year swaption and an eight-year put option on the
residual debt.
5. A bank is issuing a ﬂoating-rate loan, with a collar limiting rates between 5 % and
12 %. Suppose s is spread on the loan and p is the spread on a comparable loan (same
borrower, same collateral, same maturity, etc.) with no collar. Which of the following
statements is correct?
(A)
(B)
(C)
(D)
s > p, because the borrower is actually buying an option from the bank;
s < p, because the borrower is actually selling the bank an option;
s = p, because the bank is both selling and buying an option;
the relationship between s and p depends on the level of market interest rates.
Appendix 4A
Derivative Contracts on Interest Rates
4A.1 FORWARD RATE AGREEMENTS (FRAS)
A FRA(t, T ) is a forward contract on interest rates, whereby an interest rate is ﬁxed,
regarding a future period of time (“FRA period”), limited by two future dates, t and T .
FRAs are widely used as tools for managing interest rate risk on short maturities. The
buyer of a FRA “locks” a funding rate on a future loan: e.g., by buying a FRA(1,4) he/she
sets the interest to be paid on a quarterly loan starting in one month. This kind of contract
is also known as a “one against four months” FRA.
The key elements of a FRA are the following:
–
–
–
–
–
–
the
the
the
the
the
the
capital on which interest is computed (“notional”, N );
trade date (on which the contract is agreed);
effective date (t), on which the future loan starts;
termination date (T ) of the loan;
ﬁxed rate (FRA rate, rf );
market rate (ﬂoating rate, rm ) against which the FRA will be marked to market.
The FRA involves the payment of an interest rate differential, that is, of the difference
between the FRA rate and the market rate (on the effective date, t), multiplied by the
notional of the contract. For a FRA (1,4), e.g., if on the effective date (one month after
the trade date) the three-month market rate happens to be higher (lower) than the FRA
rate, then the seller (buyer) of the FRA will have to pay the buyer (seller) the difference
between the two rates, times the notional.
More precisely, the payment (or cash ﬂow, CF ) from the seller to the buyer is given
by:
[rm (t) − rf ] · N · (T − t)
(4A.1)
CF =
1 + rm (T − t)
(where T and t are expressed in years)
If this cash ﬂow were negative (that is, if the market rate in t were below the FRA rate)
the payment would ﬂow from the buyer to the seller. In fact, the FRA (being a forward
contract, not an option) will always be binding for both parties involved.
Note that, in (4A.1), since the payment is made in advance (that is, at time t, not on
the termination date), the interest rate differential must be discounted, using the market
rate rm , over the FRA period (T − t).
Consider the following example. On June 19, a FRA(1,4) is traded with a notional of
1 million euros, a FRA rate of 3 % and 3-month Euribor as a market (benchmark) rate.
On July 19, the 3-month Euribor is 3.5 %; this means that the seller of the FRA will have
to pay the buyer the following amount:
3
(3.5 % − 3.0 %) · 1, 000, 000 ·
12 = 1, 239.16 euros
CF =
3
1 + 3.5 % ·
12
Derivative Contracts on Interest Rates
97
4A.2 INTEREST RATE SWAPS (IRS)
An interest rate swap (IRS) is a contract whereby two parties agree to trade, periodically,
two ﬂows of payments computed by applying two different rates to the same notional
capital.
In plain vanilla swaps (the most common type of contract), one party undertakes to
make payments at a ﬁxed rate, while receiving payments at a ﬂoating rate (based on some
benchmark market rate).
In basis swaps, both payments are variable but depend on two different benchmark
rates (e.g., one party pays interest at the 3-month Euribor rate, while the other one pays
interest at the rate offered by 3-month T-bills).
The key elements of an IRS contract are the following:
– the capital on which interest is computed (“notional”, N ). Note that the notional only
serves as reference for interest computation, and is never actually traded by the two
parties;
– the trade date (on which the contract is agreed);
– the effective date (t), on which the actual swap starts;
– the termination date (T ) of the swap;
– the duration (T − t), also called the tenor of the contract;
– the m dates at which payments will take place (e.g, every six months between t and T );
– the ﬁxed rate (swap rate, rs );
– the market rate (ﬂoating rate, rm ) against which the swap rate will be traded.
An IRS can be used for several reasons:
–
–
–
–
–
to transform a ﬁxed-rate liability into a ﬂoating-rate one, or vice versa (liability swap);
to transform a ﬁxed-rate asset into a ﬂoating-rate one, or vice versa (asset swap);
to hedge risks due to maturity mismatching between assets and liabilities;
to reduce funding costs;
to speculate on the future evolution of interest rates.
The i-th periodic payment on an IRS (to be computed on each of the m payment dates)
is given by:
T −t
CFi = [rs − rm (i)] · N ·
(4A.2)
m
where rm (i) is the market rate at time i and (T − t)/m is the time between two payments
(e.g., if the contract entails four payments over two years, (T − t)/m will be equal to six
months).
If this cash ﬂow is positive, then the party who has agreed to pay the ﬁxed rate, and
receive the ﬂoating one, will make a net payment to the other one. The payment will ﬂow
in the opposite direction if the amount in (4A.2) is negative.
Consider, as an example, a 3-year IRS, with a notional N of 1 million euros, swap
rate (rs ) of 6 % and variable rate given by 6-month Euribor. Suppose the evolution of
Euribor between t and T is the one shown in Table 4A.1 (column 2): the same Table
shows the cash ﬂows due between the parties of the swap (columns 3–4) and the net
98
Risk Management and Shareholders’ Value in Banking
Table 4A.1 Cash ﬂows generated by an IRS contract
Maturities
Euribor Variable-rate cash Fixed-rate cash
ﬂows (euros)
ﬂows (euros)
Net ﬂows (euros) for
the party paying
variable and receiving ﬁxed
6 months
3.50 %
17,500
20,000
−2,500
12 months
3.80 %
19,000
20,000
−1,000
18 months
4.00 %
20,000
20,000
–
24 months
4.20 %
21,000
20,000
1,000
30 months
4.40 %
22,000
20,000
2,000
36 months
4.50 %
22,500
20,000
2,500
cash ﬂows for the party paying variable and receiving ﬁxed (column 5; negative values
denote outﬂows).
4A.3 INTEREST RATE CAPS
An interest rate cap is an option (or rather, a portfolio of m options) which, against
payment of a premium, gives the buyer the right to receive from the seller, throughout
the duration of the contract, the difference between a ﬂoating and a ﬁxed rate, times a
notional, if such difference is positive. The ﬁxed rate is called cap rate, or strike rate. No
payment takes place if the ﬂoating rate is lower than the cap rate.
The buyer of the cap hedges the risk of a rise in ﬂoating rates, which would lead to an
increase in his/her funding costs, without missing the beneﬁts (in terms of lower interest
charges) of a possible decrease. In other words, the cap sets a maximum ceiling (but no
minimum ﬂoor) to the cost of his/her debt, which will be given by the cap rate, plus the
cost of the premium (expressed on an annual basis).
The key elements of a cap contract are the following:
the capital on which interest is computed (“notional”, N );
the trade date (on which the contract is agreed);
the effective date (t), on which the option starts;
the termination date (T ) of the option;
the duration (T − t) of the contract;
the m dates at which the option could be used (e.g, every six months between t
and T );
– the cap rate (rc );
– the market rate (ﬂoating rate, rm ) against which the cap rate could be traded;
– the premium (option price), to be settled upfront or (less frequently) through periodic
payments.
–
–
–
–
–
–
The cash ﬂow from the cap seller to the cap buyer, at time i, will be:
CFi = Max 0, [rm (i) − rc ] · N ·
T −t
m
(4A.3)
Derivative Contracts on Interest Rates
99
Note that, as the cap is an option, no payment takes place at time i if the difference
rm − rc is negative.
Consider the following example. On October 1, 2007, an interest rate cap is traded,
on 6-month Euribor (rm ), with rc = 4 %, N = 1 million euros, duration (T − t) of four
years, bi-annual payments and a premium of 0.25 % per annum, to be paid periodically,
every six months.
Table 4A.2 shows a possible evolution for the Euribor, as well as the payments (from
the cap seller to the buyer) that would follow from this evolution.
Table 4A.2 An example of cash ﬂows generated by an interest rate cap
Date (i)
Euribor rm (i) Cap Rate (rc ) rm (i)−rc CFi (¤)
Periodic
Net ﬂows (¤)
premium (¤)
1/04/2008
3.00 %
4.00 %
−1.00 %
0
1,250
−1,250
1/10/2008
3.20 %
4.00 %
−0.80 %
0
1,250
−1,250
1/04/2009
3.50 %
4.00 %
−0.50 %
0
1,250
−1,250
1/10/2009
4.00 %
4.00 %
0.00 %
0
1,250
−1,250
1/04/2010
4.25 %
4.00 %
0.25 %
1,250
1,250
0
1/10/2010
4.50 %
4.00 %
0.50 %
2,500
1,250
1,250
1/04/2011
4.75 %
4.00 %
0.75 %
3,750
1,250
2,500
1/10/2011
5.00 %
4.00 %
1.00 %
5,000
1,250
3,750
4A.4 INTEREST RATE FLOORS
An interest rate ﬂoor is an option (or rather, a portfolio of options) which, against payment
of a premium, gives the buyer the right to receive from the seller, throughout the duration
of the contract, the difference between a ﬁxed and a variable rate, times a notional, if
such difference is positive. The ﬁxed rate is called ﬂoor-rate, or strike-rate. No payment
takes place if the ﬂoating rate is higher than the ﬂoor rate.
The buyer of the ﬂoor hedges the risk of a fall in ﬂoating rates, which would lead to
a decrease in the interest income on his/her investments, without missing the beneﬁts (in
terms of higher interest income) of a possible increase in rates. In other words, the ﬂoor
sets a minimum limit (but no maximum ceiling) to the return on his/her investments,
which will be given by the ﬂoor rate, minus the cost of the premium (expressed on an
annual basis).
The key elements of a ﬂoor contract are the same as for a cap. The only difference is
that the ﬁxed rate speciﬁed by the contract is now called a ﬂoor rate (rf ).
The cash ﬂow from the ﬂoor seller to the ﬂoor buyer, at time i, will be:
CFi = Max 0, [rf − rm (i)] · N ·
T −t
m
(4A.4)
Note that, as the ﬂoor is an option, no payment is made if the difference rf − rm is
negative at time i.
100
Risk Management and Shareholders’ Value in Banking
Consider the following example. On October 1, 2007, an interest rate ﬂoor is traded,
on 6-month Euribor (rm ), with rf = 2 %, N = 1 million euros, duration (T − t) of four
years, bi-annual payments and a premium of 0.25 % per annum, to be paid periodically,
every six months.
Table 4A.3 shows a possible evolution for the Euribor, as well as the payments (from
the ﬂoor seller to the buyer) that would follow from this evolution.
Table 4A.3 An example of cash ﬂows generated by an interest rate cap
Date (i)
Floor Rate (rf ) Euribor Rm (i) rf − rm (i) CF i (¤)
Periodic
Net ﬂows (¤)
premium (¤)
1/04/2008
2.00 %
3.00 %
−1.00 %
0
1,250
−1,250
1/10/2008
2.00 %
2.75 %
−0.75 %
0
1,250
−1,250
1/04/2009
2.00 %
2.50 %
−0.50 %
0
1,250
−1,250
1/10/2009
2.00 %
2.20 %
−0.20 %
0
1,250
−1,250
1/04/2010
2.00 %
2.00 %
0.00 %
0
1,250
−1,250
1/10/2010
2.00 %
1.70 %
0.30 %
1,500
1,250
250
1/04/2011
2.00 %
1.50 %
0.50 %
2,500
1,250
1,250
1/10/2011
2.00 %
1.25 %
0.75 %
3,750
1,250
2,500
4A.5 INTEREST RATE COLLARS
An interest rate collar is simply a combination of a cap and a ﬂoor having different strike
rates (namely, having rf < rc ). Namely, buying a collar is equivalent to buying a cap and
selling a ﬂoor; conversely, selling a collar involves selling a cap and buying a ﬂoor.
The collar makes it possible to constrain the variable rate within a predetermined range
(“putting a collar” to it), comprised between rf and rc .
The key elements of a collar are fundamentally the same as for a cap or a ﬂoor. The
only difference is that the contract now speciﬁes both the cap (rc ) and the ﬂoor rate (rf ).
The possible cash ﬂow from the collar seller to the collar buyer at time i, if any, will
be:
T −t
CFi = Max 0, [rm (i) − rc ] · N ·
m
while the possible cash ﬂow from the buyer to the seller is:
CFi = Max 0, [rf − rm (i)] · N ·
T −t
m
Note that those two equations are the same as (4A.3) and (4A.4) seen above. This is
consistent with the fact that a collar, as mentioned before, is simply a combination of a
long cap and a short ﬂoor.
The cash ﬂow equations indicate that the collar buyer cashes the difference between
the market rate and the cap rate when the former exceeds the latter; on the other hand,
Derivative Contracts on Interest Rates
101
he/she has to pay the difference between the ﬂoor rate and the market rate, if the latter
drops below the former. A collar can therefore be used to hedge a ﬂoating-rate liability,
ensuring that the net rate paid by its issuer (including the cash ﬂows on the collar) will
always be between rf and rc ; this means that the risk of a rate increase above rc will be
hedged but, also, that the beneﬁts of a fall in rates below rf will be forgone.
This marks an important difference with caps, where the beneﬁts from a decrease in
rates remain entirely with the buyer; however, being a combination of a long cap and
a short ﬂoor, a collar has a signiﬁcantly lower cost than a cap. In fact, the premium on
a collar is the difference between the premium on the cap (paid) and that on the ﬂoor
(received); depending on the actual values of rc and rf , as well as on the current value of
market rates, this premium could even be negative.
Finally, consider the following example. On 1 October 2007, a bank raises 1 million
euros issuing a 4-year bond indexed to the 6-month Euribor rate. To hedge against rate
increases, while limiting the cost of the hedge, the bank buys the following collar:
–
–
–
–
–
notional (N ) of one million euros;
ﬂoor rate: 3 %: cap rate: 4 %;
duration: 4 years;
benchmark rate: 6-month Euribor;
premium: 0 % per annum.
Table 4A.5 shows the cash ﬂows associated with the collar. Note that, as market rates
rise above rc (driving up the interest expenses associated with the bond), the bank gets
compensated by a positive cash ﬂow on the collar. However, when market rates fall, part
of the savings due to lower interest charges on the bond are offset by the payments due
on the collar.
Table 4A.5 An example of cash ﬂows associated with an interest rate collar
Time (i)
rf
rc
Euribor (rm (i)) rm − rc rf − rm
CF i
Periodic Net ﬂow
premium
1/04/2008
3.00 % 4.00 %
3.50 %
−0.50 % −0.50 %
0
0
0
1/10/2008
3.00 % 4.00 %
3.00 %
−1.00 %
0.00 %
0
0
0
1/04/2009
3.00 % 4.00 %
2.50 %
−1.50 %
0.50 %
−2,500
0
−2,500
1/10/2009
3.00 % 4.00 %
2.75 %
−1.25 %
0.25 %
−1,250
0
−1,250
1/04/2010
3.00 % 4.00 %
3.25 %
−0.75 % −0.25 %
0
0
0
1/10/2010
3.00 % 4.00 %
4.00 %
0.00 %
−1.00 %
0
0
0
1/04/2011
3.00 % 4.00 %
4.25 %
0.25 %
−1.25 %
1,250
0
1,250
1/10/2011
3.00 % 4.00 %
4.50 %
0.50 %
−1.50 %
2,500
0
2,500
Part II
Market Risks