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6 Summary: the ideal features of an ITR system

6 Summary: the ideal features of an ITR system

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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



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