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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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