lecture 8 notes

MSc Investment Management

Portfolio Management

Lecture 8

Yield Spread Trades - Weighting,

Financing and Applications

1. Introduction

1.1. Change in yield level vs. change in yield spread

It is commonly known that if investors are expecting interest rates to fall they should

own bonds with longer maturities, lower yields and lower coupon rates and vice

versa. Market participants, however, form expectations of the interest rate behaviour,

i.e. yield curve slope and they are not only interested in predicting whether interest

rates will rise or fall. For example, a normal (upward sloping) yield curve may be

expected to flatten and the spread between yields on short term bonds (say 2-years)

and long term bonds (say 30-years) will narrow. Alternatively, high coupon bonds

may outperform low coupon bonds of a comparable maturity if there is a change in

tax law which favours higher income for example. These kinds of scenarios differ

from simple ‘rise or fall in interest rates’ cases because they involve changes in yield

spreads and changes in the shape of the yield curve rather than just the yield level.

Yield spread changes and changes in the general level of interest rates are not linked.

This means that the scenario of flattening of the yield curve can occur in the rising

and falling interest rate environment. Consequently, taking advantage (or avoiding

losses) by using expected changes in yield spread cannot be done by buying or selling

a single fixed income security. Specifically, investor will have to follow a long-short

trading strategy, buying a security that he/she believes will appreciate more (or

depreciate less) and selling a security that he/she believes will appreciate less

(depreciate more). After creating this position, any gains or losses on this long-short

portfolio are a result of a change in the yield spread ONLY and are not related to

general levels of interest rates (rises or falls). Such a long-short strategy is commonly

known as ‘bond spread’ or ‘yield spread’ trade.

2. Weighting (constructing) a bond spread

To weight or construct a bond spread simply means to determine the quantity of

bonds to be bought relative to the quantity of bonds to be sold (or short-sold). The net

value of such a long-short portfolio should not change with variation in the level of

interest rates but it should change to investor’s advantage with the changes in the

difference in the yields on the bonds bought and sold.

Consider the data1 in Table 1. The table presents the data on a number of Treasury

bonds with different maturities and characteristics. In reality, investors will consider

on-the-run issues for yield spread trade strategy. The Table 1 also shows how much

will the $1million principal amount of each bond vary with a change in yield of one

basis point (i.e. the price value of a basis point). Further, let’s assume that investor is

expecting flattening of the yield curve (not knowing really where general level of

interest rates is moving, so it could be rising or falling). Such flattening will lead to a

decline in 168 basis points yield spread between 2-year and 30-year bond for instance

(10.62% - 8.94% = 1.68%). Based on this belief, he wants to buy a long term bond,

say a 30 year bond and sell the short term bond, say a 2-year bond (if the yield curve

is flattening, short term yield is expected to increase more or decrease less relative to

long term ones). At the same time, investor wants no exposure to general changes in

interest rates.

Table 1

Maturity Coupon

(%)

Price Yield

(%)

Price value of a basis point (per $1million

principal amount) ($)

2 yr 9% 100 3/32 8.94 177.54

3 9.5 100 16/32 9.29 246.51

5 9.62 99 5/32 9.82 392.53

7 10.37 100 22/32 10.22 481.88

20 10.75 99 16/32 10.80 809.09

30 10.62 100 10.62 897.97

Assume then that he buys $1M principal amount of 30 year 10.62% bond. As seen in

Table 1, price value of the basis point suggests that an increase (decrease) in yield of

1 bp will decrease (increase) the value of that bond by $897.97. Therefore, to

construct a yield spread trade, the appropriate amount of the short-term, 2-year issue

needs to be shorted so that increase (decrease) in its yield of 1bp will increase

(decrease) the value of the 2-year short position by exactly $897.97. The gains or

losses on the long position will then be matched by losses or gains on a short position

1 Example adapted from Garbade (1996)

whenever the yields on the two securities change by the same amount (when yields

change without altering the 2-year/30-year yield spread). Since the price value of the

basis point of a 2-year issue is $177.54, the investor needs to sell:

$897.97/$177.54 = $5.06M principal amount of a 2-year note against $1M principal

amount of purchase of 30-year bond.

Weighting rule for a bond spread: Price value of the basis points of a bond we have a

long position in is divided by a price value of the basis point of a bond we want to

take a short position in. This ratio gives us the amount to be short-sold.

The principal amounts of the 2-year and 30-year bonds must therefore be in the ratio

of 5.06 to 1. This weighting is NOT fixed for the life of any two bonds. The weighting

needs to be adjusted from time to time to maintain zero risk position in relation to the

change in the level of interest rates. Note that weighting is given in relative NOT

absolute values, so investor who is long $2M 30-year bonds will short $10.12M of the

2-year bond.

Table 2 shows the initial net portfolio value of the positions for different changes in

interest rates. The values shown in the table exclude accrued interest.

Table 22

Initial Value

Short: $5.06M 9% 2-year bond, yield 8.945%, quoted at 100.0937

Long: $1M 10.625% 30-year bond, yield 10.622%, quoted at 100

Net Value

-5 064 744

1 000 000

-4 064 744

I. Value if both yields rise by 10bps

Short: $5.06M 9% 2-year bond, yield 9.045%, quoted at 99.9156.

Long: $1M 10.625% 30-year bond, yield 10.722%, quoted at 99.1064

Net Value

Change

-5 055 729

991 064

-4 064 665

+79

II. Value if both yields fall by 10bps


Long: $1M 10.625% 30-year bond, yield 10.522%, quoted at 100.9046.

Net Value

Change

-5 073 704

1 009 046

-4 064 658

+86

III. Value if 30yr bond yield falls by 10bps



Net Value

Change

-5 064 744

1 009 046

-4 055 698

+9 046

IV. Value if 2yr bond yield rises by 10bps


Long: $1M 10.625% 30-year bond, yield 10.622%, quoted at 100.

Net Value

Change

-5 055 729

1 000 000

-4 055 729

+9 015

V. Value if 30yr bond yield rises by 10bps



Net Value

Change

-5 064 744

991 064

-4 073 680

-8 936

VI. Value if 2yr bond yield falls by 10bps


Long: $1M 10.625% 30-year bond, yield 10.622%, quoted at 100.

Net Value

Change

-5 073 704

1 000 000

-4 073 704

-8 960

2 Table adapted from Garbade (1996)

The net values in Table 2 are all negative because the value of a short position in a 2

year bond exceeds the value of the long 30-year bond position. Thus, establishing the

bond spread position generates cash rather than uses cash. Conversely, liquidating

position uses cash because the funds needed to buy back 2 year bond exceed the

amount of cash produced from the sale of 30 year bond. In this example financing

considerations (i.e. how do we finance the long and short position) are ignored but we

will consider that issue in later sections.

Analysis of scenarios in Table 2: if the yields on two securities rise or fall by 10bps

(case I and II) an investor (almost) breaks even; s/he earns around $9000 if the yield

curve flattens (case III and IV) and looses around $9000 if yield curve steepens by

10bps (case V and VI). So, we can conclude that selling a $5.06M of a short term, 2-

year bond and buying $1M of a long term, 30-year bond is the profitable spread

position if the flattening of the yield curve is expected. If steepening was expected, it

would be profitable to buy a short term bond and short-sell a longer term one.

Final note on weighting: You can weight a bond spread using duration as follows: the

product of bond duration and bond value should be the same for both sides of the

spread position, such as:

)(

)(

:

)()(

111

222

2

1

22221111

APD

APD

Q

Q

then

DAPQDAPQ

Note: Q is quantity of a bond, P is price of a bond, D is duration and A is accrued

interest

Weighting with duration is more complicated to compute but it will give identical

results to weighting with price value of a basis point if the yields on the two bonds are

the same and will give quite similar results if the yields are similar.

3. Financing considerations

The above analysis indicates that if an investor has properly weighted a bond spread,

s/he only needs to wait until the yield spread moves in the desired direction (say, if

s/he expects the yield curve to flatten – it does so) then simply unwind the trades and

make profits. This is the case only if the spread changes quickly enough and the

position is liquidated equally quickly. If a position is held for longer periods of time

(say, days or weeks) the financing of the position needs to be taken into account. In

other words, there could be income and expenses from financing which can enhance

or reduce the profits. Therefore, financing bond spreads is a very important issue in

this strategy.

3.1. A simple analysis of financing a bond spread

Our starting point of the analysis is the earlier example where investor holds a $1M

long position in 30-year, 10.62% coupon bond and $5.06M short position in 2-year,

9% coupon bond, expecting the yield curve to flatten (i.e. expecting to make profits

from such trade as per Table 2, scenario III and IV).

Assume that a 30-year bond can be financed in the market by REPO (repurchase

agreement) at 6.5% interest and that 2-year bond can be borrowed in the market for

reverse repurchase agreement by accepting a rate of 6.5% on funds lent against the

notes borrowed. Therefore, for simplicity, we assume that the long and the short

position are both being financed at 6.5%.

Then, the question is: How much does the value of our initial spread position change

over an interval of one day, net of financing income and costs, assuming that bond

yields don’t change? If we assume that yields do not change, since in our example the

prices of both bonds are close to par value, this would be equivalent to assuming that

quoted bond prices do not change. Therefore, there is four parts of answer to the

above financing question:

1. The increase in the value of the bonds held long due to accumulation of

accrued interest:

dayperM 10.291$1$%625.10365

1

2. The decrease in the value of the bonds held short due to accumulation of

accrued interest:

dayperM 67.1247$)06.5$%9365

1(

3. The cost of financing the 30-year bonds held long (i.e. the cost of borrowing

the money to buy them) at REPO rate of 6.5%:

dayperM 55.180$)1$%5.6360

1(

4. The income from financing the 2-year bonds sold short position at a rate of

6.5%:

dayperM 61.913$06.5$%5.6360

1

Note: REPO market uses 360 days count, whereas accrued interest in calculated based

on 365 days in a year.

The Net Financing Cost from 1- 4 above is -$223.51 per day.

From Table 2, we saw that a 10bps change in 2-year/30-year yield spread was

providing profit of around $9000, so 1bps change in spread will earn investor a profit

worth around $900. If we include a financing cost here, this means that the spread

position will show a net loss unless the spread narrows by 1bpt in 4 days (4days*(-

223.51) = -$894.04, which is still less than $900 profit).

3.2. Problems with the simple analysis from 3.1.

The example assumed that the cost of financing the long position in the Repo

market is the same as the rate on money lent in the reverse Repo market. In

reality the former will be higher than the latter, which increases the net

financing cost.

We have assumed that if yields don’t change - then quoted prices don’t

change as the bonds were priced close to a par. However, if a bond is priced at

a substantial premium or discount, then assumption of unchanged yields

doesn’t mean that quoted prices will remain unchanged. Premium or discount

price will approach par value over time even if yields do not change (see the

Bond Price Volatility file on CitySpace, Theorem 1).

This simple analysis assumes that only the principal amount of the bond

should be financed, it ignores the cost of financing accrued interest or

financing a premium bond in the REPO market.

3.3. A more advanced analysis of financing of a bond spread: including accrued

interest and premium bond financing

Consider a spread position where investor buys 12% coupon bond expiring on

15/4/2011 and short-sells a weighted amount of 7% coupon bond expiring also on

15/4/2011. These transactions were completed on 8th

February and the settlement is

on 9th

February. Table 3 shows the prices, yields, values of a basis point and financing

rates for these bonds.

Table 3

Coupon 7% bond 12% bond

Quoted price (Feb 9) 98 106

Yield (Feb 9) 8.6% 9.03%

Value of a basis point $180 $190

Financing rate 6% (reverse REPO) 6.5% (REPO)

Implied quoted price (Feb 10) 98.0044 (at 8.6% yield) 105.9923(at 9.03% yield)

Accrued interest (Feb 9) 3.0458 5.0431

Accrued interest (Feb 10) 3.0660 5.0775

From Table 3, the relative weighting on the bond spread trade is 1.056 to 1

(=190/180=1.056). So for example, if an investor buys $10M principal amount of the

12% bond, he will have to short-sell $10.56M principal amount of 7% bond. In order

to compute the daily net financing position, we have to compute the following:

a) The change in the values of the bonds over one day interval if the yields do not

change, which is composed of 1) accumulation of accrued interest and 2)

change in bond’s price as it approaches maturity (since one bond is a premium

and the other is a discount bond, both prices will change towards a par value

even though yields have not changed). Therefore, we have:

Accumulation of Accrued interest for 7% bond: 0.0202% of principal amount

(= 3.0660-3.0458)

Accumulation of Accrued interest for 12% bond: 0.0344% of principal amount

(=5.0775-5.0431)

Change in price for 7% bond: +0.0044% of principal amount

(=98.0044-98)

Change in price for 12% bond: -0.0077% of principal amount

(=105.9923-106)

The daily net change for 7% bond: -$2597.76 (=-$10.56M*(0.0202%+0.0044%))

The daily net change for 12 % bond: +$2670 (= $10M*(0.0344%-0.0077%))

And

b) The cost and income of financing long and short positions over a one day

interval:

The cost of borrowing to finance $10M long position per day is:

- dayperM 94.2004$)0431.5106(10$%5.6360

1

The income from financing $10.52M short position per day is:

dayperM 40.1778$)0458.398(56.10$%6360

1

Using all accumulations of accrued interest from a) and financing expense and income

from b), we obtain: -$2597.76 +$2670-$2004.94+$1778.40 = -$154.3. Overall, the

Net Financing Cost for the bond spread is actually a loss of -$154.3 per day.

If we have used simpler analysis in this example (assuming different Repo and

reverse repo rates but ignoring financing of accrued interest and premium bond), we

would have had:

1. The increase in the value of the bonds held long due to accumulation of

accrued interest:

dayperM 67.3287$10$%12365

1

2. The decrease in the value of the bonds held short due to accumulation of

accrued interest:

dayperM 21.2025$)56.10$%7365

1(

3. The cost of financing the 12% coupon bond held long at REPO rate of 6.5%:

dayperM 56.1805$)10$%5.6360

1(

4. The income from financing the 7% coupon bond short position at a rate of 6%:

dayperM 1760$56.10$%6360

1

Simple analysis shows the Net Financing GAIN from 1-4 above of +$1216.9 per day.

We can see clearly then that the simple analysis would have overstated results as it is

ignoring the losses arising from the change in bonds price as it approaches maturity

and the greater cost of financing the premium bond.

4. Applications of Bond Spread trading

As noted before, a bond spread has two main characteristics: 1) its value should not

vary with changes in the general level of interest rates and 2) its value should vary

with change in a specified yield spread.

4.1. Yield curve trades

This is one of the most common bond spreads. Here, the investor anticipates a

flattening or steepening of the yield curve. If the flattening is anticipated, he will buy

longer maturity bonds and sell a weighted amount of shorter maturity bonds. If the

seepening is anticipated, he will buy shorter maturity bonds and sell a weighted

amount of longer maturity ones. One of the key things is to identify which sectors of

the curve to buy and sell. The yield spreads do not need to move in the parallel

manner for different segments of the yield curve (1mth T-bill/1year T-bond; 2-year

bond/5 year bond; 7-year bond/30 year bond; 1mth T-bill/30 year bond are all

different spreads which do not have to move in the parallel manner). This means that

an investor must be careful in identifying which part of the curve he expects to see

flattening or steepening. It is possible that while one segment of the curve is flattening

the other one may remain unchanged or even steepening.

4.2. Coupon Spreads

They generally involve bonds of comparable maturity or duration having very

different coupon rates. If the interest rates (yields) are expected to decrease, then we

know that the lower coupon bond offers greater volatility for investor, hence they are

expected to outperform higher coupon bonds.

5. Restrictions for Bond Spread trading

In all the previous strategies, we have assumed that investors can short-sell and that

they can finance the bond purchase in the REPO market. However, in reality, many

institutional investors can neither sell bonds nor borrow money to finance bonds.

These restrictions limit the applicability of bond spreads but they do not eliminate it.

An investor who cannot sell a bond short against the purchase of a weighted amount

of another bond, will actually sell a bond he already owns and replace it with a

weighted amount of the second bond. This is actually representing a bond swap.

If an investor cannot finance bonds in the REPO market, he will be unable to execute

a bond spread or a bond swap that uses cash. Suppose for example that an investor

owns $20 million market value of a 25-year bond and believes that the yield curve

will steepen but doesn’t have any feelings about interest rates going up or down. He

would like to sell his bond and swap into a weighted amount of, say 5-year bond.

However, unless he can borrow cash, he will be unable to switch into 5-year bonds on

a weighted basis because he will have to invest more than $20 million in the bonds.

This will suggest that institutional investors will generally be restricted from

shortening up on the yield curve on a weighted basis in anticipation of the yield curve

steepening. On the other hand, they will be able to extend on the curve on a weighted

basis in expectation of the yield curve flattening.

PART 2:

Hedging with bonds

1. Introduction

A ‘hedge’ in the context of this lecture is discussed in view of taking a strategic

position (long or short) in the fixed income security (or securities) that will reduce the

uncertainty about prospective future income from another fixed income security. If

investor anticipates rising interest rates (in which case the bond prices will drop), s/he

might hedge a long position in e.g. a 7 year bond by short-selling 20 year bond

(longer term bonds have greater volatility, so if short-sold, there will be gains on the

short position and losses on the long position), so the loses on a long position will be

offset by the gains on the short position. However, this definition doesn’t determine

the size of the hedge, i.e. how much of a 20 year bond one should short-sell, given the

principal amount of the 7-year issue. For example, a short position in $2M of 20 year

bond is not adequate to hedge $60M of 7-year bonds. On the other hand $180 M of a

20year bond would be too much of a hedge for that position. In this part of the lecture

we will discuss a conventional approach to hedging with bonds as well as more

complex hedging approach.

2. A conventional approach to hedging

Conventional approach to hedging IS actually the yield spread trade. Let us look at

this example here. Consider an investor who has a long position in $100M principal

amount of 2-year bonds and wants to hedge against higher interest rates by selling 7-

year bonds short. Consider then the following information for the two bonds:

Table 4

Maturity Coupon Quoted Price Yield Price value of

a basis point

2 years 7% 99 7.9% $180

7 years 9% 102 8.5% $400

20 years 10% 106 9.3% $1000

The data in the table indicates that the change in yield by 1bp will change the value of

$1M 2-year bond by $180 and will change the value of $1M 7-year bond by $400.

Hence, a 1bp change in yield will change the value of $100M principal amount of a 2-

year bond by $18000. Applying the weighting principal used in yield spread trades

discussed in Part 1 of the lecture, we obtain the amount of a 7-year bond that should

be sold to complete this hedge:

45.0400

180

Since the principal amount of a 2-year bond to hedge is $100M, then $100M *0.45 =

$45 million principal amount of 7-year bond should be sold to hedge $100 million

long position in 2-year bonds. Or in general, the equation for a simple hedge is:

1

2

12 Q

V

VQ

Note: it is very important to label correctly Bond 1 and Bond 2!

2.1. Problems with the conventional approach to hedging (yield spread trades)

There are two major problems with the above approach, both related to the

assumptions on which the approach is based:

1. Volatility of yields on the two bonds is assumed to be the same

Conventional hedge assumes that the yield volatility of the two bonds used in the

hedge is identical (i.e. if the yield of a short term bond changes by 1bp, the yield on

the long term bond will change by the same amount). However, in reality this will not

be the case. Consider the case in which the yield changes on the 2-year and 7-year

bonds are positively correlated but the yield on the 7-year bond changes only half as

much as that of the 2-year bond. For example, if the yield on the 2-year bond rises by

1bp than the yield on the 7-year bond rises by ½ bps. It can be said than that the

volatility of the yield of the 2-year bond is double the volatility of the yield of the 7-

year bond. In such cases, it would be incorrect to calculate the hedge using the

conventional method as in section 2. Using our example from Table 4 1bp yield

change on the 2-year bond will produce a change in value of $18000 on $100M

principal amount of those bonds. However, this will be accompanied by only ½ bp

change in yield on the 7-year bond and $9000 change in value of the $45 million

principal amount of those bonds:

1/2bp*$400*$45 = $9000

Therefore, the short position in 7 year bond hedges only half of the risk of the long

position in 2 year bond.

2. Correlation of the yield changes on two bonds is assumed to be perfect

Suppose that the 2-year and 7-year bond have equal yield volatility but the changes in

their yields are uncorrelated. This means that knowing that the yield on the 2-year

bond went up by 1bp for example tells us nothing about the contemporaneous change

in yield on the 7-year bond. In this case, it will not be possible to use 7-year bonds for

hedging purposes as the gains or losses on the 2-year bonds will be unrelated to losses

or gains on the 7-year bonds.

Both of these problems are corrected when one applies the optimal hedging approach,

as discussed in Section 3 below.

3. Optimal hedging with one hedge bond

The size of a bond hedge is affected by three factors:

1. The volatility of each bond involved in the hedge (measured through the Price

Value of a Basis Point or Duration for example)

2. The volatility of the yield on each bond

3. The correlation between changes in yields on pairs of bonds

The example from Table 5 below will be used in explaining the influence of these

factors.

Table 5 3

2-year 3-year 4-year 5-year 7-year 10-year 20-year 30-year

St. Deviation 20.3bp 20.5bp 21.2bp 21bp 20.8bp 20.3bp 19.2bp 18.3bp

Correlation with

2-year 1

3-year 0.984 1

4-year 0.973 0.983 1

5-year 0.956 0.970 0.988 1

7-year 0.927 0.945 0.972 0.985 1

10-year 0.921 0.939 0.965 0.978 0.993 1

20-year 0.891 0.909 0.940 0.953 0.973 0.982 1

30-year 0.886 0.904 0.933 0.949 0.969 0.982 0.988 1

Values in Table 5 show that the yield volatility varies significantly across different

maturities of bonds. The most volatile appears to be the 4-year bond. Also, it can be

noted that although some correlation coefficients are very high, there is no perfect

correlation between changes in yields for the bonds in the example (as the

conventional hedge assumes!). The highest correlations are observed for the more

‘near-by’ issues (7-year and 10-year bond or 5-year and 4-year bond for example) and

the lowest for more ‘distant’ issues (2-year and 30-year bond for example). The

greater the difference in maturity (or duration) is, the lower the correlation between

the issues.

3.1. Calculation of Optimal Hedge with One Hedge Bond – Risk Minimisation

The purpose of hedging is to minimise the risk, i.e. minimise the standard deviation

(variance) of the value of the bond portfolio position. Let us denote the bond that we

want to hedge as Bond 1 and the bond with which we want to hedge as Bond 2. The

principal amount of Bond 1 in the hedge can be denoted as 1Q and the principal

amount of Bond 2 as 2Q . Also, let 1V be the price value of a basis point of Bond 1 and

2V is the price value of a basis point of Bond 2. If the yields on two issues change by

1y and 2y (not that these yield changes are unlikely to be the same in reality), the net

value of the long-short bond portfolio position ( W ) changes by:

3 Garbade (1996)

222111 yVQyVQW

Therefore, it can be seen that the change in the value of security as calculated as the

principal value times the price value of the basis point times the change in the yield on

a security.

The variance or uncertainty depends on the principal amounts of both securities, price

value of a basis point for both securities, on the variances and correlation between the

changes in yield of the two securities. Therefore, the variance of W is:

2,121212122

22

22

21

21

21

2 2 VVQQVQVQW

(1)

Normally, we will know the principal value of bond 1 (as we already have it and we

want to hedge it), 1Q , and we will need to find which value of bond 2, 2Q , will

minimise the variance from equation (1). To do this we will need to find the first

derivative of variance with respect to 2Q , 2

2

Q , set it equal to zero and solve for

2Q , as shown in the following steps:

2,12121122

222

2

222 VVQVQ

Q

= 0

Then the amount of bond 2, 2Q , that should be used to hedge 1Q is given as:

1

22

112,1

2 QV

VQ

(2)

Example

We will use securities from Table 4. We want to hedge a long position in $100M

principal of 2-year bonds by selling 7-year bonds short. Therefore from table 4 we

have:

1Q =$100M

1V =$180 per $1M principal amount

2V =$400

From table 5 we have:

1 =20.3pb

2 =20.8bp

2,1 =0.927

Using these values we can calculate the optimal amount of 7-year bond that should be

sold short to hedge $100M long position in the 2-year bond:

MMQV

VQ 71.40$100$

)8.20)(400(

)3.20)(180)(927.0(1

22

112,1

2

Please note that this optimal hedge is over $4M smaller than the conventional hedge

of $45M calculated earlier. The reasons for this are:

Optimal hedge estimates lower amount for the 7-year bonds to be short-sold

partly because the yield on 7-year bonds is more volatile than the yield on 2-

year bonds.

The size of the hedge depends on the level of correlation of yield changes

between two bonds. Conventional hedge was assuming perfect positive

correlation (correlation = 1), so when correlation is lower than that, as it is in

this case (correlation = 0.927) it will lead to a smaller hedge position.

The conventional approach to bond hedging (which uses only the price value

of the basis point) is interpreted as a special case of equation (2): if securities

have equal volatilities of yields and perfect correlation coefficient equal to 1.

Then, optimal hedge becomes the conventional hedge.

There are two additional implications of the above analysis of optimal hedge which

are important and also counterintuitive: optimal hedges are not symmetric and they

may not be transitive. Each of those characteristics will now be explained in turn.

3.1.1. Optimal Hedges are not symmetric

As we have seen above, the optimal hedge for $100M long position in 2-year bonds

was a short position in $40.71M 7-year bonds. Let’s see how the situation changes if

we reverse this problem and consider the optimal 2-year hedge for a short position in

$40.71M of 7-year bonds.

This reverse hedging problem can be solved using equation (2) as before. However,

this time, bond 1 is the 7-year bond (with 1Q = -$40.71M) and bond 2 is the 2-year

bond. We use the same Price Value of a Basis Point, correlation and standard

deviation data as before, to obtain:

MMQV

VQ 93.85$)71.40$(

)3.20)(180(

)8.20)(400)(927.0(1

22

112,1

2

long position in 2-year

bond will be an optimal hedge for the short position of -$40.71M in 7-year bond.

We would have intuitively expected here that the answer is that long position of

$100M 2-year bonds is the optimal hedge for this short position of -$40.71M in 7 year

bond. In general, intuition suggests that the composition of a hedged position should

not depend on which bond is labelled as bond 1 and which as bond 2, but we can see

from the analysis that labelling bonds correctly is of crucial importance.

Therefore, this analysis shows that optimal hedges are not symmetric. This

characteristic can be understood by noting the difference between the two hedging

problems. In the first problem, we asked which quantity of 7-year bonds that

minimises the risk of the existing $100M long position in 2-year bonds. In the second

case, the problem is reversed and we search for a quantity of 2-year bonds that will

minimise the risk of the short position of -$40.71M in 7-year bonds. These two

problems are not the same, they are reverse, as in the first one we can only buy or sell

7-year bonds and in the second one we can only buy or sell 2-year bonds, so there is

no reason why they should lead to the same optimal hedge solution.

3.1.2. Optimal Hedges May Not be Transitive

Consider the following example, which is divided into two parts:

Part 1: Hedging a short position of -$40.71M 7-year bonds with 30 year bonds

As we said earlier that labelling of bonds is crucial: 7-year bond is bond 1 and 30-year

bond is bond 2.

From Table 4 we have:


2V =$1000


1 =20.8bp

2 =18.3bp

2,1 =0.969

From equation (2) we have:

MMQV

VQ 93.17$71.40$

)3.18)(1000(

)8.20)(400)(969.0(1

22

112,12

of 30-year bonds

Part 2: Hedging a long position $100M 2-year notes with 30-year bonds

In this part, 2-year bond is Security 1 and 30-year bond is Security 2.



2V =$1000


1 =20.3pb

2 =18.3bp

2,1 =0.886

From equation (2) we have:

MMQV

VQ 72.16$100$

)3.18)(29.1057(

)3.20)(84.179)(886.0(1

22

112,1

2

of 30-year bonds

In part 1, we see that the optimal 30-year bond hedge for a short $40.71 M position in

7-year bond is a long position in $17.93M of 30-year bonds. Intuitively, this implies

that the optimal 30-year hedge for $100M 2-year bonds should be a long position

$17.93M 30-year bonds. However, in part 2 it is shown that this is not the case and

that the optimal hedge is actually a short position in $16.72M 30-year bonds.

To generalise this counterintuitive result, lets consider three bonds, A, B and C. Bond

C may not be an optimal hedge for position in bond A even though it is an optimal

hedge for position in bond B and bond B is optimal hedge for position in bond A. In

different words, this means that optimal hedges are not transitive. Such a non-

transitivity can be explained through analysis of the correlation coefficients between

bond yield changes. Less than perfect positive correlation (<1) implies that there is

some independence in the changes in yields on two bonds. However, the correlations

between yield changes of bond A and C and B and C and A and B may all be

different, i.e. there is no requirement that the degree of independence between

securities A and C is determined by the independence between securities A and B and

between B and C. Each of the pairs of bonds should be examined separately if used

for hedging purposes.

4. Optimal hedging with two hedge bonds

The previous analysis was based on a single bond hedging principal; however there is

nothing to suggest that using one bond is better than using two, three or more bonds

for hedging purposes. In particular, the analysis was looking at the optimal size of a

single bond hedge but not at the optimal number of hedge bonds. We will now look at

the case in which two bonds are used to determine an optimal bond hedge.

4.1. Calculation of Optimal Hedge with Two Hedge Bonds – Risk Minimisation

The same rationale will be applied in this section as in the section for hedging with a

single bond: the aim is to minimise the risk, i.e. minimise the standard deviation

(variance) of the value of the long-short position. However in case of two hedge

bonds, we have positions in three securities, i.e. 32,1 andi . The principal amounts

in those three securities are denoted with iQ and the Price value of a basis point

with iV . If the yields on the three securities change by 2,1 yy and 3y respectively,

the net value of the position changes by:

333222111 yVQyVQyVQW

It can be seen that this just an extension of the equation used in single bond hedging

case. The variance depends on the amount invested in each bond, the price value of a

basis point for each bond and also on the statistical structure of change in yields

(standard deviation of yields and correlations between yield changes for pairs of

bonds). Therefore, the variance of W can be written as:

3,23232323,13131312,121212123

23

23

22

22

22

21

21

21

2 222 VVQQVVQQVVQQVQVQVQ (4)

Normally, we will be given the value of bond that we want to hedge, 1Q ,and we will

have to find the values of bond 2 and bond 3, 2Q and 3Q , that jointly minimise the

variance. In order to do this, partial derivatives of equation (4) with respect to 2Q and

3Q should be found, i.e. 2

2

Q and

3

2

Q , then set to zero, which will allow us to

solve for 2Q and 3Q . That is shown in the following steps:

From equation (4), we have:

3,2323232,12121122

222

2

2222 VVQVVQVQ

Q

=0

and

3,2323223,13131123

233

3

2222 VVQVVQVQ

Q

=0

Solving for 2Q and 3Q gives us:

1

222

3,2

113,23,12,1

2)1(

)(Q

V

VQ

(5)

and

1

332

3,2

113,22,13,1

3)1(

)(Q

V

VQ

(6)

Let us look at the example of the application of hedging with two bonds.

Example 1: Hedging with Shorter and Longer Bonds

Suppose that we want to hedge a position in $100M of 7-year bonds (bond 1,

1Q =100) with 2-year notes (bond 2) and 30-year bonds (bond 3). We will use data for

price value of basis points from Table 4 and data for standard deviation and

correlation between changes in yields for these bonds from Table 5, so:

886.0

969.0

927.0

3.18

3.20

8.20

1000

180

400

3,2

3,1

2,1

3

2

1

3

2

1

bp

bp

bp

V

V

V

Using these figures in equations (5) and (6) gives us the optimal amount of 2-year

bond and 30-year bond to be sold short to hedge a $100M long position in 7-year

bond:

amountprincipalMQ 51.72$1003.20180)886.01(

8.20400)886.0969.0927.0(22

and

amountprincipalMQ 23.31$1003.181000)886.01(

8.20400)886.0927.0969.0(23

This means that some (but not all) variation of a 7-year bond is hedged by the

variation in the 2-year bond and some (but not all) by the variation in the 30-year

bond. It is preferable to use both shorter and longer maturity bonds (relative to the

bond we want to hedge), as in this example.

Example 2 – Hedging with Two Shorter Bonds

Consider a different hedging problem now. In this example, we should find an

optimal hedge a position, when we want to hedge $10M principal amount of 30-year

bonds with 2-year bonds and 7-year bonds. Security 1 then is a 30-year bond ( 1Q =

10), security 2 is 2-year bond and security 3 is 7-year bond. Then, from Table 4 and

Table 5 we have:

927.0

969.0

886.0

8.20

3.20

3.18

400

180

1000

3,2

3,1

2,1

3

2

1

3

2

1

bp

bp

bp

V

V

V

From equations (5) and (6) we have:

bondsyearamountprincipalMQ

237.4$10

3.20180)927.01(

3.181000)927.0969.0886.0(22

and

bondsyearamountprincipalMQ

709.23$10

8.20400)927.01(

3.181000)927.0886.0969.0(23

This shows that the optimal hedge for a $10M long position in 30-year bond is a short

position of -$23.09M in 7-year bonds and a long position of $4.37M in 2-year bonds.

In comparison to all previous examples, it is surprising that we do not have a short

position on both bonds used for the hedge. This anomaly can be explained by

observing the consequences of hedging a long position of $10M 30-year bonds with

only 7-year bonds. It can be calculated by using equations for single bond hedge that

the optimal hedge of a $10M 30-year bond with 7-year bond is a short position of -

$21.31M principal amount of the 7-year bonds. While this is optimal single bond

hedge, in the sense of minimising the variance of the change in net worth of the

hedged position, it can lead to some significant yield curve risk. Specifically, a short

7-year/long 30-year bond position will lose value if the yield curve steepens. This

yield curve risk is reduced when we use two bonds to hedge by ‘overhedging’ 30-year

bonds with 7-year bonds and by taking additional long position in 2-year bonds. If the

yield curve then steepens, there will be a gain on the long 2-year/short 7-year portion

of the hedge that will be used to offset the loss on the short 7-year/long 30-year

portion of hedge.

This interpretation of results in Example 2 leads us to re-interpret the results of 2-year

and 30-year hedge of a 7-year bond position from Example 1.

Re-interpretation of Example 1: Hedging With Shorter and Longer Bonds

It was shown in Example 1 that the optimal hedge of $100M long position in 7-year

notes was a short position of -$72.51M 2-year bonds and short position of -$31.23M

30-year bonds. This hedge with two bonds can also be viewed as a device to limit the

yield curve risk. If the hedge consisted of only 2-year bonds held short, the position

would lose value if the yield curve steepened. If the hedge consisted of only 30-year

bonds held short, the position would lose value if the yield curve flattened. By

dividing the hedge between both 2-year notes and 30-year bonds, the hedge position is

less sensitive to changes in slope of the yield curve.

5. Some General Comments on Hedging

Although we have examined cases of hedging with one and two bonds, this can be

extended to hedging with any other arbitrary number of bonds. However, the most

interesting aspect of multiple bond hedging is not on the analytical side but rather the

question of ‘how many different hedge bonds are enough?’

By comparing the analysis of single bond and two bond hedges, roughly speaking we

can say that a one bond hedge can protect an existing position against changes in the

general level of interest rates but may still expose investors to losses due to changes in

the slope of the yield curve. On the other hand, a two bond hedge is hedging the risk

of changes in the slope of the curve as well as changes in the levels of interest rates.

This implies that as we extend the number of bonds with which we hedge we are able

to protect our security against more types of changes in interest rates. For example a

three bond hedge may protect investor from changes in the 1) level, 2) slope and 3)

curvature of the yield curve. Therefore, the issue of how many bonds would be

sufficient for a good hedge is the matter of the number of different ways the yield

curve can fluctuate.

References

Garbade, K. (1996), Fixed Income Analytics, The MIT press

lecture 8 notes

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