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6.5 Application of the Bond Immunization Theorem

I

n this topic, we apply the insight from the bond immunization theorem to the Zero-Coupon Bond example developed in Topic 6.3.

Recall that in this example, there were three securities having the following cash flows:

Security

Year 1

Year 2

Year 3

Year 4

Security I

0

160

200

250

2-Year Zero

0

100

0

0

3-Year Zero

0

0

100

0

The initial position was:

Endowment

Duration

Cash

5300

0

Security I

-14

3

2-Year Zero

51

2

3-Year Zero

0

3

At the set of yield curve realizations considered possible, the end-of-year position values are:

Yield Curve at End of Year 1: Value of Position

(1, 2, and 3 year spot rate) at End of Year 1

5%, 5%, 5% $3785.69

15%,15%,15% $4693.45

25%,25%,25% $5329

35%,35%,35% $5784.62

45%,45%,45% $6117.61

That is, this position is exposed to shifts in the yield curve. In particular, it is hurt by a decrease in interest rates. The objective for this example was to create a hedge that ensures the end-of-year value does not fall below $5,000.

You can see that the yield curve shifts are not exactly parallel because the spot curve at the beginning of Period 1 was 25%, 25%, 25%, 25%, which shifts to 25%, 15%, 15% 15%, and so on. As a result, the bond immunization theorem will not hold exactly. Instead, we will see whether the insights from this theorem are useful.

Consider buying 84 3-year zeroes and selling 51 2-year zeroes. This results in a duration for the final position that equals zero:

Security

Final Position

Duration

PV@25%

Dur*PV

Cash

$4,263.20

0

$4,263.20

0

Security I

-14

3

-14*$307.20

-42*$307.20

2-Year Zero

0

2

0*$64

0*$64

3-Year Zero

84

3

84*$51.20

252*$51.20

Position

0

Even though the assumptions for the bond immunization theorem do not hold exactly, the value of this zero-duration position is still protected against shifts in the yield curve:

Yield Curve at End of Year 1 Value of Position

(1, 2, and 3 year spot rate) at End of Year 1

5%, 5%, 5% $5251.60

15%,15%,15% $5314.27

25%,25%,25% $5329

35%,35%,35% $5319.90

45%,45%,45% $5299.61

This achieves our desired investment objective.

Implementing Duration Solutions in the Marketplace

The investment example worked out above is a variation of a Financial Trading System (FTS) case. FTS is an electronic market system that allows you to trade a wide range of securities, including fixed-income securities. Bond Tutor can be used in conjunction with this trading system to provide a sophisticated trading support system.

In the full FTS trading case, there are two sides to the market that have offsetting positions so that there are gains to exchange at the market clearing price. The trading objective is to manage the downside exposure of the position. An additional security is added to the market and what can and cannot be traded is varied.

By trading in this kind of market, you can see how the market lets traders with different exposure profiles hedge risks to make themselves strictly better off.

In the next topic, Fisher-Weil duration, you will see an alternative definition to what hasa been applied to date.