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Convexity

Why It Matters

Convexity is often used to describe how the value of a fixed coupon bond alters with respect to interest rates. This explanation of convexity considers the value of a bond with a principal amount of \$10,000,000, a coupon of 5% and a maturity of 10 years. This bond pays the investor \$500,000 every year and returns the principal at maturity.

The market value of the bond is the sum of its discounted cash flow values. If different interest rates are used to discount those cash flows the value of the bond will change.

The higher the interest rate the lower the value of the bond becomes. Let's look at this bond's value at interest rates between 0% and 10%.

Interest rate      Bond value

0%                  \$14,500,000

1%                  \$13,335,878

2%                  \$12,284,601

3%                  \$11,333,993

4%                  \$10,473,307

5%                  \$9,693,043

6%                  \$8,984,793

7%                  \$8,341,109

8%                  \$7,755,378

9%                  \$7,221,731

10%                \$6,734,944

What is immediately clear is that its value declines as interest rates increase. This is straight forward bond math; as interest rates increase the present value of future cash flows falls.

But what is not quite so clear is the rate at which the bond falls in value. Let's now look at the change in the value of the bond for each 1% increase in interest rates.

Interest rate      Bond value                Change in value

0%                   \$14,500,000

1%                   \$13,335,878                \$1,164,122

2%                   \$12,284,601                \$1,051,277

3%                   \$11,333,994                \$950,608

4%                   \$10,473,307               \$860,686

5%                   \$9,693,043                 \$780,264

6%                   \$8,984,794                 \$708,249

7%                   \$8,341,109                  \$643,685

8%                   \$7,755,379                  \$585,730

9%                   \$7,221,732                  \$533,647

10%                 \$6,734,945                  \$486,787

The change in value between 1% and 2% is \$1,164,122 and between 9% and 10% is \$486,787.

This means the rate at which the bond changes in value for each small increase in interest rates is not constant. This is the bond's convexity.

The bond increases in value at a faster and faster rate as interest rates decline and loses value at a slower and slower rate as interest rates rise. This has some very important implications in financial markets. Let's look at some of these.

The reduced rate of loss is the dealer's friend.

In our example convexity works in favour of the dealer. As interest rates fall he makes money at a faster and faster rate. As they increase the rate of loss gets slower and slower.

When interest rates are low a small change in rates leads to significant changes in the profit and loss in a trader's portfolio.

In this case a dealer holding \$10m of this bond would experience a loss of \$1,051,277 if interest rates moved from 1% to 2%. But a loss of less than half as much (\$486,787) is experienced if interest rates move from 9% to 10%.

Hedge ratios need adjustment.

Sometimes dealers buy one bond and sell another. This can be to hedge the interest rate risk between the two bonds or speculate on their relative values. Whatever the reason the dealer needs to think about convexity, this is why.

If one bond is trading at a yield of say 4% and the other at a yield of 6% a 1% change in their respective yields will lead to different changes in value. From our earlier table an increase in yield from 4% to 5% means a loss of \$780,264 and an increase in yield from 6% to 7% a loss of \$643,685.

The hedge ratio would therefore be \$643,685/\$780,264 = 0.82. This means every \$0.82m of the 4% bond would carry the same interest rate risk as \$1m of the 6% bond.

Convexity can work against you.

In our example convexity is seen as advantageous to the dealer. This is not always the case. A dealer with a short (sold) position would experience the opposite to the dealer with the long (bought) position.

That's smaller increases in profits as rates rise and accelerating losses as rates fall. This is sometimes referred to as negative convexity. Holders of bonds that have issuer call options or pre-payment options face similar risks. When interest rates fall the bond can be repaid and any anticipated gain is not forthcoming.

This means that you need to think very carefully about products that present you with negative convexity. Are you being adequately compensated for the additional risk that you are taking on?

First published by Barbican Consulting Limited 2007

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