Back-to-Basics: Understanding Convexity

Teri Geske
Senior Vice President, Product Development

Convexity is:

1. used with effective duration to estimate how a bond's (or portfolio's) price is affected by changes in interest rates.
2. a way to measure option risk.
3. computed in different ways by different systems.
4. a number that people often find difficult to explain (and therefore try to avoid in client presentations)
5. all of the above.

If you chose (a), (b), (c) or (d) you are correct. Those of you who chose (e) can go to the head of the class. No matter which answer you chose, it is important to remember that convexity is a critical concept in fixed income that is used throughout BondEdge and is worth taking the time to understand. Not only can convexity be interpreted by itself, when examined in connection with effective duration it can help us to gain an intuition for the price behavior of all different types of securities.

This brief article attempts to provide some insights about how to understand convexity. Since we said that all of the multiple choices offered above are correct, let's explore them one by one:

Answer (a) - "Convexity is used in conjunction with effective duration to describe how a bond's (or portfolio's) price is affected by changes in interest rates." Effective duration is the average percentage change in a bond's price given +/- 100bp shifts in the underlying Treasury spot curve (assuming a constant OAS). We can use effective duration to estimate the change in a bond's (or a portfolio's) price for a given change in rates, but duration alone does not give as accurate a prediction as duration and convexity together can provide.

Recall that effective duration is computed from the actual prices for the bond under +/-100bp yield curve shifts, and those prices inherently reflect the impact of any embedded options (e.g. calls, puts, prepayments, etc.). However, the average change in price under both positive and negative shifts in rates can conceal the lopsided profile of the bond's upside potential and downside risk. For example, consider a callable corporate bond priced at $101.75, callable at 101.940. Its Effective Duration is 1.57 and its convexity is -1.143. In this example, the bond's current price is very close to its call price; therefore, it would only increase in value by 0.40% if rates fell 100bps. However, if rates rise 100bps its price would drop by 2.76%. The average change is 1.57% (its effective duration), but this average by itself does not give us enough information.

Using effective duration only, we would use the following formula to estimate the bond's new prices given a +/- 1% change in interest rates: Price = (Effective Duration x i), where i = the change in interest rates. This would produce price estimates of $100.153 and ($103.345) for a 1% rise (decline) in rates. The actual prices for +/-100bp changes are $98.937 and 102.191, respectively, not particularly close to these estimates. If we use effective duration and convexity together to predict the prices, we use the formula: Price = (Effective Duration x i) + (Convexity x i²). This approach produces price estimates of $99.007 and 102.202, only pennies away from the actual prices.

Answer (b) - "Convexity is a way to measure option risk." - In the example above, the bond was priced just below its call price; a slight rally would cause it to be called. BondEdge shows the value of the embedded call option would more than double, from $3.45 to $8.31, if rates declined 1%, and would fall to $1.26 given an 1% rise in rates. This bond's large, negative convexity reminds us that the option value is highly sensitive to changes in interest rates, and has a large impact on the security's price. If we change the call price from approximately $102 to $104, the bond's convexity improves to -0.694 (from -1.143), indicating the fact that the value of the option would be less sensitive to further changes in rates if the bond's current price were still $2 away from the call price.

Negative convexity reflects the fact that most embedded options (except puts) are disadvantageous to the bond investor. The more negative the convexity, the greater the impact of the embedded option (calls, prepayments, etc.) on the price behavior of the security. As interest rates fall, a negatively convex bond's price movement is limited by the fact that the call (or prepayment) option is becoming more valuable. As rates rise, the option loses value and the bond's price declines in response to the higher rate environment. In the example above, we saw how relying on effective duration alone to predict price behavior for a bond with negative convexity caused us to understate the price decline for a rise in rates and overstate the rise in price in a rally. In contrast, an option-free bond has positive convexity (>0.00), which means its price will rise more than (and decline less) than effective duration alone predicts given a rise (and fall) in rates. To compensate the investor for accepting option risk, bonds with negative convexity typically have higher yields than similar but option-free securities.

Answer (c) - "Convexity is computed in different ways by different systems." In BondEdge, convexity is defined as a "contribution to price change" - in other words, it is the amount of the change in a bond's price (or a portfolio's market value), given a +/- 100bp change in interest rates, which is not explained by effective duration alone. Calculated this way, convexity is easily combined with effective duration to estimate change in value for a given change in interest rates; in fact, when considering a 1% change in rates, we can simply add effective duration and convexity together to estimate the percentage change in price for a security or portfolio. (For changes in rates other than 1%, we must use the formula shown in (b) above).

Convexity is sometimes calculated as the change in effective duration for a given change in interest rates. In this case, convexity is referred to as "duration drift". This way of computing convexity can be used to show how much a callable bond's effective duration will shorten as interest rates fall and lengthen when interest rates rise, due to the fact that as interest rates fall the likelihood of the bond being called (or of a mortgage-backed security being pre-paid) increases, and vice versa.

When comparing convexity numbers from different sources it is important to know whether the value is computed as a "contribution to price change" or a "duration drift" measure; the "duration drift" form of convexity is approximately twice as large as the convexity expressed as a "contribution to price return".