# Measuring the duration and convexity of bond risk

## What are duration and convexity?

Duration and convexity are two tools used to manage the risk exposure of fixed income investments. Duration measures the sensitivity of a bond to changes in interest rates. Convexity is related to the interaction between bond prices and their yields because it has undergone changes in interest rates.

For coupon bonds, investors rely on an indicator called duration to measure the sensitivity of bond prices to changes in interest rates. Since coupon bonds make a series of payments throughout their life cycle, fixed-income investors need to measure the average maturity of the bond’s promised cash flow as a summary statistics of the bond’s effective maturity. Duration achieves this, allowing fixed income investors to more effectively measure uncertainty when managing their investment portfolios.

### Key points

• For coupon bonds, investors rely on an indicator called “duration” to measure the sensitivity of bond prices to changes in interest rates.
• Using gap management tools, banks can equate the duration of assets and liabilities, effectively protecting their overall positions from changes in interest rates.

## Bond maturity

In 1938, Canadian economist Frederick Robertson Macaulay referred to the concept of effective maturity as the “term” of a bond. In doing so, he suggested calculating this duration as the weighted average of the maturity time of each coupon or principal payment paid by the bond. The Macaulay duration formula is as follows:

D = ∑ i = 1 T t ∗ C (1 + r) t + T ∗ F (1 + r) t ∑ i = 1 TC (1 + r) t + F (1 + r) t where: D = bond Macaulay duration T = number of periods before maturity i = ith time period C = periodic coupon payment r = periodic yield to maturity F = face value at maturity begin{aligned} &D = frac{sum_ {i=1}^T{ frac{t*C}{left( 1+r right)^t}} + frac{T*F}{left( 1+r right)^t} } {sum_{i=1}^T{ frac{C}{left( 1+r right)^t}} + frac{F}{left( 1+r right)^t} } \ textbf{where:}\ &D = text{Macauley duration of the bond}\ &T = text{terms before maturity}\ &i = text{the} i^{th } text {time period}\ &C = text{regular interest payment}\ &r = text{regular yield to maturity}\ &F = text{face value at maturity}\ end{aligned}

Where:D=A generation=1Ton(1+r)TonC+(1+r)TonFA generation=1Ton(1+r)TonTonC+(1+r)TonTonFD=Macaulay DurationTon=Number of periods before expirationA generation=This A generationTonH periodC=Regular interest paymentsr=Periodic Yield to MaturityF=Face value due

## Duration of fixed income management

Duration is essential for managing fixed income portfolios for the following reasons:

1. This is a simple summary statistics of the effective average maturity of a portfolio.
2. It is an important tool to protect the investment portfolio from interest rate risk.
3. It estimates the interest rate sensitivity of the portfolio.

The duration indicator has the following properties:

• The duration of a zero coupon bond is equal to the time to maturity.
• Keep the maturity unchanged. When the coupon rate is higher, the duration of the bond is lower. This is due to the impact of the higher coupon rate paid in advance.
• Keeping the coupon rate constant, the duration of a bond usually increases with maturity. But there are exceptions, such as deep discount bonds and other instruments, whose duration may decrease as the maturity schedule increases.
• When other factors remain unchanged, when the yield to maturity of the bond is low, the duration of the coupon bond is higher. However, for zero coupon bonds, regardless of the yield to maturity, the duration is equal to the time to maturity.
• The continuous duration of the level is (1 + y) / y. For example, at a rate of return of 10%, a perpetual payment of \$100 per year will be equal to 1.10 / .10 = 11 years. However, with an 8% rate of return, it will be equal to 1.08 / .08 = 13.5 years. This principle is obvious, and maturity and duration can vary greatly. For example: the term of perpetual bonds is unlimited, while the term of a tool with a 10% yield is only 11 years. The present value-weighted cash flow at the early stage of the perpetual term dominates the calculation of the duration.

## Duration of gap management

The maturity dates of many banks’ assets and liabilities do not match. Bank liabilities, mainly deposits owed to customers, are generally short-term and have a short duration. In contrast, bank assets mainly include outstanding commercial and consumer loans or mortgage loans. These assets tend to last longer and their value is more sensitive to interest rate fluctuations. During periods of unexpected surge in interest rates, if the value of a bank’s assets falls more than the value of its liabilities, the bank’s net assets may fall sharply.

A technique called gap management is a widely used risk management tool in which banks try to limit the “gap” between the maturity of assets and liabilities. Gap management relies heavily on Adjustable Rate Mortgage (ARM) as a key component of shortening the duration of the bank’s portfolio. Unlike traditional mortgage loans, when market interest rates rise, the value of ARM will not fall because the interest rate they pay is tied to the current interest rate.

On the other side of the balance sheet, the introduction of longer-term bank certificates of deposit (CD) with a fixed maturity can help extend the duration of bank liabilities and also help narrow the duration gap.

## Understanding gap management

Banks use gap management to balance the duration of assets and liabilities, effectively protecting their overall positions from changes in interest rates. Theoretically, the size of the bank’s assets and liabilities is roughly equal. Therefore, if their durations are also equal, then any change in interest rates will affect the value of assets and liabilities to the same extent. Therefore, changes in interest rates will have little or no final impact on net worth. Therefore, net asset immunity requires zero investment portfolio duration or gap.

Institutions with fixed future obligations, such as pension funds and insurance companies, differ from banks in that their operations focus on future commitments. For example, pension funds are obliged to maintain sufficient funds to provide workers with an income stream after retirement. As interest rates fluctuate, the value of the assets held by the fund and the interest rate at which these assets generate income fluctuate. Therefore, the portfolio manager may wish to protect (immunize) the future accumulated value of the fund at a certain target date from changes in interest rates. In other words, immunization protects assets and liabilities that match maturity, so banks can fulfill their obligations regardless of changes in interest rates.

## Convexity of fixed income management

Unfortunately, when used as a measure of interest rate sensitivity, duration has limitations. Although the linear relationship between bond price and yield change is calculated statistically, in fact, the relationship between price and yield change is convex.

In the figure below, the curve represents the change in price when the rate of return changes. The straight line tangent to the curve represents the price change estimated by the duration statistics. The shaded area shows the difference between the estimated duration and the actual price change. As mentioned above, the greater the change in interest rates, the greater the error in estimating bond price changes.

Convexity is a method of measuring the curvature of bond price changes relative to changes in interest rates. It solves this error by measuring the change in duration when interest rates fluctuate. The formula is as follows:

C = d 2 (B (r)) B ∗ d ∗ r 2 where: C = convexity B = bond price r = interest rate d = duration begin{aligned} &C = frac{d^2left( B left(r right )right)}{B*d*r^2} \ &textbf{where:}\ &C = text{convexity}\ &B = text{bond price} \ &r = text{interest rate}\ &d = text{duration}\ end{aligned}

C=Seconddr2d2(Second(r))Where:C=ConvexitySecond=Bond pricer=interest rated=period

Generally speaking, the higher the coupon, the lower the convexity, because 5% bonds are more sensitive to changes in interest rates than 10% bonds. Due to the bullish nature, if the yield falls too low, the callable bond will exhibit negative convexity, which means that when the yield falls, the duration will be shortened. Zero coupon bonds have the highest convexity, and this relationship is valid only when the comparison bonds have the same duration and yield to maturity. It is clearly pointed out that high-convex bonds are more sensitive to changes in interest rates, so when interest rates change, prices should fluctuate more.

The situation with low convexity bonds is just the opposite. When interest rates change, their prices do not fluctuate much. When drawing a figure on a two-dimensional graph, this relationship should produce a long oblique U shape (hence the term “convex”).

Low-interest and zero-coupon bonds tend to have lower yields, showing the highest interest rate volatility. Technically speaking, this means that the modified bond duration requires greater adjustments to keep up with higher price changes following changes in interest rates. Lower coupon rates result in lower yields, and lower yields result in higher convexity.

## Bottom line

Changing interest rates have brought uncertainty to fixed income investments. Duration and convexity allow investors to quantify this uncertainty and help them manage fixed income portfolios.

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