Utilizing Effective Duration and Convexity for Option-embedded Bonds

Yield duration and convexity assume predictable bond cash flows. However, bonds with embedded options, e.g., callable or puttable bonds, have future cash flows which are uncertain. The option is exercised based on market interest rates relative to the coupon interest paid or received. Bonds with embedded options do not have clear-cut yield-to-maturity. For such bonds, traditional measures like Macaulay and modified durations do not accurately capture interest rate risk. Instead, “effective duration” is used, which focuses on how the bond’s price reacts to changes in a benchmark yield curve, like the government par curve. Finally, while the interest rate risk estimates of yield and convexity are only useful for small changes in yields, the estimates from effective duration and convexity are useful for both small and large yield changes.

Effective Duration

Effective duration serves as an essential tool in measuring the bond’s price sensitivity to changes in the benchmark yield curve. The formula for calculating effective duration (EffDur), which is very similar to the one for determining the approximate modified duration, is expressed as:

\[\text{EffDur} = \frac{\left( {PV}_{-} – {PV}_{+} \right)}{2 \times (\Delta\text{Curve}) \times \left( {PV}_{0} \right)}\]

Where:

• \({PV}_{-}\) and\(\ {PV}_{+}\): The present values are calculated using option pricing models when the interest rate is decreased and increased, respectively.
• \(\Delta\)Curve: The change in the benchmark yield curve.
• \({PV}_{0}\): The present value at the current interest rate.

Effective Duration for Callable and Non-Callable Bonds

Non-callable bonds are consistently priced higher than callable bonds due to the value of the embedded call option held by the issuer. When interest rates are low, the value of the call option in callable bonds rises; this limits the bond’s price appreciation. Effective durations of callable and non-callable bonds are similar when yields are high. However, for low interest rates, callable bonds have a lower effective duration because of the presence of the call option.

Effective Duration for Putable and Non-Putable Bonds

Putable bonds allow investors to sell them back to the issuer before maturity at face value. This provision protects investors from benchmark yield increases that could lower the bond’s price below par. A putable bond’s price is always greater than its non-putable counterpart due to the value of the embedded put option. Put options minimize bond sensitivity, especially during rising interest rates, in terms of effective duration. When rates are lower than the bond’s coupon rate, the put option’s value is limited, hence the bond’s market reaction to yield changes resembles non-putable bonds. As benchmark rates rise, the put option’s value rises, protecting investors from price declines.

Effective Convexity

Effective convexity analyzes the second-order effects of shifts in the benchmark yield curve. It helps in understanding the potential changes in curve shape and the transition into negative regions, particularly when the value of the embedded call option increases. The formula for calculating effective convexity (EffCon) is:

\[EffCon\ = \frac{\left\lbrack \left( PV_{-} + PV_{+} \right) – 2 \times PV_{0} \right\rbrack}{(\Delta Curve)^{2} \times PV_{0}}\]

When the benchmark yield falls, non-callable bonds’ price-yield curve steepens, suggesting positive convexity. The curve of callable bonds flattens and even turn negative when the benchmark yield falls. Both bond types have positive convexity at high benchmark yields. As yields fall, the callable bond may become negative convexity due to the embedded call option’s value.

On the other hand, putable bonds are characterized by positive convexity. The embedded put option protects investors, especially during rising interest rates. This option lets investors sell the bond back to the issuer at par, limiting price losses.

Effective duration and convexity are also relevant for mortgage-backed securities (MBSs). MBSs cash flows depend on homeowners’ refinancing decisions, especially prevalent in areas where refinancing is common during low-interest-rate scenarios.

Example: Interest Rate Sensitivity of a Callable Bond

A portfolio manager is contemplating investing in a callable bond and approaches you, the lead analyst in the fixed-income team, to assess its interest rate sensitivity. The manager has a preference for bonds with a duration between 6 and 7 years and a positive convexity. The full price of the callable bond is 105.50 per 100 of par value. When the government par curve shifts by 30bps, your option valuation model indicates the new full prices for this callable bond are 103.40 when raised and 107.350 when lowered. Therefore:

• \(PV_{0} = 105.500\)
• \(PV_{+} = 103.400\)
• \(PV_{-} = 107.350\)
• \(\Delta\)Curve \(= 0.0030\)

Using the provided data, you calculate the effective duration and effective convexity for this callable bond as:

\[EffDur\ = \frac{\left( PV_{-} – PV_{+} \right)}{2 \times \Delta\text{Curve} \times PV_{0}}\]

\[EffDur = \frac{(107.350 – 103.400)}{2 \times 0.0030 \times 105.500} = 6.2401\]

\[EffCon\ = \frac{\left( PV_{-} + PV_{+} \right) – 2 \times PV_{0}}{\Delta\text{Curve}^{2} \times PV_{0}}\]

\[EffCon\ = \frac{(107.350 + 103.400) – 2 \times 105.500}{(0.0030)^{2} \times 105.500} = – 263.296\]

Based on your analysis, you advise the portfolio manager to exercise caution when considering this callable bond. Although its effective duration is within the desired range, the bond exhibits negative effective convexity. A bond displaying negative effective convexity will experience a more pronounced decrease in its price due to a rise in the benchmark yield compared to the price increase resulting from a decrease in the benchmark yield.

Question

An investor plans to allocate $500,000 in two-year Mega-Corp bonds. One of the bonds, Bond X, is standard, while Bond Y has an embedded put option. Which duration metric is best suited to assess the interest rate risk for these bonds?

1. Money duration
2. Effective duration
3. Macaulay duration

Solution

The correct answer is B.

Bond Y has an embedded put option, introducing optionality to the bond. Bonds with embedded options, such as callable or putable bonds, have uncertain future cash flows because the exercise of the option depends on market interest rates relative to the bond’s coupon interest. Since these bonds do not have clearly defined yields-to-maturity, traditional measures like Macaulay and modified durations are not suitable. Instead, effective duration, which measures a bond’s price sensitivity to changes in a benchmark yield curve (like the government par curve), is the appropriate measure for bonds with embedded options.Therefore, effective duration is the best measure to use. Effective duration is also applicable for bonds without embedded options, enabling the investment advisor to compare the interest rate risks of both Bond X and Bond Y.

A is incorrect. While money duration measures the sensitivity of the bond’s price to interest rate changes, it does not specifically cater to bonds with embedded options.

C is incorrect. Macaulay duration provides the weighted average time until a bond’s cash flows are received, but it does not adjust for embedded options.