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Convexity

The prior tutorial introduced the concept of price volatility. Price volatility is the sensitivity of a bond’s value to changes in market interest rates. An approximate measurement of a bond’s price volatility can be derived by calculating its effective duration. Effective duration is the percentage a bond’s value will change if its yield-to-maturity moves 1% (corresponding to a 1% movement in market interest rates). It is calculated by comparing the new values of a bond if the bond’s yield-to-maturity is both increased and decreased by any small amount (Reference the prior tutorial for a comprehensive review).

The effective duration formula will derive a close approximation of a bond’s percentage change in value if the actual percentage change in market interest rates is small. However, the use of effective duration, alone, would result in an inaccurate measurement of price volatility if the actual percentage change in market interest rates is moderate to large. Why? Recall that an inverse relationship exists between a bond’s market value and yield-to-maturity. The use of effective duration, alone, must be limited to the measurement of small movements in interest rates because it assumes that the relationship between a bond’s market value and yield-to-maturity is linear as shown in the graph below:

In reality, the relationship between a bond’s market value and yield-to-maturity is positively convex (curved) rather than linear. The departure of the price/yield curve from the linear line (which represents effective duration) is known as a bond’s convexity. The convex relationship between a bond’s price and yield-to-maturity is shown in the graph below:

It is important to note that the convex relationship between a bond’s market value and yield-to-maturity is the premise of two fundamental behaviors:

1. The actual change in a bond’s market value will be largely determined by the initial level of market interest rates. If a bond’s initial yield-to-maturity is large (corresponding to high market interest rates), a small percentage change in rates will not have a significant impact on the bond’s market value. In contrast, if a bond’s initial yield-to-maturity is small (corresponding to low market interest rates), the same percentage change in rates will have a larger impact on the bond’s market value.

2. Because the price/yield curve is positively convex (a positive slope), a specified movement in market interest rates would result in greater price appreciation, assuming rates adjusted downward, relative to the price depreciation that would occur if rates adjusted upward. This behavior is depicted in the graph above. If an increase/decrease in interest rates causes a bond’s yield-to-maturity to both increase (R₊₁) and decrease (R_-1) by the same amount from the initial yield-to-maturity (R₀), the bond will experience greater price appreciation (P_-1) relative to price depreciation (P₊₁). This behavior is considered attractive to investors who purchase fixed-income securities.

Because the relationship between a bond’s market value and yield-to-maturity is positively convex, a convexity adjustment must be applied to the effective duration formula when measuring a bond’s change in value driven by moderate to large changes in market interest rates. An approximate measure of convexity can be derived by calculating effective convexity using the formula below:

Effective Convexity = (V₂ + V₁ – V₀) / [2(V₀)(∆y²)]

Where:
V₂ = The bond’s value if its yield decreases 0.50%
V₁ = The bond’s value if its yield increases 0.50%
V₀ = The bond’s value at its present yield
∆y² = The percentage change in yield (0.005 or 0.50%) squared.

Consider the following example: Calculate the effective duration and effective convexity of a semi-annual coupon bond with five years remaining until maturity, a market value of $95.735 (per $100 of face value), a 5% coupon rate, and a 6% yield-to-maturity. Use the bond’s effective duration and effective convexity to determine the percentage change in the bond’s dollar value if interest rates move 1.4%.

Using an HP 12C:

First, Calculate V₁ (By calculating the bond’s value if its yield increased 0.50%):

Enter Payment (as the annualized coupon rate):   PMT = 5
Enter Yield (annualized YTM):    I = 6.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012005 f PRICE

V₁ = $93.683

Second, Calculate V₂ (By calculating the bond’s value if its yield decreased 0.50%):

Enter Payment (as the annualized coupon rate):    PMT = 5
Enter Yield (annualized YTM):    I = 5.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012005 f PRICE

V₂ = $97.840

Third, Calculate the bond’s effective duration using V₁ and V₂:

Effective Duration = (V₂ – V₁) / [2(V₀)(∆y)]

Effective Duration = (97.840 – 93.683) / [2 (95.735)(0.005)]

Effective Duration = 4.342%

Four, Calculate the bond’s effective convexity using V₁ and V₂:

Effective Convexity = (V₂ + V₁ – 2V₀) / [2(V₀)(∆y²)]

= (97.840 + 93.683 – (2)95.735) / [2 (95.735)(0.000025)]

Effective Convexity = 11.0717

The effective duration of the bond equates to 4.342%. The bond’s effective convexity equates to 11.0717. The bond’s effective duration and effective convexity can now be applied to the following formula to derive both the bond’s percentage change in price if rates both increase and decrease by 1.4%.

%∆ Price = - 1(∆rates) (D_E) + (C_E) (∆rates)²

Where:
∆rates = Change in interest rates
V₀ = The bond’s value at its present yield
D_E = Effective Duration
D_E = Effective Convexity

Assuming interest rates increase by 1.4%.

%∆ Price = -1(.014)(4.342) + (11.0717)(+.014)²

%∆ Price = -(.06079) + (.00217)

%Change in Price = -.05862 or -5.862%

Assuming interest rates decrease by 1.4%.

%∆ Price = -1(-.014) (4.342) + (11.0717) (-.014)²

%∆ Price = +(.06079) + (+.00217)

%Change in Price = +.06296 or +6.296%

Summary

The effective duration formula, alone, will derive a close approximation of a bond’s percentage change in value if the actual movement in market interest rates is small. In such circumstances, the convexity adjustment would be insignificant (approximately equal to zero). However, for moderate to large movements in market interest rates, the convexity adjustment must be incorporated into the price volatility equation to derive an accurate percentage change in the value of the bond.

Questions:

1. Calculate the effective duration and effective convexity of a semi-annual coupon bond with seven years remaining until maturity, a market value of $111.69 (per $100 of face value), a 7% coupon rate, and a 5% yield to maturity.

Using an HP 12C:

First, Calculate V₁ (By calculating the bond’s value if its yield increased 0.50%):

Enter Payment (annualized coupon rate):   PMT = 7
Enter Yield (annualized YTM):    I = 5.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012007 f PRICE

V₁ = $108.618

Second, Calculate V₂(By calculating the bond’s value if its yield decreased 0.50%):

Enter Payment (annualized coupon rate):    PMT = 7
Enter Yield (annualized YTM):    I = 4.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012007 f PRICE

V₂ = $114.870

Third, Calculate the bond’s effective duration using V₁ and V₂:

Effective Duration = (V₂ – V₁) / [2(V₀)(∆y)]

Effective Duration = (114.870 – 108.618) / [2 (111.69)(0.005)]

Effective Duration = 5.596%

Four, Calculate the bond’s effective convexity using V₁ and V₂:

Effective Convexity = (V₂ + V₁ – 2V₀) / [2(V₀)(∆y²)]

= (114.870 + 108.618 – (2) 111.69) / [2 (111.69)(0.000025)]

Effective Convexity = 19.355

2. Calculate the effective duration and effective convexity of a semi-annual coupon bond with three years remaining until maturity, a market value of $97.379 (per $100 of face value), a 7% coupon rate, and an 8% yield-to-maturity. Use the bond’s effective duration and effective convexity to determine the percentage change in the bond’s value if interest rates increase by 1.7%.

Using an HP 12C:

First, Calculate V₁ (By calculating the bond’s value if its yield increased 0.50%):

Enter Payment (annualized coupon rate):   PMT = 7
Enter Yield (annualized YTM):    I = 8.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012003 f PRICE

V₁ = $96.100

Second, Calculate V₂(By calculating the bond’s value if its yield decreased 0.50%):

Enter Payment (annualized coupon rate):    PMT = 7
Enter Yield (annualized YTM):    I = 7.5%
Enter Settlement Date (as m/d/yr):    1.012000 ENTER
Enter Maturity Date (as m/d/yr):    1.012003 f PRICE

V₂ = $98.679

Third, Calculate the bond’s effective duration using V₁ and V₂:

Effective Duration = (V₂ – V₁) / [2(V₀)(∆y)]

Effective Duration = (98.679 – 96.100) / [2 (97.379)(0.005)]

Effective Duration = 2.648%

Four, Calculate the bond’s effective convexity using V₁ and V₂:

Effective Convexity = (V₂ + V₁ – 2V₀) / [2(V₀)(∆y²)]

= (98.678 + 96.100 – (2) 97.379) / [2 (97.379)(0.000025)]

Effective Convexity = 4.107

Five, Determine the bond’s percentage change in price if interest rates increase 1.7%:

%∆ Price = -1(+.017)(2.648) + (4.107)(+.017)²

%∆ Price = -(.04502) + (.001187)

%∆ in Price = -.04383 or -4.383%