4.1.5 Duration and Convexity

5. Duration and Convexity

5.1 Duration

Duration is the sensitivity of bond’s full price to changes in the bond’s YTM or in benchmark interest rates.

• Longer time-to-maturity usually leads to higher duration.
• Higher coupon rate leads to lower duration.
• Higher yield-to-maturity leads to lower duration.

5.1.1 Yield Duration

Macaulay duration: the average time the bond holder has to wait before receiving the present value.

$$\text{MacDur}=\frac{{\sum }_{t=1}^{n}t\times PVC{F}_t}{\sum PVC{F}_t}$$

For a plain bond, the Macaulay duration is less than or equal to its maturity.

For a zero coupon bond, the Macaulay duration equals to its maturity.

Consider a two-year bond that provides annual coupons at the rate of $6\%$. The YTM of the bond is $5\%$.

$N=2$, $1/Y=5\%$, $PMT=6$, $FV=100$ $\to PV=-101.8594$

$$\text{MacDur}=1\times \frac{\frac{6}{1+5\%}}{101.8594}+2\times \frac{\frac{106}{(1+5\%{)}^2}}{101.8594}=1.9439$$

Modified duration: provides a linear estimate of the percentage price change for a bond given a change in yield.

$$\text{ModDur}=-\frac{\Delta P/P}{\Delta y}=\frac{\text{MacDur}}{1+y/m}$$

$$\Delta P\approx -\text{ModDur}\times \Delta y\times P$$

The Macaulay duration applies in the situation where $y$ is measured with continuous compounding.

$$\Delta P\approx -\text{MacDur}\times \Delta y\times P$$

Dollar duration: a measure of the dollar change in a bond’s value to a change in the yield.

$$\text{DollarDur(D)}=-\frac{\Delta P}{\Delta y}=\text{ModDur}\times P$$

$$\Delta P\approx -\text{DollarDur(D)}\times \Delta y$$

DV01: describes the impact of a one-basis-point($0.0001$)change in interest rates on the value of a portfolio.
$$\text{DV01}=-\frac{\Delta P}{10,000\times \Delta y}=\frac{\text{DollarDur}}{10,000}=\text{ModDur}\times P\times 0.0001$$

5.1.2 Curve Duration

The One Factor Assumption: assumes that all interest rates move by the same amount, which means the shape of the term structure never changes(parallel shift).

Curve Duration: used for bonds with embedded option due to uncertain future cash flow.

Effective Duration: describes the percentage change in the price of a bond, due to a small change in all rates.

$$D^E=-\frac{\Delta P/P}{\Delta r}=\frac{P_{-\Delta y}-P_{+\Delta y}}{2P_0\Delta y}$$

• $P_0$: initial observed bond price
• $\Delta y$: change in required yield

Consider a portfolio consists of a Treasury bond with a face value of USD 1 million paying a $10\%$ per annum coupon semi-annually. The tenor of this bond is one year. Suppose that spot rates are as shown in the table below. Calculate the DV01 and effective duration, if spot rates move $5$ bp.

Maturity(Years)	Rate(%)	+5bp Rate(%)	-5bp Rate(%)
0.5	7.0	7.05	6.95
1.0	7.5	7.55	7.45

The value of the bond is:
$\frac{50,000}{1.035}+\frac{1,050,000}{1.0375^2}=1,023,777.32$

The rates increase by five basis points, the value of the bond:
$\frac{50,000}{1.03525}+\frac{1,050,000}{1.03775^2}=1,023,295.72$

$DV01^{\prime }=\frac{1,023,777.32-1,023,295.72}{5}=96.32$

The rates decrease by five basis points, the value of the bond:
$\frac{50,000}{1.03475}+\frac{1,050,000}{1.03725^2}=1,024,259.26$

$DV01^{\prime \prime }=\frac{1,024,259.26-1,023,777.32}{5}=96.39$

The two estimates of DV01 differ slightly because the bond’s price is not exactly a linear function of interest rates. We can get a good estimate of DV01 by averaging the two estimates:

$DV01=(96.32+96.39)/2=96.355$

$$D^E=\frac{P_{-\Delta y}-P_{+\Delta y}}{2P_0\Delta y}=\frac{1,024,259.26-1,023,295.72}{2\times 1,023,777.32\times 0.05\%}=0.9412$$

5.1.3 Limitations of Duration

Duration provides a good approximation of the effect of a small parallel shift in the interest rate term structure.

But these equations cannot be relied upon, if the change in the bond yield arises from a non-parallel shift in the interest rate term structure or the change is large.

5.2 Convexity

5.2.1 Convexity

Convexity(凸度) measures the non-linear relationship of bond prices to changes in interest rates and the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes.

The duration plus convexity approximation fits a quadratic function and captures some of the curvature, which provides a better approximation.

• Duration overestimates(高估) the magnitude of price decreases
• Duration underestimates(低估) the magnitude of price increases
  
  $$\text{MacConvexity}=\frac{{\sum }_{t=1}^n{t}^2\times {PVCF}_t}{\sum {PVCF}_t}$$

Consider a bond that provides annual coupons at the rate of $6\%$. The maturity is 2 years. The YTM is $5\%$. Calculate the Macaulay convexity of it.

$N=1$, $1/Y=5$, $PMT=6$, $FV=100\to PV=-101.8594$

$$\text{MacCovexity}=1^2\times \frac{\frac{6}{(1+5\%)}}{101.8594}+2^2\times \frac{\frac{106}{(1+5\%{)}^2}}{101.8594}=3.8317$$

$$\text{ModifiedConvexity}=\frac{\text{MacConvexity}}{(1+y/m{)}^2}$$

Consider a bond a bond’s Macaulay Convexity is $8.13904$. If the bond is compounded semi-annually and YTM is $5.2455\%$, calculate the modified convexity of it.
$$\text{ModifiedConvexity}=\frac{8.13904}{(1+5.2455\%/2{)}^2}=7.7283$$

Effective convexity measures the sensitivity of the duration measure to changes in interest rates. The effective convexity($C^E$) of a position worth $P$ can be estimated as

$$C^E=\frac{1}{P}\left[\frac{P^{+}+P^{-}-2P}{(\Delta y{)}^2}\right]$$

5.2.2 Price Approximation

Duration provides a good approximation when there is small parallel shift in the interest rate term structure. However, it will provide a poor approximation if there’s non-parallel shift or the change is large.

The effect of parallel shifts of interest rate term structure can be more accurate by adding convexity analysis to the analysis of duration.

With continuous compounding

$\Delta P/P\approx -\text{MacDur}\ast \Delta y+\frac{1}{2}\ast \text{MacConvexity}\ast (\Delta y{)}^2$

$\Delta P\approx -\text{MacDur}\ast P\ast \Delta y+\frac{1}{2}\ast \text{MacConvexity}\ast P\ast (\Delta y{)}^2$

With discrete compounding frequencies
$\Delta P/P\approx -\text{ModDur}\ast \Delta y+\frac{1}{2}\ast \text{ModConvexity}\ast (\Delta y{)}^2$

$\Delta P\approx -\text{ModDur}\ast P\ast \Delta y+\frac{1}{2}\ast \text{ModConvexity}\ast P\ast (\Delta y{)}^2$

Suppose a bond with modified duration of $31.32$ and modified convexity of $667$, when its yield is expected to fall by $50$ bps, what should be the expected percentage price change?

$\Delta P/P=-31.32\ast (-0.0050)+\frac{1}{2}\ast 667\ast (-0.0050{)}^2=16.49\%$

5.2.3 Negative Convexity

A callable bond gives the issuer the right to redeem all or part of the bond before the specified maturity date.

Most mortgage bonds are negatively convex, and callable bonds usually exhibit negative convexity at lower yield.

In a vanilla bond(without embedded options) we can typically use modified and effective interchangeably. When the bond contains embedded options, we prefer effective duration and effective convexity.

Putable bonds often have higher positive convexity, especially when interest rates are high.

The security with more convexity outperforms the less convex security in both bull (rising price) and bear(failing price) markets.

The bigger the volatility of the interest rate, the greater the gains from the positive convexity.

• Long volatility of the interest rate $\to$ choosing a security with positive convexity
• Short volatility of the interest rate $\to$ choosing a security with negative convexity

5.2.4 Portfolio Calculations

In regard to both modified(effective) duration and convexity, portfolio duration and convexity equal the weighted sum of individual, respectively, durations and convexities where each component’s weight is its value as a percentage of portfolio value.
$$D_{\text{portfolio}}=\sum _{i=1}^n{w}_i\times D_i$$

$$C_{\text{portfolio}}=\sum _{i=1}^n{w}_i\times C_i$$

$w_i$: the bond’s market value to the whole portfolio value

DV01: The DV01 for a portfolio is simply the sum of the DV01s of the components of the portfolio.

5.2.5 Barbell vs. Bullet Portfolio

Barbell portfolio: securities in this portfolio concentrate in short and long maturities but less intermediate maturities.

Bullet portfolio: has more exposure at intermediate maturities.

For bonds with same duration, the one that has the greater dispersion of cash flows has the greater convexity.

A manager purchase $1 million Bond B. The coupon payments are semi-annual. Using A and C to construct a portfolio with the same cost and duration.

Bond	Coupon-Semi	Maturity	Price	Yield	Duration	Convexity
A	2%	5	95.3889	3%	4.7060	25.16
B	4%	10	100	4%	8.1755	79
C	6%	30	115.4543	5%	14.9120	331.73

$$\{\begin{array}{l}{V}_A+V_C=1\text{million}\\ \frac{V_A\times 4.7060}{1\text{million}}+\frac{V_C\times 14.9120}{1\text{million}}=8.1755\end{array}\to \{\begin{array}{l}{V}_A=0.66\text{million}\\ V_C=0.34\text{million}\end{array}$$

${\text{Convexity}}_{A+C}=0.66\times 25.16+0.34\times 331.73=129.39$

Advantage for barbell portfolio:

• These two strategies will have the same duration and different convexity.
• The barbell strategies produces a better result when there is a parallel shift in the yield curve.

Disadvantage for barbell portfolio: The bullet investment would perform better than the barbell investment for many non-parallel shifts.

5.3 Hedging

5.3.1 Duration Hedging

To construct a portfolio that can hedge a small change in interest rates.

$\Delta$(Price change of underlying asset) + $\Delta$(Price change of hedging instrument) = 0

If $\text{DV01}$ is expressed in terms of a fixed face amount, hedging a position of $F_A$ face amount of security A required a position of $F_B$ of security B where:

$$F_B=\frac{F_A\times {\text{DV01}}_A}{{\text{DV01}}_B}$$

5.3.2 Duration and Convexity Hedging

We can make both duration and convexity zero by choosing $P_1$ and $P_2$ so that:

$V\times D_V+P_1{D}_1+P_2{D}_2=0$

$V\times C_V+P_1{C}_1+P_2{C}_2=0$

The position is hedged against relatively large parallel shifts in the term structure. However, it will still have exposure to non-parallel shifts.

An investor has a bond position worth USD $20,000$ with a duration of $7$ and a convexity of $33$. Two bonds are available for hedging. Bond A has a duration of $10$ and a convexity of $80$. Bond B has a duration of six and a convexity of $25$. How can a duration plus convexity hedge be set up?

$$\{\begin{array}{l}10P_A+6P_B+20,000\times 7=0\\ 80P_A+25P_B+20,000\times 33=0\end{array}\to \{\begin{array}{l}{P}_A=-2,000\\ P_B=-20,000\end{array}$$

A short position A of USD $2,000$ and a short position B of USD $20,000$ are required.