fixed income 12: Yield-Based Bond Convexity and Portfolio Properties

What is convexity?

measures the curvature of the bond price–yield relationship.

Convexity vs duration

Convexity

• captures the second-order effect of yield changes.

• more accurate for large yield changes

Duration

• captures the first-order (linear) effect.

• assumes the price–yield relationship is a straight line.

• Uses the tangent line to the price-yield curve.

• accurate for small yield changes:

What is the duration-plus-convexity approximation formula?

$$\%\Delta P=-\left(\text{AnnModDur}\right)\cdot\Delta\text{yield}+\frac12\cdot\text{AnnConvexity}\cdot\left(\Delta\text{yield}\right)^2$$

Does convexity increase or decrease estimated bond prices?

Increase.

• Price rises larger when yields fall.

• Price declines smaller when yields rise.

cash-flow-based convexity formula

$$\text{(Time to receipt of CF) × (Time to receipt of CF+1) × (Weight of CF) × (1+Periodic YTM)}^{-periods}$$

$$\frac{t\cdot\left(t+1\right)\cdot w}{\left(1+r\right)^{m}}$$

What is the approximate convexity formula?

$$\text{ApproxCon}=\frac{PV^{-}+PV^{+}-2PV_0}{PV_0\cdot\left(\Delta Yield\right)^2}$$

• $PV_0$ ​ = Current price

• $PV_+$ ​ = Price if yield increases

• $PV_-$ ​ = Price if yield decreases

when will a fixed-rate bond have greater convexity:

1. the longer its time-to-maturity,

2. the lower its coupon rate, and

3. the lower its yield-to-maturity.

4. greater dispersion of cash flow

What is Money Convexity (MoneyCon)?

The convexity effect expressed in currency units.

What is the formula for Money Convexity?

$$\text{MoneyCon=AnnConvexity×}PV_{Full}$$

What does Money Duration measure?

first-order price change in currency units.

What formula combines Money Duration and Money Convexity?

$$\%\Delta P=-\left(\text{MoneyDur}\right)\cdot\Delta\text{yield}+\frac12\cdot\text{MoneyCon}\cdot\left(\Delta\text{yield}\right)^2$$

What are the two methods for calculating portfolio duration and convexity?

1. Weighted average of aggregate portfolio cash flows (theoretical method)

2. Weighted average of individual bond durations and convexities (practical method)

What weights are used when calculating portfolio duration and convexity and why?

Market value weights.

• Interest rate risk depends on market value exposure.

What is the formula for portfolio duration?

$$D_{p}=\sum_{}^{}w_{i}\cdot D_{i}$$

• $w_i$ ​= market value weight

• $D_i$ = duration of bond i

What is the formula for portfolio convexity?

$$C_{p}=\sum_{}^{}w_{i}\cdot C_{i}$$

• $w_i$ = market value weight

• $C_i$ = convexity of bond i

What is a parallel shift?

All yields across maturities move by the same amount and in the same direction.

• made when using portfolio duration and convexity

• not common in reality

What happens in reality instead of a parallel shift?

1. Steepening

• Long-term yields rise relative to short-term yields.

2. Flattening

• Difference between long- and short-term yields narrows.

3. Twisting

• Different maturities move by different amounts or directions.