Fixed Income: Convexity


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Interest rate risk
  • Duration

Today:

  • More interest rate risk
  • Convexity
  • Callable bonds

Convexity

Interest rate risk and duration

  • Duration allows a linear approximation of the price-yield relationship
    • Where and why is it a bad approximation?

Moving beyond the linear approximation

  • Can we improve on the relationship?
    • Hint: think back to your math classes

\[ P(y + \Delta y) \approx P(y) + \frac{dP}{dy} \cdot \Delta y + 0.5 \cdot \frac{d^2P}{dy^2} \cdot (\Delta y)^2.\]

Expressed in returns, rather than prices:

\[ \frac{\Delta P}{P(y)} \approx \frac{1}{P}\cdot\frac{dP}{dy} \cdot \Delta y + 0.5 \frac{1}{P} \frac{d^2P}{dy^2} \cdot (\Delta y)^2. \]

Convexity

  • Convexity captures curvature of the pricing function
    • second derivative of price w.r.t. yield, scaled by price.

\[ \text{convexity} = \frac{1}{P} \cdot \frac{d^2 P}{dy^2}\]

  • For coupon bonds,\[ \text{convexity} = \frac{1}{(1+y/m)^2} \left[\sum_{i=1}^T \frac{i(i+1)}{m^2} \cdot \frac{PV(CF_{t_i})}{P} \right]. \]

Price and return approximations

The second-order price approximation is:

\[ P(y + \Delta y) \approx P(y) - \text{mduration}\cdot P(y)\cdot \Delta y + 0.5 \cdot \text{convexity}\cdot P(y) \cdot (\Delta y)^2.\]

The second-order return approximation is:

\[ \frac{\Delta P}{P(y)} \approx -\text{mduration} \cdot \Delta y + 0.5\cdot\text{convexity}\cdot (\Delta y)^2.\]

  • Let’s take a look at today’s notebook to see how this approximation performs.

Desirability of convexity

Positive convexity is desirable for investors

  • For a fixed rate change magnitude, bond prices rise when rates fall by more than they fall when rates rise
  • Example: coupon bonds

Negative convexity is undesirable for investors

  • Instead, bond issuers like negative convexity
  • Examples: callable bonds, mortgages

Callable Bonds

Call Schedules

Callable bond: the issuer has the right to call (repurchase) the bond at specified times at pre-determined price(s)

  • usually a call schedule with call prices at specified call dates
    • first call price may be at premium over par value
    • call prices step down toward par later in call schedule
    • investors may be protected against call for an initial window
  • issuers usually offer a higher coupon as compensation for the call option

Interest rate risk

If rates fall,

  • the bond price rises,
  • the PV of future payment obligations for the firm may exceed the call price of the bond,
  • the issuer benefits from calling the bond and reissuing debt at a lower coupon rate.

This creates a ceiling for the bond value at the call price.

Callable vs. straight bond prices

Interest rate risk

At low interest rates, callable debt exhibits negative convexity.

  • For a fixed rate change magnitude, bond prices rise when rates fall by less than they fall when rates rise
  • this is undesirable for investors (hence higher coupon rates as compensation)

An aside: Yield to Call

We can calculate the IRR of paying today’s price and receiving cash flows to a call date:

\[ P = \sum_{t=1}^{T_{\text{call}}}\frac{C}{(1+\frac{y_{\text{call}}}{m})^t}+\frac{\text{Call Price}}{(1+\frac{y_{\text{call}}}{m})^{T_{\text{call}}}} \]

  • \(y_{\text{call}}\): the annual yield-to-call
  • \(T_{\text{call}}\): number of periods until the assumed call date
  • \(m\): number of payments per year

Estimating Duration and Convexity

Modified Duration

  • Suppose we observe prices at three yields
    • \(P_0 \equiv P(y_0)\)
    • \(P_{+} \equiv P(y_0 + \Delta y)\)
    • \(P_{-} \equiv P(y_0 - \Delta y)\)

An empirical estimate of modified duration at \(y_0\) is:

\[ \widehat{\text{mduration}} = \frac{1}{P_0} \frac{P_{-}-P_{+}}{2\Delta y}.\]

Convexity

An empirical estimate of convexity at \(y_0\) is:

\[ \widehat{\text{convexity}} = \frac{1}{P_0} \frac{(P_{-} -P_0)-(P_0-P_{+})}{(\Delta y)^2}.\]

For next time: Credit Risk





BUSI 448