- CFA Exams
- 2024 Level I
- Topic 7. Fixed Income
- Learning Module 13. Curve-Based and Empirical Fixed-Income Risk Measures
- Subject 1. Curve-Based Interest Rate Risk Measures

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##### Subject 1. Curve-Based Interest Rate Risk Measures PDF Download

**Callable Bonds**

A callable bond exhibits positive convexity at high yield levels and negative convexity at low yield levels. Negative convexity means that for a large change in interest rates, the amount of the price appreciation is less than the amount of the price depreciation.

**Putable Bonds**

The difference between the value of a putable bond and the value of an otherwise comparable option-free bond is the value of the embedded put option.

**Effective duration**and

**effective convexity**describe the interest rate risk of bonds with embedded options.

**Effective Duration**

Although they appear similar, the approximate Modified Duration and Effective Duration are different. The Modified Duration is a yield duration statistic that measures interest rate risk with reference to a change in the yield-to-maturity (ΔYield) of the bond. On the other hand, effective duration is a

*curve duration statistic*that measures interest rate risk in terms of a parallel shift in the benchmark yield curve (ΔCurve).

Effective Duration is the best duration measure of interest rate risk when valuing bonds with embedded options because such bonds do not have well-defined internal rates of return (yield-to-maturity). Therefore, yield durations statistics such as Modified and Macaulay Durations do not apply.

**Effective Convexity**

_{-}+ PV

_{+}- 2 x PV

_{0})/[(ΔCurve)

^{2}x PV

_{0}]

Effective convexity includes the effects of yield changes on the cash flows. Effective convexity requires an adjustment in the estimated bond values (V-, V+) to reflect any change in estimated cash flows due to the presence of embedded options.

When bonds include embedded options, effective convexity may even be negative. This is in fact just what investors need if they are constructing an estimate that falls within a region of positive convexity, as may be the case with callable bonds.

- When the call option is out of money, the callable bond has positive convexity.
- When the call option is near the money, the effective convexity becomes negative. This is because the callable bond is capped by the price of the call option.
- Putable bonds always have positive convexity.

Since these measures are "effective" they must recognize that a shift in interest rates may bring about a change in cash flows, because embedded options may be exercised. In contrast, modified duration and convexity do not allow for the fact that the cash flows for a bond with an embedded option may change due to the exercise of the option.

In order to calculate effective duration and convexity, we use the binomial model to find both V -and V+. An important assumption is that OAS stays constant when interest rates shift.

A pricing model can be used to estimate the price change resulting from a change in the benchmark yield curve instead of the bond's own yield-to-maturity. V- and V+ are adjusted to reflect any changes in the cash flows (due to embedded options) that result from the change in benchmark yield curve.

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**User Contributed Comments**
6

User |
Comment |
---|---|

shiva5555 |
Of course, it's so obvious. |

johntan1979 |
The point where a callable bond's curve changes from convex to concave (negative convex) is when coupon rate = required yield |

Amrokken |
Shiva5555 it can be obvious for you and not for others, thanks Jonathan 1979 for clarifying. |

GBolt93 |
Are putable bonds very rare in actual practice? I've heard of callable bonds fairly frequently in school and generally, but this is my first time ever coming across putable bonds. |

ashish100 |
shiva if its so obvious, why are you here? just go take the exam without studying you pothead |

khalifa92 |
differentiate Stocks: called when prices go up to protect the issuer put when prices go down to protect the holder Bonds: called when market yield is lower to refinance at a lower rate. put when market yield is higher to re-invest at higher rates. |

I am happy to say that I passed! Your study notes certainly helped prepare me for what was the most difficult exam I had ever taken.