# Callable Bond: Negative Convexity and Yield to Worst

Drag the yield on a callable bond: the call caps the price, convexity turns negative and yield to worst replaces yield to maturity.

## About this tool

A callable bond lets the issuer redeem it early at a fixed call price. Investors are short that option, so when rates fall the price stops near the call price instead of rising with the straight bond. This lab prices the call, shows where convexity turns negative, compares yield to maturity with yield to worst, and marks the dates on which the issuer would call.

## Frequently asked questions

### What is negative convexity?

The price–yield curve bends the wrong way: as yields fall the price rises ever more slowly because a call becomes likelier, so a fall in yield helps less than a rise hurts. The lab measures it as effective convexity, from prices at ±25 bp.

### What is yield to worst?

The lowest yield an investor can earn over maturity and every possible call date. For a bond trading above the call price it is usually the yield to the first call, which is why yield to maturity overstates the return.

### When does the issuer call?

When refinancing is cheaper than the coupons: if the remaining bond is worth more than the call price at the current yield, the issuer gains by redeeming. The timeline marks each call date this way and highlights the first one.

### What is effective duration?

Duration measured from the actual price change when the whole yield curve moves by a small amount, here 25 bp each way. It is needed because the cash flows of a callable bond change with yields, so the usual duration formula does not apply.

## Learn the theory

- [Callable bonds and negative convexity](https://quantsandbox.dev/guide/fixed-income/callable-bonds): The issuer's call option, price compression near the call price, yield to call and negative convexity.
- [Duration and convexity](https://quantsandbox.dev/guide/fixed-income/duration-convexity): Macaulay and modified duration, first- and second-order approximations of interest-rate risk.

Open the interactive tool: https://quantsandbox.dev/callable-bond