Yield to call and yield to worst

How do you calculate the yield to call?

A callable bond can be redeemed early at the issuer’s choice. If it is called, the investor receives the call price, not par, and the coupons stop on that date. The yield to call is the yield on that assumption.

$$P = \sum_{t=1}^{N} \frac{PMT}{(1 + r)^{t}} + \frac{K}{(1 + r)^{N}}, \qquad YTC = m \times r$$

$P$ is the flat price, $K$ the call price, $N$ the number of periods to the call date and $m$ the periods a year. It is the yield-to-maturity calculation with two changes: the horizon is the call date and the final payment is the call price.

Many bonds have a call price schedule : a list of dates, each with its own price, usually falling toward par. You compute one yield to call per date.

For the other dates, change only N and FV (and recompute I/Y). The coupon and price stay in memory.

Which yield is the yield to worst?

The investor does not choose when the bond is called, the issuer does. So the cautious quote is the lowest yield the investor could end up with. That is the yield to worst .

$$YTW = \min\left(YTC_1,\, YTC_2,\, \ldots,\, YTM\right)$$

Callable bond yield measuresthe issuer can end the bond early, so one yield is not enough

• Yield to maturity Memorize assumes the bond is never called
• Yield to call Memorize one per call date; call price is the final payment
• Yield to worst Memorize the lowest of the yields above
  • Premium bond worst case is usually an early call
  • Discount bond worst case is usually the YTM
• Option-adjusted yield Use it prices the call with an option model

In the schedule above, the yields run 4.842%, 4.865%, 4.887% and 5.215%. The lowest is the first call, so the yield to worst is 4.842%.

Which date is worst depends on the price:

• Premium bond. A call at a price below the market price is a loss of the premium, so an early call is usually worst.
• Discount bond. A call at par or above is a gain, so the issuer is unlikely to call, and the yield to maturity is usually the worst.

Go deeper Why yield to worst is crude

Yield to worst treats the most pessimistic outcome as certain. It gives no weight to how likely each call is, and it ignores the value of the call to the issuer. It is widely quoted because it is simple and conservative. The option-adjusted approach below, and the option-adjusted spread in the next lesson, price the call with a model instead.

For a putable bond the logic flips. The investor holds the put, so there is a yield to put, and the worst case is the lowest of the yield to maturity and the yields to each put date. The effective duration of callable bonds, covered in the curve-based risk measures module, also depends on the same option.

What are the option-adjusted price and yield?

The embedded call has a value to the issuer, so the investor pays less for a callable bond than for an otherwise identical option-free bond. Value of the call = option-free price − callable price.

An option pricing model, plus an assumption about future interest rate volatility, produces that value. Add it to the flat price of the callable bond and you have the option-adjusted price : the price of the same bond with the call removed.

$$P_{OA} = P_{\text{callable}} + V_{\text{call}}, \qquad V_{\text{call}} = P_{\text{option-free}} - P_{\text{callable}}$$

The option-adjusted yield is the market discount rate at which the bond is priced at that option-adjusted price.