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Bond yield and callable bonds

Given the price, the yield is the root of the price equation; for a callable bond you price to the WORST call date for the buyer.

The formulas

Yield equation

$$P=Fr{a}_{\overline{n}|i}+C{v}^n$$

solve for i

Current yield

$$\frac{Fr}{P}$$

an approximation only — ignores the capital gain or loss

Callable rule

$$\text{premium bond}\Rightarrow \text{price to the EARLIEST call}$$

Callable rule

$$\text{discount bond}\Rightarrow \text{price to the LATEST call}$$

Where it comes from

1. Price is a strictly decreasing function of the yield, so the price equation has exactly one positive root, found by bisection or by a financial calculator's IRR.
2. The issuer chooses the call date, and will choose the one WORST for the investor, so the investor must assume that date when deciding a price.
3. A premium bond loses value as it approaches redemption, so early call hurts most; a discount bond gains, so late call hurts most.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Price above par ⇒ yield below coupon rate. The two always sit on opposite sides.
• Callable: premium → earliest call, discount → latest call. 'The issuer is not your friend.'

Traps

• Quoting the current yield as the yield to maturity.
• Pricing a callable premium bond to maturity and overpaying.

Related

• Bond pricing
• Internal rate of return

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