The Four Yields

The four definitions

Nominal yield is the coupon rate: annual interest divided by par. It is fixed for the life of the bond and takes no account of what you paid.

Current yield is annual interest divided by the current market price. It accounts for the price you paid but ignores the gain or loss at maturity, and ignores the timing of the cash flows. A $1,000 par bond with a 6 percent coupon trading at $800 has a current yield of $60 over $800, which is 7.5 percent.

Yield to maturity is the total annualized return if you buy at the current price, hold to maturity, and reinvest the coupons at the same yield. It accounts for the coupon, the price paid and the gain or loss at maturity, and it is the number the market means when it says 'yield.' It is also the number that is genuinely hard to compute by hand, which is why the exam mostly asks you to order it rather than to produce it.

Yield to call is the same calculation run to the call date and the call price instead of the maturity date and par.

The ordering rule

Learn this as a picture rather than as two lists.

For a discount bond — one trading below par — you get the coupon plus a capital gain at maturity, so each successive measure that captures more of the return is larger. Nominal is the smallest, then current, then yield to maturity, then yield to call is largest of all, because a call brings the gain forward and earns it over a shorter period.

For a premium bond — one trading above par — you get the coupon minus a capital loss at maturity, so each successive measure is smaller. Nominal is largest, then current, then yield to maturity, then yield to call is smallest of all, because a call forces the loss to be absorbed over fewer years.

The compression that survives exam pressure: for a discount bond the yields ascend in the order nominal, current, YTM, YTC; for a premium bond they descend in that same order. Note that the sequence of names does not change — only the direction does.

At par, all four are equal to the coupon, assuming the call price is par.

Approximating yield to maturity

You will not have a financial calculator. The approximation the exam tolerates is straightforward and worth practising until it is quick.

Annual interest, plus or minus the annualized gain or loss to maturity, divided by the average of price and par.

Worked example. A 6 percent bond, ten years to maturity, trading at $800. Annual interest is $60. The gain to maturity is $200 spread over ten years, which is $20 a year. The numerator is therefore $80. The average of $800 and $1,000 is $900. Eighty over nine hundred is about 8.9 percent. Note that this comfortably exceeds the current yield of 7.5 percent, which exceeds the nominal 6 percent — the ordering rule holding, as it must.

Premium example. A 6 percent bond, five years to maturity, trading at $1,100. Annual interest $60, annualized loss $100 over five years, or $20 a year, so the numerator is $40. The average of $1,100 and $1,000 is $1,050. Forty over 1,050 is about 3.8 percent, below the current yield of 5.45 percent, which is below the 6 percent nominal.

The approximation is not exact — it treats the reinvestment crudely — but it lands close enough to select among four multiple-choice options, which is its only job.

Yield to worst, and why premium bonds are quoted to the call

Yield to worst is the lowest of the yields computable across all the possible redemption dates: maturity and every call date. It is what a dealer must disclose, because it is the return a buyer is actually assured of in the least favourable outcome.

For a premium bond, yield to call is lower than yield to maturity, so yield to worst is the yield to call — usually to the nearest call date. For a discount bond, yield to call is higher, so yield to worst is the yield to maturity. This is the mechanical reason that a premium bond is quoted to the call and a discount bond is quoted to maturity, and MSRB Rule G-15 requires municipal confirmations to disclose yield computed to the lower of call or maturity.

Call risk deserves its own moment because it is asymmetric and customers routinely misunderstand it. An issuer calls when rates have fallen. That is precisely when the holder would most like to keep a high coupon and least wants to reinvest. So the call takes away the best outcome and leaves the worst — and the reinvestment problem it creates has a name, reinvestment risk, which the exam pairs with call risk in the same breath. Call protection, a stated number of years during which the bond cannot be called, is what a buyer pays up for.

One class of bond is free of reinvestment risk entirely: a zero-coupon bond, because there are no coupons to reinvest. That is the answer whenever a question asks which instrument best locks in a known future sum for a known future need, such as a college tuition bill.

The yield curve and taxable equivalent yield

The yield curve plots yield against maturity for bonds of similar credit quality. A normal or positive curve slopes upward: longer maturities yield more, compensating for greater uncertainty. A flat curve shows little difference across maturities. An inverted curve slopes downward, with short maturities yielding more than long — historically associated with tight monetary policy and with expectations of falling rates ahead.

Taxable equivalent yield converts a tax-exempt municipal yield into the taxable yield that would leave an investor equally well off. Divide the municipal yield by one minus the investor's marginal tax rate. A 4 percent municipal to an investor in the 32 percent bracket is equivalent to 4 divided by 0.68, which is about 5.88 percent taxable.

Run in reverse — multiply a taxable yield by one minus the tax rate — it gives the after-tax yield of a corporate bond for comparison with a municipal. Module 4 does the full municipal tax treatment, including the state-level layer that changes the arithmetic for an in-state buyer.