CFA Program Level I

Yield to maturity, current yield and yield to worst: CFA Level I

Yield to maturity is the discount rate that makes the present value of all remaining coupons and principal equal the bond's price. Current yield is only the annual coupon divided by the price, and yield to worst is the lower of the yield to maturity and the yield to call. A bond-equivalent yield is a semiannual rate times two, so convert it before comparing it with an annual-compounding rate.

The yield measures, from simplest to fullest

Coupon rate: the annual interest rate set at issue. It does not change.

Current yield: annual coupon / current price. FINRA's example: a bond paying USD45 a year that trades at 103 (USD1,030) has a current yield of 45 / 1,030 = 4.37%, against a 4.5% coupon rate.

Yield to maturity (YTM): the discount rate at which the present value of all coupons and principal equals the price. It assumes the bond is held to maturity and payments are made on time. It is usually computed assuming coupons are reinvested at the same rate, which rarely happens, so it is an estimate, not the realised return.

Price and yield move in opposite directions. A bond priced above par has a YTM below its coupon rate; a bond priced below par has a YTM above it.

Worked example: YTM of a premium bond

The SEC's corporate bond bulletin uses a 10-year bond with a 4.00% coupon paid semiannually (USD20 every six months on USD1,000 face) priced at 110, or USD1,100.

YTM is the rate r that solves: 1,100 = sum of 20 / (1 + r/2)^k for k = 1 to 20, plus 1,000 / (1 + r/2)^20. Solving gives r = 2.84%, the figure the SEC reports.

The YTM is below the 4.00% coupon because the buyer pays USD100 above the face value that will be repaid.

Callable bonds: yield to call and yield to worst

Yield to call is computed like YTM, but uses the first call date in place of maturity and the call price in place of par.

Yield to worst is the lower of YTM and yield to call. FINRA calls it the most conservative potential return and says to check it for every callable bond.

Example: a callable bond has a YTM of 5.80% and a yield to call of 4.90%. Its yield to worst is 4.90%. If instead the yield to call were higher than the YTM, the yield to worst would be the YTM.

Converting between compounding bases

US Treasury yields are quoted as bond-equivalent yields: a semiannual rate stated on a simple annualised basis. They exclude compounding.

Effective annual yield = (1 + BEY/2)^2 - 1. Treasury's example: 8.00% becomes (1.04)^2 - 1 = 8.16%. For a 6.00% BEY: (1.03)^2 - 1 = 6.09%.

Continuously compounded yield = 2 x ln(1 + BEY/2). For a 6.00% BEY: 2 x ln(1.03) = 5.91%. More frequent compounding needs a lower stated rate for the same growth, so the continuous rate is the lowest of the three.

Where the exam catches people

Believing current yield is a total return measure: it ignores the gain or loss from price to par and the reinvestment of coupons.
Believing yield to worst is always the yield to call: it is whichever of YTM and yield to call is lower.
Believing a bond-equivalent yield is an effective annual rate: it is twice the semiannual rate, so its effective annual equivalent is higher.
Believing the YTM you buy at is the return you will earn: it assumes holding to maturity, no default and reinvestment at the same rate.
Believing a premium bond's YTM exceeds its coupon rate: paying above par lowers the yield below the coupon.

Written from CFA Institute's documentation

Try 3 questions on this

Pick an answer to see why each option is right or wrong.

A government bond's yield to maturity is quoted as 7.40 percent on a bond-equivalent basis, a semiannual rate stated on a simple annualized basis. To compare it with an investment that compounds once a year, an analyst converts it to an effective annual yield. The effective annual yield is closest to:
Right. In the game that’s Renown for your Fixed Income building.
Not that one: B is right. In the game your Fixed Income building would lose a little Renown, and this question would come back later.
Your answerA.7.40%7.40% is the quote itself. A bond-equivalent yield is not an effective annualized yield, because it leaves out the compounding of the two semiannual periods: (1.037)^2 - 1 = 7.54%.The answerB.7.54%A bond-equivalent yield is a semiannual rate stated on a simple annualized basis, so it excludes the effect of compounding. Convert with APY = (1 + I/2)^2 - 1 = (1 + 0.0740/2)^2 - 1 = 1.037^2 - 1 = 1.075369 - 1 = 7.54%.Your answerC.7.68%7.68% is e^0.0740 - 1, which treats 7.40 percent as a continuously compounded rate. The quote compounds twice a year, so the conversion is (1 + 0.0740/2)^2 - 1 = 7.54%.
Written from home.treasury.gov/policy-issues/financing-the-government/interest-rate-statistics/interest-rates-frequently-asked-questions
Talbridge Ltd's fixed-coupon bond trades below par and is callable at 102 per 100 of par on its first call date in two years. Its yield to maturity is 5.10 percent, and its yield to call, measured to that first call date at the call price, is 6.40 percent. The bond's yield to worst is closest to:
Right. In the game that’s Renown for your Fixed Income building.
Not that one: A is right. In the game your Fixed Income building would lose a little Renown, and this question would come back later.
The answerA.5.10 percentYield to worst is whichever of a bond's yield to maturity and yield to call is lower, giving the most conservative potential return on a callable bond.Your answerB.5.75 percent5.75 percent is the average of 5.10 and 6.40 percent. Yield to worst is a comparison, not an average: it is the lower of the yield to maturity and the yield to call, here 5.10 percent.Your answerC.6.40 percent6.40 percent is the yield to call. A call is not automatically the worst case: yield to worst is whichever of the two yields is lower, and here the yield to maturity, 5.10 percent, is lower.
Show the full explanation and doc quotes
Here the yield to maturity, 5.10 percent, is below the yield to call, 6.40 percent, so the yield to worst is 5.10 percent.
Written from www.finra.org/investors/insights/bond-yield-return
A four-year zero-coupon government bond has a yield of 5.00 percent stated on a coupon-equivalent basis, that is, with semiannual compounding. Expressed on a continuously compounded basis, the bond's yield is closest to:
Right. In the game that’s Renown for your Fixed Income building.
Not that one: A is right. In the game your Fixed Income building would lose a little Renown, and this question would come back later.
The answerA.4.94 percentThe continuously compounded yield y and the coupon-equivalent yield y_ce are linked by y = 2 ln(1 + y_ce/2). Here y = 2 x ln(1 + 0.05/2) = 2 x ln(1.025) = 2 x 0.024693 = 0.049385, or about 4.94 percent.Your answerB.5.00 percentYields stated on different compounding bases are not interchangeable. The same discount factor implies a continuously compounded yield of 2 ln(1.025) = 4.94 percent, below the 5.00 percent semiannual figure.Your answerC.5.06 percentThis computes (1.025)^2 - 1 = 5.06 percent, the annually compounded equivalent. A continuously compounded yield is 2 ln(1.025) = 4.94 percent; it is lower, not higher, than the semiannual rate.
Show the full explanation and doc quotes
With more frequent compounding, a lower stated rate produces the same growth, so the continuous rate sits below the semiannual one.
Written from www.federalreserve.gov/pubs/feds/2006/200628/200628pap.pdf