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Call me weird, but I like math. I looked at a few books on bond investing, but I couldn't find one that described how to compute the yield-to-maturity of a bond. Can anyone help me? Is there an empirical formula for this calculation?
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Hi Adam,
I just happen to be looking for the same info. though I don't love math. I found the following on Post #121346 on the Information Desk/Ask a Foolish Question Board. Heres the link:http://boards.fool.com/Message.asp?id=1010001047405001
Hope this helps,
Ginny
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AdamSmyth:

YTM calculations use the standard future value-present value formula:

FV=PV(1+r)^t

For YTM calculations, you solve for r:

(1+r)^t=FV/PV

1+r = (FV/PV)^(1/t)

r = [(FV/PV)^(1/t)] - 1

For example, you can buy a $1000 for $425 that matures in 14 years:

1000=(425)(1+r)^14

(1+r)^14 = 1000/425 = 2.353

1+r = (2.353)^(1/14) = 1.063

r = 6.3%

The bond would yield 6.3% if held to maturity.


Pat
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Next question is: you purchase the bond on Aug 24. It matures in 14 years on Nov 15.

Presumably time t is now 14 yrs plus a decimal representing the interval between Aug 24 and Nov 15.

How is this calculated? Do you use calendar days or business days?

Is published YTM based on todays date as the purchase date or the closest quarter, most recent dividend payment or year? Does this small difference make a large difference? Is there a convention?
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pauleckler:

The time period can be adjusted to any value that is convenient; t can be 14.56 years, if that is the time left to maturity. The YTM calculation will then account for the total time to maturity.


Any YTM rate would have to be based on the most recent bond quote, as the yield changes constantly. You bring up a good point in making sure you know the time frame for the YTM quote.


Pat
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YTM calculations use the standard future value-present value formula

I don't believe the computation is quite that easy. Let me rephrase the problem somewhat differently using a concrete example. Suppose we have a $1000.00 10-year government bond with a coupon rate of 8% that pays interest semi-annually. The price of this 10-year bond is $1102.85. What is the yield-to-maturity?

The yield-to-maturity computation is complicated because you have to factor in (1) the $102.85 capital loss you absorb since you only receive $1000.00 face value at maturity, and (2) the future value of the $40.00 interest payments you receive semi-annually are discounted at the YTM rate.

The answer to this problem, by the way, is 6.58%, but I'm trying to figure out how to compute this value in an empirical formula.
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It is more complicated, but is similar to calculating the discount rate of corporate debt. You set the present value of the bond equal to a series of future cash flows and solve for YTM. It would look like:

1102.85=[40/(1+YTM)^1] + [40/(1+YTM)^2] +

......+ [40/(1+YTM)^20]


$40 is the semi annual interest payment, with 20 pay periods in the calculation. This does not take into account tax concequences.

Pat
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Next question is: you purchase the bond on Aug 24. It matures in 14 years on Nov 15.

Presumably time t is now 14 yrs plus a decimal representing the interval between Aug 24 and Nov 15.

How is this calculated? Do you use calendar days or business days?

Is published YTM based on todays date as the purchase date or the closest quarter, most recent dividend payment or year? Does this small difference make a large difference? Is there a convention?


There are, unfortunately, many conventions for counting the days. The Securities Industry Association (SIA) publishes a whole book on how to do these calculations. The International Swap Dealers Association (ISDA) publishes their handbook that specify still other ways to do yield calculations. Looking throught the SIA Handbook's examples, the examples labeled "Corporate Bond" almost all use a 30/360 day count method.

The fraction you need is the fraction of a compounding period from Aug 24 to Nov 15. The number of days between 8/24 and 11/15 is (11-8)*30+(15-24)=81. The full compounding period is half of a 360 day year, or 180 days. So the fraction is 81/180=0.45.

To calculate the yield, you need to know the "full price" for the bond. Bond prices are quoted as "flat prices," but you actually pay the flat price plus the accrued interest. How much accrued interest is there? From 5/15 to 8/24 is (8-5)*30+(24-15)=99 days, so 99/180 = 0.55 times the semiannual coupon. Add this accrued interest to the flat price. Now you guess yield, and for the guessed yield you calculate the price as the sum of the coupons and principal discounted by (1+Y/2)^(N+0.45), when N will range from 0 for the coupon on Nov 15 to 28 for the principal at maturity. You guess new yields until you find one the generates the proper price.

Later in this thread you posited a variation of "suppose a government bond." Looking through the examples again, I see 30/360 day count method used for "Agency Bonds" and "Municipals," but "Actual/Actual" used for Treasuries. In Actual/Actual you count the actual calendar days from Aug 24 to Nov 15 (83) and the actual calendar days from May 15 to Nov 15 (184) and calculate the fraction as 83/184=0.451087.

Additional twists in the rules come into play if one of the dates is 31st or the last day of February, or the first or last coupon is a different length than the rest, or the yields are "adjusted" to account for dayroll conventions (when do you get paid if a regular pay date is a weekend of holiday?), or one of the more arcane daycount methods is specified.
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