What is the formula to calculate the yield to maturity?

If I have a corporate bond with the face value of 1,000 with a coupon rate of 9 and a current market value of 850 for 10 years what the yield to maturity
 

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Last updated April 6, 2020 Views 12,960 Applies to:
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See my previous posting to answer your previous question.

The purpose of this posting is to pull together a number of corrections.

My previous methods -- IRR and RATE -- assume that the coupon frequency is once per year.

Apparently it is more common to assume (if not known otherwise) that the coupon frequency is twice per year.

In that case, my previous methods calculate a periodic YTM, which must be annualized.

Consequently, the IRR model would have 21 periods numbered 0 to 20.  The periodic coupon payment is 45 instead of 90, to wit:  9%*1000/2.  The denominator is the coupon frequency:  2 per year for semiannual coupons.  And the final cash flow (period 20) is 1045 (1000 + 45).

Thus, YTM is:

=(1+IRR(A1:A21))^2 - 1

Likewise, using RATE, YTM is:

=(1+RATE(20,45,-850,1000))^2 - 1

where 20 is the number of coupons (10 years times 2 per year).

Both methods result in (annual) YTM of about 11.9047462769355%.

The method of annualization above is based on "Approximation of Yield Maturity".ppt (click here) [1] and other references.

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In contrast, the Excel YIELD function always returns an annual YTM.  For the semiannual coupons, the usage for your example is:

=YIELD(DATE(2013,1,1),DATE(2023,1,1),9%,850/10,1000/10,2,0)

The only difference is 2 in the 6th parameter (coupon frequency).  See the YIELD help page for alternatives.

That returns about 11.5700794317906%.

Note the large difference compared to the annualized IRR and RATE results above.  That is due to a difference of opinion about how to annualize the periodic YTM.

For semiannual coupons, apparently the YIELD function effectively multiplies the periodic YTM by 2.

(Note that 2*IRR(A1:A21) is indeed about 11.5700794317906%).

I would disagree, based on time-value principles.

However, the Excel YIELD function result agrees with the HP 12C result.  And I have a lot of confidence that the HP 12C financial methods were well-researched.

So I suspect that simply multiplying the periodic YTM by the number of coupons per year follows bond industry conventions.  (TBD)


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[1] http://foba.lakeheadu.ca/hartviksen/2039/Approximation%20Yield%20to%20Maturity.ppt

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Ellen wrote:
My school book shows that the yield to maturity = dollar amount of annual interest + face value - market value / the number of period of times which = market value + face vale / 2 = 60 + 1000 - 900 / 10 which again = 900 + 1000 / 2 which equals 0.074 or 7. 4. I have no idea where the number 2 comes from, and why it is used. Can you please explain this to me?

I believe your presentation of the formula is mangled.  But it appears that you are referring to a formula that approximates yield to maturity, to wit:


where:
coupon = periodic interest payment
principal = face value (normally)
price = current market value (normally)
numberOfCoupons = number of interest payment over the full term

(I think the approximation works best when the bond is sold at a discount; that is, the price is less than face value.)

In your original example, if the coupon frequency is once per year (my original assumption):
coupon = 90 (9% of face value)
principal = 1000 (face value)
price = 850
numberOfCoupons = 10 (10 years times 1 coupon/year)

The result of that approximation is about 11.35%.  Compare with the "exact" result of about 11.61% shown in my previous posting.

The best explanation of the formula that I found is "Approximation of Yield Maturity".ppt (click here) [1].  However, even that explanation needs to be "taken with a grain of salt"; that is, interpreted carefully, not read literally.

I have only seen that approximation used with examples of bonds that pay interest semiannually (twice per year, not annually as I assume).

However, the 2 in the denominator is not based on that assumption.  Instead, the denominator is simply the average of the principal and price, according to the aforementioned reference.

When using Excel, I see no point in using the approximation since we can "easily" compute the YTM "exactly".

However, you could set up the approximation calculation as follows:
A1, price:  850
A2, annual rate:  9%
A3, term (years):  10
A4, coupon frequency:  1
A5, face value:  1000
A6, approximate periodic YTM:
=( A2*A5/A4 + (A5-A1)/(A4*A3) ) / AVERAGE(A1,A5)

Note that if the coupon frequency is not 1, the approximate periodic YTM must be annualized as follows, according to the aforementioned reference (and others) [2]:

A7, YTM:
=(1+A6)^A4 - 1

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[1] http://foba.lakeheadu.ca/hartviksen/2039/Approximation%20Yield%20to%20Maturity.ppt

[2] But the Excel YIELD function appears to annualize effectively by =A4*A6.  See my subsequent "corrections" posting.

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