Feb 20

Yield to maturity - 9 minute screencast

by David Harper, CFA, FRM, CIPM

FRM |

A bond's yield to maturity (yield) is its internal rate of return (IRR): the rate that discounts future cash flows to the current market price. An important assumption is that coupons are reinvested at the same YTM; if they are not, realized yield will vary from YTM (this is reinvestment risk).

In this tutorial, I calculate yield with both Excel and the TI BA II+. In both cases, we are just using time value of money functions. So, in each case we are solving for the unknown yield (rate in Excel, I/Y in the calculator) given four inputs:

ytm2

Here is the screencast:

Comments

  1. William Little 20 Feb 2008

    Quote from your article: “An important assumption is that coupons are reinvested at the same YTM; if they are not, realized yield will vary from YTM (this is reinvestment risk)”.

    NO ...
    When calculating the yield to maturity of a bond or other project which is characterised by a cash flow, the YTM or IRR does not have its coupon reinvested and if so, it is then a different project.
    If coupons are reinvested, you cannot call the results, YTM or IRR since they then are a modified IRR ie MIRR.
    For YTM or IRR projects, the coupons can be spent, given away or put in a tin can - it does not affect the YTM or IRR.

  2. David Harper, CFA, FRM, CIPM 20 Feb 2008

    Hi William,

    I disagree. The yield to maturity assumes reinvestment of coupons @ YTM. If the coupons are, for example, put in a tin can (@ say 0%), the realized return, as reflected by the dollar return at the end of the period, will be less than the YTM. This is a known weakness of YTM. I am not saying reinvestment changes the definition of YTM, but rather, the calculation of YTM implies reinvestment at YTM. I am referring to the difference between (ex ante) YTM and (ex post) realized yield.

    But please clarify if you disagree, I admit i’ve long assumed this and it’s been a while since i saw an example....

    David

  3. David Harper, CFA, FRM, CIPM 20 Feb 2008

    William,

    I was curious, so i recreated Fabozzi’s example (Fixed Income Analysis, 2nd edition).

    This is read/write version that illustrates the point:
    http://www.editgrid.com/bt/frm_2008/YTM_reinvested

    His numbers are inconvenient, but the baseline is investing in an 8% CD for 8 years. His point is that we should focus on dollar return; i.e., $82.21 in this case

    Then compare two cases: first, without reinvesting coupons. If you don’t reinvest coupons, your dollar return is a shortfall (61.83).

    Second, assume coupons reinvest at 8% (the YTM). Under this case, dollar return matches. Hence Fabozzi on YTM “it does consider reinvestment income; however, it assumes that the coupon payments can be reinvested at an interest rate equal to the yield to maturity” (p.122(

  4. William Little 21 Feb 2008

    David,

    For the example you quote:

    If the coupons are paid out on the coupon date - every six months, then the the IRR of this Bond is 8%. These coupons take no further part in the Bond performance.
    Now to define YTM.: The YTM is the discount rate which causes the sum of the present values of all the future cash flows of the Bond to equal the initial value of the Bond (94.17). This discount rate is also called the IRR.
    Reinvesting coupons at any rate does not change the value of the above defined YTM.

    Alternatively, if the coupons are reinvested externally until maturity at the the same rate as the YTM or IRR (8%), then the overall rate of return for this mixed internal/external investment project is also 8%.
    This is a modified IRR called MIRR and represents the yield on the initial investment (94.17) until maturity.

    For these two cases, the dollar values generated differ and occur at different times.

    If the reinvestment rate were less than the YTM then the MIRR will also be less than the YTM.

    All I am saying is that for a Bond, if you reinvest coupons then you get a MIRR - only if you reinvest at the same rate as the YTM will the MIRR have the same value as the YTM.

  5. David Harper, CFA, FRM, CIPM 26 Feb 2008

    William,

    Thanks for your clarifications. Can I assume we agree about my original statement: realized yield will equal YTM only if coupons are reinvested at YTM. My point was meant to be about realized yield: that ex ante YTM only matches ex post REALIZED yield if coupons are reinvested at YTM. (that’s just my way to speak the spreadsheet outcome above). If you still disagree with that, I’d still (sincerely) hope to understand where i may have gone wrong..

    And, to the degree i am correctly following you, I think you are improving on that statement by adding: the YTM (as disinct from the realized yield) does not require a reinvestment assumption. When you add a reinvestment to the discounting, you are calculating an MIRR. If the reinvestment happens to be the YTM, then it’s an IRR (put another way, IRR is narrow case of MIRR?). If I got that right, thanks for your help on this because I really hadn’t thought of it this way. If not, i’ll take more help…

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