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# Interpretation of Yield-To-Maturity

#### filip313

##### New Member
Subscriber
Hi,

I have a question on the assumptions behind Yield-To-Maturity.

I have read the Yield-To-Maturity (YTM) chapter on the Tuckman (chapter 3 on my edition) that explains why YTM is a measure of the realized return to maturity of a bond. My understanding of the explanation is as follow:
If every coupon received between now and maturity is reinvested at the YTM rate, at maturity, the future values of all these coupons + face value will be equal to the future value of the current price of the bond invested at the YTM rate. In other words, the YTM concept assumes that the prevailing rate in the market will be equal to YTM for the entire life of the bond.

My question is:
Is the concept of YTM consistent with the concept of implied forward curve?
In other words, does the YTM concept assumes that the 1 year rate in 1 year time will be equal to the current 1 year implied forward rate or does it assume that it will be equal to the current 1 year spot rate?
The no-arbitrage theory tells me that the prevailing rate in 1 year should be equal to current implied forward rate, hence how can I assume, as the YTM concept seems to do, that the prevailing rate remains the same for the entire life of the bond?

Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @filip313 Super good question! I think the answer is, no. YTM is not congruent with the implied forward curve, unless the curve is flat. See below, I did this quickly (I may clean up and add to our R28 Tuckman XLS which captures most of Tuckman's extant scenarios. My below is dead simple (XLS is here https://www.dropbox.com/s/iymllky1gcif19d/0112-bond-forward.xlsx?dl=0)

This is 10.0% coupon on a 3-year bond when the spot rate is up steep at 1.0%, 2.0%, and 3% (annual compound freq). The price of this bond is $120.18 and the yield is 2.88%. My Cum'l row shows how the cumulative future cash position is$130.87. Just as you suggest, the yield is only realized the return if the coupons are re-invested at the yield. If each of the two $10.00 coupons are reinvested at the 2.88% yield, the buyer holds$130.87 at the end of three years; same as (130.87/120.18)^(1/3)-1 = 2.88%.

Below that I calculate the return under the assumption of realized forwards. Specifically, the first $10.00 coupon is reinvested at the original F(1,3) = 4.01% and the second$10.00 coupon is reinvested at the original F(2,3) = 5.03%. This bond returns 3.0%; i.e., the original 3-year spot rate. This confirms Tuckman's "Which of the following two strategies is more profitable, rolling over one-period bonds or investing in a long-term bond and reinvesting coupons at prevailing short-term rates? As just demonstrated, if forward rates are realized, the two strategies are equally profitable" and shows how a realized forward is different than -- in this case of the upward sloping spot rate curve, more profitable than -- realizing the yield. This is intuitive: yield is a single factor, a (complex) average of the spot rates; it's not realistic to expect it to stay constant if the forwards are realized in a non-flat curve. Great question on your part, I hope that helps!

#### filip313

##### New Member
Subscriber
Thanks David.
It is clear now. The worked example makes it really easy to understand!