# Yield to maturity as discount rate

Assume that face value of a bond is equal to 1000. The coupon rate is 3% and yield to maturity is 4%.How can we correlate coupon rate and YTM in order to explain the state of current bond price. (Maturity 10years-Redemption value is 1000).

My approach: $P_0=r C a(n,YTM) + \frac{P}{(1+YTM) ^{n}}= 0.03 * 1000 a(10,0.04) + \frac{1000}{(1+0.04) ^{10}}= 918.88$ \$.

The yield to maturity or equivalent the return that investors expect from the bond is 0.04. The bond price is less than the face value beacuse the coupon rate is 0.03. Investors are not interested in buying the bond at 1000 since they can earn 0.04 from a bond with the same characteristics and price. So the Market squizzes the price to 918.88

Can we generalize that YTM is the discount rate for bond with same duration and characteristics(coupon rate and risk)?

• What you describe is a bond with 10 year "maturity". You use the term "duration" but this term has a different meaning in Finance, it is not the same as maturity at all. As a result your question is very confusing to me... Of course if two bonds have exactly the same maturity coupon and risk etc. they are essentially the same and will have the same YTM (discount rate) by the "law of 1 price". But this is just obvious. – Alex C Jan 27 '18 at 23:55
• Agree with Alex C above. Furthermore, the term discount rate has a more specific meaning than how you are using it. In YTM is more like the IRR. In some instances the discount rate and the YTM will be identical but only in the event that the yield curve is flat (at 4% in your case). Also, since you are assuming that the coupon is annual and that you are pricing it at inception (or right after a coupon payment with 10Yrs to maturity). – AlRacoon Jan 28 '18 at 20:52