Present Value and YTM
Present Value and YTM
We've introduced the relation between the present value of the bond and the YTM of it. In this section, we'll switch the gear and look deeper into this the relation.
In the graph below, the Y axis is the present value of a bond, and the X axis is the YTM of the same bond. This annual coupon bond has a face value of 1000 and coupon rate is 10%. The maturity of the bond is sliding between 1 and 30 years.
What can we learn from the graph?
(1) In general the present value of the bond goes in the opposite direction with the YTM. The higher the YTM is, the lower the PV of the bond is.
(2) The shorter the maturity of the bond, the flatter the curve is. In other words, if the maturity is short, then the PV fluctuate less compared with when the maturity is long. This fluctuation of bond price given the variation of the YTM is called the interest risk of the bond. Clearly, the shorter the maturity, the lower the interest risk of the bond.
(3) No matter how the maturity changes, the curve always goes through one point: (10%, 1000). This implies that, when the YTM is 10%, the PV of the bond is always 1000, regardless of the change of maturity.
Then you may realize, wait, isn't 10% the coupon rate and the 1000 the face value of the bond?
Nice catch. Indeed it is no coincidence.
We have the following rules, which you can confirm from the graph above as well:
(1) if YTM > Coupon Rate, then PV < FV
(2) if YTM = Coupon Rate, then PV = FV
(3) if YTM < Coupon Rate, then PV > FV