There is a five year, 5% annual coupon bond and the YTM’s of very similar 4 and 7 year bonds are 4.783% and 5.336% respectively. We are told to estimate the YTM on the 5 year as the yield on the 4 year + 1/3rd of the difference between the YTM of the 4 and 7 year:
4.783% + [(5-4)/(7-4)] * (5.336% - 4.738%)
My question:
Why do we use the difference between the 5 year and the 4 year, and the 4 year and the 7 year? Let me state a previous problem to show the root of my confusion:
Estimate the value of a 4% annual pay bond w/ 5 years to maturity. (all bonds have same rating)
4 year annual 5% coupon bond: YTM = 4.738
6 year annual 4% coupon bond: YTM = 5.232
6 year annual 6% coupon bond: YTM = 5.284
To solve, take the average of the two 6 year bonds, then average that with the 4 year bond. Plug in that average YTM into the calculator along with the attributes of the five year bond in question.
There we took the average YTM of the similar bonds and applied that YTM to the five year bond. The difference between the bond in question and the reference bonds were both one year (5 and 6, 5 and 4) but with the other problem, we are taking the difference between the bond in question and the shorter term reference bond and then the difference between the two reference bonds, not the bond in question and the longer term reference bond.
The first problem says all of the bonds are very similar, meaning same coupon, credit risk, etc. So you find the difference in the two YTMs and average it over the difference in the maturities (3 years) and multiply this “maturity premium” by the number of years (1) from the “base” bond (4-year bond) here. Then add this to the “base” YTM. It’s a linear equation-- start by breaking it into pieces to gain some insight.
The second problem requires a different approach since the coupons are different. You want a 4% coupon bond with 5-year maturity. If you average the two six year bonds, you have a 5% coupon and the YTM. Since the average of the two 6-year bonds gives a 5% coupon, you can now average it with the 4-year bond since they have the similar coupons. Now you have the yield for a 5-year bond (average between 4-year and 6-year). Plug in the YTM, 4% coupon, face value, and maturity to solve for PV.
At least thats the way it works for me. Let me know if this helps!
So, the solution for the second example: ’ To solve, take the average of the two 6 year bonds, then average that with the 4 year bond.’ is consistent with the first example.
In the second example we can simply take the average of 6 and 4 because we are looking for 5.
In the first example we are looking for 5, and have 4 and 7 at our disposal so we have to use this formula:
But I think you do need the coupon for the second question, since they are asking you for the price of the security, not only the yield. Without the coupon, you can’t estimate the price. I think you are right in the regard that I didn’t need to “make” matching coupons to solve, though.
For finding the yields, you’ve used the same equition, but carried through some algebra. You’ve used a weighted average in both questions.