One bond, with a face value of 1000 dollars and annual coupons at a rate of 7.6 percent effective, has a price of 1146.17 dollars. A second bond, with a face value of 1000 dollars and annual coupons at a rate of 6.1 percent effective, has a price of 1050.43 dollars. Both bonds are redeemable at par in the same number of years, and have the same yield rate. Find the yield rate. ?
What is the formula to find the yield rate? I can't find any that include the variables that I was given in the question?
The prices of the two bonds are $$ \begin{align} 1146.17&=76\,a_{\overline{n}|r}+1000\,v^n \tag 1\\ 1050.43&=61\,a_{\overline{n}|r}+1000\,v^n \tag 2\\ \end{align} $$ and solving the system we find $$ \begin{align} a_{\overline{n}|r}&=6.38267=\frac{1-v^n}{r}\tag 3\\ v^n&=0.661087=\frac{1}{(1+r)^n}\tag 4 \end{align} $$ From $(3)$ and $(4)$ we find $$ r=\frac{1-0.661087}{6.38267}\approx 5.31\%\tag 5 $$ and from $(4)$ and $(5)$ we have $$ n=-\frac{\log 0.661087}{\log(1+0.0531)}\approx 8\tag 6 $$