# How to derive the relationship between log yield and log price?

Usually, people write $y_t^{(n)}=-\frac{p_t^{(n)}}{n}$ where $y, p$ and log yield and log price respectively. My question is how do one derive this expression?

Note that $e^{-Y_t^{(n)}\cdot n}=P_t^{(n)}$ if $Y,P$ are the continuous yield and price of the one dollar $n$ period bond respectively at time $t$. Now, if we take logs, we get $$-Y_t^{(n)}\cdot n=p_t^{(n)}.$$ Therefore, we have $Y_t^{(n)}=-\frac{p_t^{(n)}}{n}$ which differs from the equation from the first paragraph. What went wrong?

• I believe Cochrane and Piazzesi define the "log yield" using the first expression. However it is not the same as the log of the yield, which is what you are assuming in your second derivation. Aug 1 '16 at 20:53
• IMHO "continuous time yield" would be a better name for this than "log yield". Aug 1 '16 at 21:06