# How to compute par yield from zero rate curve?

How does one calculate the below two-year par yield given the zero rate curve: Assume the following two-year zero rate curve, with continuous compounding:

2.0% @ 0.5 year
2.5% @ 1.0 year
3.0% @ 1.5 years
3.5% @ 2.0 years


What is the two year par yield?

I know we can use the zero rate curve to derive a theoretical price of the bond but don't know how that can be used for calculating the par yield.

For simplicity, let us assume continuously compounded zero rates and periodically compounded par yields. If you have to work with continuous rates, you may adapt the formulas accordingly.

Using the zero rate discount factors $$D(T) \equiv e^{-r(T)T}$$, the present value of a coupon bearing bond is

$$$$PV=\sum_i^N c D(t_i) + D(t_N)$$$$

For a coupon bearing bond, we can relate the coupon rate of a par bond (!) to the yield structure as:

\begin{align} 100\%=&\sum_i^N \frac{c}{(1+y_T)^{t_i}} + \frac{1}{(1+y_T)^{t_N}}\\ \Leftrightarrow c=&y_T \end{align}

If your bond pays at semi-annual frequency, then $$y_T$$ is the corresponding semi-annual yield rate, and your annual yield would of course be $$\tilde{y}=(1+y)^2-1$$.

We have thus established that the coupon rate of a par bond reflects the yield information. Thus, all you now need to do is to find coupon rates such that your hypothetical bonds are priced at par:

\begin{align} 1&=0.5c_T\sum_i^N D(t_i) + D(t_N)\\ \Leftrightarrow c_T&=2\frac{1-D(t_N)}{\sum_i^N D(t_i)} \end{align}

In your case, the discount factors are \begin{align} D(t_{0.5})=e^{-0.5*0.020}&=0.9900498\\ D(t_{1.0})=e^{-1.0*0.025}&=0.9753099\\ D(t_{1.5})=e^{-1.5*0.030}&=0.9559975\\ D(t_{2.0})=e^{-2.0*0.035}&=0.9323938\\ \end{align}

And hence your semi-annual coupons are $$y_{0.5}=0.02010033$$, $$y_{1.0}=0.02512526$$, $$y_{1.5}=0.03012471$$, $$y_{2.0}=0.03508591$$

For the annualised yields, we then obtain

$$$$y_{ann,i}=(1+0.5*c_i)^2-1$$$$

As a sanity check, you may want to compute the present value of a 2-year bond using the coupon of 3.508591% (annual) and the corresponding yield of 3.539366% (annualised).

HTH

PS: ultimatively, you can also invert the ‚standard‘ boot strapping equation:

$$$$\frac{1-D(t_N)}{\sum_i^N D(t_i)}$$$$

in order to quickly arrive at the par rates.