Par Rate and Deriving the Par Curve from the Spot Curve

If you were to calculate the par rate of a bond that is between coupon payment dates, which price do you think you would set to par?

Correct! The interest that accrues between coupon dates is not used in time value of money calculations. Setting the full price to par, would have a corresponding yield to maturity that is different than the coupon rate. Therefore, the flat price would be set to par.

Incorrect. The interest that accrues between coupon dates is not used in time value of money calculations. Setting the full price to par, would have a corresponding yield to maturity that is different than the coupon rate. Therefore, the flat price would be set to par.

The __par curve__ is the set of bond yields for hypothetical benchmark securities priced at par. The par curve would be easy to build if bonds were trading at or very close to par. In reality, bond prices fluctuate based on market interest rates and regularly trade at a premium or discount to par. Therefore, the par curve must be derived from the spot curve. Spot rates, which are yields of zero-coupon bonds, are used to derive the par curve using the equation below. $$\displaystyle 100 = \frac{PMT}{(1+Spot\: Rate_{1})^{1}}+\frac{PMT}{(1+Spot\: Rate_{2})^{2}}+...+\frac{PMT+100}{(1+Spot\: Rate_{n})^{n}}$$ Why do you think spot rates are used to derive the par curve?

Incorrect. Since spot rates are yields of zero-coupon bonds, they do not have reinvestment rate risk. Coupon paying bonds do possess reinvestment rate risk.

Great job. Since spot rates are yields of zero-coupon bonds, they do not have reinvestment rate risk. Coupon paying bonds do possess reinvestment rate risk.

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Investment bankers that underwrite bonds construct par yield curves to assist in valuing new issues. As with all yield curves, it is important a par curve be built with bonds that possess the same credit risk, tax status, coupon payment periodicity, currency, and liquidity. A __par rate__ is determined by pricing a bond along the term to maturity structure at par. In other words, the yield to maturity is equal to the bond's coupon rate.

Flat price

Full price

Spot rates have the same risk as par rates

Spot rates do not have reinvestment rate risk

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