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Explain what a par rate is and outline how a bond's par rate is found.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.d

The idea

The par rate for a maturity is the coupon rate at which a bond of that maturity prices at exactly par, so its coupon equals its yield. Because the price must equal the face value, the coupon c on a bond paying n annual coupons must satisfy Par bond price = c × (DF1 + … + DFn) + DFn. Setting this price to 1 per unit of face value and solving for c gives Par rate = (1 − DFn) ÷ (DF1 + … + DFn). With semiannual coupons the same sum runs over every half-year discount factor and the result is multiplied by two. Equivalently, the par rate is a weighted average of the spot rates, the weights being the discount factors. A bond with a coupon above the par rate trades above par, and one below it trades below par. The standard mistake is to take the final spot rate as the par rate. They agree only on a flat curve.

Formulas

Par rate from discount factors

c = (1 − DFn) ÷ (DF1 + … + DFn)

DF1
the discount factor for the first coupon date
DFn
the discount factor for the final date n

Annual coupons give the annual par rate directly. With semiannual coupons, sum every half-year factor and double the result.

A practice question

From the question bank, with the reason each wrong option is wrong.

The par rate for a given maturity is best defined as:

  1. the coupon rate at which a bond of that maturity is priced at its face value. - correct
  2. the yield on a zero-coupon bond of that maturity.Why not: Describes the spot rate, which relates to a single cash flow rather than a coupon bond.
  3. the rate for a loan that starts at that maturity.Why not: Describes a forward rate, which applies to a future period.
  4. the average coupon rate of all bonds issued at that maturity.Why not: Not a defined rate; the par rate is solved from the discount factors, not from market coupons.

Par rate means the coupon at which the bond's present value equals its par value, so its coupon equals its yield. It is found from the discount factors.

A harder one

The discount factors for years 1, 2 and 3 are 0.9709, 0.9070 and 0.8163 (the 3-year spot rate is 7.0%). The 3-year par rate for annual coupons is closest to:

  1. 7.00%Why not: Reports the 3-year spot rate as the par rate; they differ when the curve is not flat.
  2. 6.82% - correct
  3. 5.00%Why not: Averages the 1-, 2- and 3-year spot rates, ignoring the discount-factor weights.
  4. 7.50%Why not: Spreads the 3-year growth over 3 years with no compounding, which is not the par coupon formula.

Par rate = (1 - DF3) / (DF1 + DF2 + DF3) = (1 - 0.8163) / 2.6942 = 6.82%. It is a weighted average of spot rates, below the 3-year spot rate on an upward-sloping curve.

The hardest tier

The continuously compounded spot rates are 1% for 1 year, 4% for 2 years and 9% for 3 years. The par rate for a 3-year bond with annual coupons, expressed as an annually compounded rate, is closest to:

  1. 9.00%Why not: Takes the 3-year continuous spot rate as the par rate.
  2. 8.84% - correct
  3. 9.42%Why not: Converts the 3-year spot to an annual rate (e^0.09 - 1) and calls that the par rate.
  4. 4.67%Why not: Averages the three spot rates without discount-factor weights.

Discount factors e^(-s t): 0.9900, 0.9231, 0.7634. Par rate = (1 - 0.7634) / 2.6765 = 8.84%. The par rate sits below the 3-year spot rate on an upward-sloping curve.

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