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3.4 Bond Price Quotation

3.4.1 Spot Rates

Because the market discount rates for the cash flows with different maturities are rarely the same, it is fundamentally better to calculate the price of a bond by using a sequence of market discount rates that correspond to the cash flow dates. These market discount rates are called 'spot rates".

  • Spot rates are yields-to-maturity on zero-coupon bonds maturing at the date of each cash flow. Spot rates are sometimes called "zero rates." The General formula for calculating a bond price given the sequence of spot rates:

PV = Int/(1+Z1)1 + Int/(1+Z2)2 + …+ (Int + FV)/(1+ZN)N Where: Z1, Z2, and ZN are Spot rates at period 1, 2, and 3 respectively.

Bond price (or value) determined using the spot rates is sometimes referred to as the bond's "no-arbitrage value." If a bond's price differs from its no-arbitrage value, an arbitrage opportunity exists in the absence of transaction costs.

3.4.2 Flat Price (Clean Price), Accrued Interest, and the Full Price (Dirty Price) of a Bond.

Bond price consists of two components: Flat (clean) price (Pc) and Accrued interest (AI). The sum of flat price and accrued interest is the full (dirty) price (Pf)

Pf = Pc + AI While bond dealers usually quote the flat price, buyers pay the full price for the bond on the settlement date.

  • The reason to use the flat price for the quotation is to avoid misleading investors about the market price trend for the bond. If the full price were quoted by dealers, investors would see the price rise day after day even if the yield-to-maturity did not change. That is because the amount of accrued interest increases each day. Then, after the coupon payment is made, the quoted price would drop dramatically. Using the flat price for quotation avoids that misrepresentation.
  • If a bond is sold on the coupon payment date, its accrued interest is equal to zero (i.e., full price = clean price).

AI is the proportionate share of the next coupon payment (PMT).

AI = t/T x PMT Where t is the number of days from the last coupon payment to the settlement date; T is the number of days in the coupon period.

Conventions for quotes and calculations There are 2 conventions to count days in the bond market:

  • Actual/Actual Method: the actual number of days is used, including weekends, holidays, and leap days. This convention is common for government bonds.
  • The 30/360 day-count convention: assumes that each month has 30 days and that a full year has 360 days. 30/360 is common for corporate bonds.

Calculation of Bond Prices

  • PV would typically be close in value to Pc but conceptually, it's not the same as a flat price.
  • The price is in euros per 100 euros of par value.
  • There are 23 semiannual coupon payments (periods) to maturity, 11 years, and one six- month period.
  • The coupon payment is 3 euros per 100 euros of value (6% x 100/2).
  • T is 180 days because the six-month period is 180 days in the 30/360 convention.
  • t is 89 days = 3 months x 30 days minus 1 day.
  • The accrued interest does not depend on YTM.

Lesson Wrap-Up

This lesson should leave you able to explain the bond price quotation in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain spot rates in your own words.
  2. Explain flat price (clean price), accrued interest, and the full price (dirty price) of a bond. in your own words.
  3. State one exam-style risk, valuation, or market implication of the bond price quotation.