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Define a bond's spread, interpret it, and explain its derivation from the bond price together with a term structure of rates.

FRM® Part I · Valuation and Risk Models · Bond Yields and Return Calculations · FRM1.VR.11.b

The idea

A bond's spread is the single constant added to every rate on a benchmark term structure so that the present value of the bond's promised cash flows equals its market price. The pricing equation is Price = Σ CFt ÷ (1 + rt + s)^t. Here CFt is the cash flow on date t, rt is the benchmark spot rate for that date and s is the spread, found numerically. A bond priced exactly on the curve has a spread of zero. A positive spread means the market pays less than the curve value, compensating investors for credit, liquidity and other risks the benchmark does not carry. This differs from the simple yield spread, which subtracts one benchmark yield from the bond's YTM and ignores the curve's shape. For a zero-coupon bond the two coincide, because Spread = YTM − spot rate. A frequent error is adding the spread to only one rate rather than to every spot rate.

Formulas

Bond price with a spread over the benchmark spot curve

P = Σ CFt ÷ (1 + rt + s)ᵗ

CFt
the promised cash flow at date t
rt
the benchmark spot rate for date t
s
the spread, the same for every date
t
the date of the cash flow, in years

The spread is found numerically by solving for s at the market price. It is added to every spot rate, not just one.

A practice question

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

Which description best fits a bond's spread when it is derived from the bond price together with a term structure of rates?

  1. The bond's coupon rate minus the benchmark spot rate for its final maturity.Why not: Compares a coupon with one spot rate; it does not solve the pricing equation using the market price.
  2. The constant added to every benchmark spot rate so that the discounted cash flows equal the bond's market price. - correct
  3. The difference between the dealer's bid and ask prices, expressed in basis points.Why not: A transaction-cost measure that has nothing to do with discounting cash flows.
  4. The bond's YTM minus the YTM of one benchmark bond, with no reference to the rest of the curve.Why not: This is the simple yield spread; the term-structure spread is solved using the whole curve.

The spread s solves: market price = sum of cash flow / (1 + r_t + s)^t, where r_t are the benchmark spot rates. It is one number that closes the gap between the curve value and the market price.

A harder one

A bond has a spread of 120 bp over the benchmark spot curve. The benchmark curve then shifts up by 50 bp in every maturity while the bond's market price does not change. The bond's new spread is closest to:

  1. 170 bpWhy not: Adds the curve shift to the spread; a higher benchmark curve with the same price means the spread has shrunk.
  2. 70 bp - correct
  3. 120 bpWhy not: Assumes the spread is unchanged because the price is unchanged.
  4. 50 bpWhy not: Mixes up the spread with the size of the curve shift.

The price fixes the total discount rate at each maturity, which is spot rate plus spread. If the spot rates rise 50 bp and the total must stay the same, the spread falls by 50 bp: 120 - 50 = 70 bp.

The hardest tier

A 3-year bond pays a 5% annual coupon on face value 100 and trades at 98.00. The benchmark spot curve (annual compounding) is 2.0% for one year, 2.6% for two years and 3.1% for three years. What is the bond's spread over the curve, closest to?

  1. 74 bpWhy not: Takes YTM minus the coupon rate, which is not the constant added to the spot curve.
  2. 374 bpWhy not: Subtracts only the 1-year spot rate from the YTM, using one rate instead of the whole curve.
  3. 761 bpWhy not: Treats the percentage shortfall of price below the curve value as if it were a yield spread.
  4. 268 bp - correct

Curve value = 5/1.020 + 5/1.026² + 105/1.031³ = 105.46, above the price of 98.00. Solve 98.00 = 5/(1.020+s) + 5/(1.026+s)² + 105/(1.031+s)³ numerically: s = 268 bp.

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