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Assess the return a bond investment is expected to deliver and name its components.

FRM® Part I · Financial Markets and Products · Corporate Bonds · FRM1.FM.17.i

The idea

The return expected from a bond investment has three components: coupon income, capital gain or loss from the change in price, and reinvestment income on the coupons received. Yield to maturity is a promised return. It assumes the bond is held to maturity, no default occurs and every coupon is reinvested at the YTM itself. Realised return depends on the holding period. Price moves with the yield and spread, and the price is pulled to par over time. Over a horizon, Total return = (coupons + reinvestment income + ending price − beginning price) ÷ beginning price. Annualise it by taking the root of the terminal wealth ratio. For credit bonds, the expected return is lower than the promised yield by the expected credit loss. Expected credit loss = probability of default × (1 − recovery rate). Do not treat the quoted yield as the return you will earn.

Formulas

Total return on a bond over a horizon

total return = (coupons + reinvestment income + P₁ − P₀) ÷ P₀

coupons
the coupons received over the horizon
reinvestment income
what those coupons earn once reinvested
P₁
the price at the end of the horizon
P₀
the price paid at the start

Any horizon, and the return is not annualised. The quoted yield to maturity is earned only if the bond is held to maturity, does not default and coupons are reinvested at the YTM.

Expected credit loss on a bond, per year

expected credit loss = PD × (1 − recovery rate)

PD
the annual probability of default
recovery rate
the share of the claim recovered after default

Subtract it from the promised yield to get the expected return. The trap is treating the quoted yield as the return you will earn.

A practice question

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

What are the three components of the return an investor expects from holding a bond?

  1. Coupon income, capital gain or loss, and reinvestment income. - correct
  2. Coupon income, default premium and inflation compensation.Why not: Lists compensation components of the yield, not the components of the return over time.
  3. Principal, face value and redemption premium.Why not: Lists terms of the bond, not return components.
  4. Credit spread, liquidity premium and term premium.Why not: Lists components of the yield, not of the return.

Return = coupons + reinvestment income on coupons + price change (capital gain or loss) over the holding period.

A harder one

A bond has a promised yield of 7.2%, an annual probability of default of 2% and an expected recovery of 40%. What is the approximate expected return after expected credit loss?

  1. 5.2%Why not: Subtracts the full default probability and ignores recovery.
  2. 6.0% - correct
  3. 6.4%Why not: Subtracts probability times the recovery rate instead of the loss given default.
  4. 7.2%Why not: Treats the promised yield as the expected return.

Expected credit loss = PD x (1 - recovery) = 2% x 60% = 1.2%. Expected return = 7.2% - 1.2% = 6.0%.

The hardest tier

An investor buys a 3-year, 8% annual-coupon bond at par (100). Coupons are reinvested at 5%, and the bond is sold at 102 after three years (just after the third coupon). What is the annualised total return?

  1. 8.57%Why not: Assumes the coupons are reinvested at the 8% coupon rate instead of 5%.
  2. 8.01%Why not: Ignores reinvestment income; coupons are added without growth.
  3. 8.36% - correct
  4. 8.67%Why not: Takes simple average of the cash return over three years, ignoring compounding and reinvestment.

Coupons grow to 8 x 1.05² + 8 x 1.05 + 8 = 25.22. Terminal wealth = 25.22 + 102 = 127.22. Annualised return = (1.2722)^(1/3) - 1 = 8.36%.

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