Calculating Bond Excess Returns from Yield Data

Summary

The document examines how quarterly excess returns on Treasury bills and long-term bonds are calculated in an appendix to a 1988 asset-pricing paper. It describes the maturities involved and asks how the paper’s formulas relate to a direct holding-period return calculation: compare the price of a security at the start and end of a quarter, then subtract the contemporaneous short bill rate.

The author proposes prices based on inverse yield expressions and asks whether those produce the right bill and bond excess returns. No resolution or empirical evidence is included. The question highlights an important distinction for fixed-income return construction: a bond’s yield is not generally its price, and a holding-period return depends on repricing the remaining cash flows as maturity shortens and yields change. The document therefore identifies a calculation issue rather than supplying a validated formula.

Key ideas

• The question concerns quarterly excess returns on bills and long-term Treasury bonds.
• A holding-period excess return compares an asset’s price return with the risk-free return over the same period.
• Yield data alone cannot generally be treated as inverse bond prices without specifying cash flows and maturity conventions.
• The document proposes candidate calculations but does not resolve whether they match the cited paper.

Tags

# Excess Returns of Bonds


# Excess Returns of Bonds












In the appendix of the 1988 paper "A Capital Asset Pricing Model with Time-Varying Covariances" by Bollerslev et al, the excess returns of bills and bonds are shown to be calculated as follows:

where $r_t^f$ is the yield on 3-month Treasury bills, $r_t^{bill}$ yield on 6-month Treasury bills and $r_t^{bond}$ 20-year Treasury bonds and $t=Quarter$.

I have trouble in understanding (A1) and (A2) since I would have thought that we would compute them like:

$$P_{t-1}^{6M} = \frac{1}{(1+r_{t-1}^{bill})^2}$$ in the next quarter the price is then given by $$P_{t}^{3M} = \frac{1}{(1+r_{t}^{f})}$$ From simple price return calculation $r_t = (P_t/P_{t-1}) - 1$ I would have written the one-quarter excess holding yields as $$y_t^{bill} = [(P_{t}^{3M}/P_{t-1}^{6M}) - 1 - r_t^f] = [\frac{(1+r_{t-1}^{bill})^2}{(1+r_{t}^{f})}- 1 - r_t^f]$$

and for the 20-year bonds $$y_t^{bond} = (r_{t-1}^{bond} / r_{t}^{bond}) + r_{t-1}^{bond} - 1 - r_t^f$$ with $P_t \propto 1/r_t^{bond}$

What am I am misunderstanding here? Thanks in advance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.