Deriving T-Note Total Return from Beginning and Ending Yields

Summary

The document explains a formula for estimating a one-period total return on a ten-year Treasury note using its yield at the start and end of the period. The formula values the note’s coupon stream as an annuity discounted at the ending yield, adds the discounted principal repayment, subtracts the initial investment, and includes the coupon received during the period at the starting yield.

This decomposition clarifies why the calculation can use two yield observations under the stated setup: the starting yield determines the coupon amount, while the ending yield is used to value the remaining cash flows. The explanation is concise and offers no worked numerical example or discussion of alternative conventions. Its interpretation depends on the assumed ten-year maturity and coupon and reinvestment conventions embedded in the formula; the document does not address how actual bond returns may differ when those assumptions do not match the instrument or measurement period.

Key ideas

• The formula values coupon payments as an annuity discounted at the ending yield.
• It adds the discounted principal repayment to the present value of coupons.
• The starting yield determines the coupon amount and the coupon received during the period.
• The explanation assumes a ten-year note and does not discuss alternative bond-return conventions.

Tags

# T-note returns from T-note yields ... derivation of Damodaran's formula


# T-note returns from T-note yields ... derivation of Damodaran's formula












Damodaran's historical data on 10-year T-note returns (found here) uses the following formula to calculate the 1-period total return on a T-note ($R_1$) given the 10-year constant maturity yield-to-maturity in the prior year ($Y_0$) and the current year ($Y_1$).

$R_1=(Y_0*\frac{1-(1+Y_1)^{-10}}{Y_1}+\frac{1}{(1+Y_1)^{10}})-1+Y_0$

Where can I find a derivation or description of this formula? It seems very odd to me that the only two data points I would need to calculate the total return to a T-note are the beginning and ending yield-to-maturities.

## Answer by Joel (score 2, accepted)

https://quant.stackexchange.com/a/3945

First term in parens is the annuity formula applied to a 10 year stream of coupon payments at rate y0, second term in parens is discounted one dollar of principal payable at ten year maturity. Then subtract $1 invested today and add a current cash coupon at rate y0.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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