How to calculate US treasury total return from yield?

Question

I'm struggling to understand the meaning of US treasury total return.

What is easily available to get is yield data. Yield can be directly translated to the bond price at that time. In other words, even if yield uses % unit, this represents the percentage of discount from the principal. (here)

However, I don't think it means any sort of return. I found that there are some sources (e.g. Merrill) which report US treasury index value, price return, and total return. (I think what 'total' means is that they take account of coupons and everything.)

I'd like to know how to translate [yield ---> index ---> return]. If possible, I want to know deep level technical (mathematical) details, as well as intuition.

p.s. @haginile commented on a similar question here, but he didn't explain why it is hard to calculate return from yield. Also his blog link is broken.

Answer 1

Let's start with a single bond. The total return from time $t_0$ to time $t_1$ can be easily calculated as follows:

$$ R = \frac{\text{ending price} + \text{ending accrued interest} + \text{coupon payments between $t_0$ and $t_1$}}{\text{starting price} + \text{starting accrued interest}} - 1. $$

(This is no different from how you'd calculate the total return on a stock or any other assets: $(P_1 + \text{dividend}) / P_0 - 1$).

If you know exactly who the bond is (i.e., if you know the coupon and maturity), then given the yields to maturity at both $t_0$, and $t_1$, it is trivial to calculate the corresponding prices and accrued interests. From there, computing total return is also trivial.

The problem is, what if you don't know the coupon and maturity of the bond? In that case, approximations or assumptions are involved. Fortunately, a bond's total return can be (well) approximated from: $$ R = \text{yield income} - \text{duration}\cdot \Delta y + \frac{1}{2} \cdot \text{convexity} \cdot (\Delta y)^2, $$ where yield income can be approximated by $\text{yield}_0 \times \Delta t$. For instance, if yield is 5%, then yield income for a month is simply $5\% / 12$. If you have duration and convexity statistics, then you can approximate the total returns pretty well.

Now let's go to the index level. For an index, you basically repeat the total return calculation above for every single bond included in the index. The total return of the overall index is simply the market-value weighted average of the constituents' returns: $$ R_\text{index} = \sum_{i=1}^N w_i R_i. $$ (This is also similar to how an equity index total return is calculated.)

At this point, we may have several problems:

1. Do we have all the constituents of the index?
2. Do we know their market values?
3. Do we have each bond's pricing information?

For the US Treasury index, these are probably not issues. But for most other indices, you probably don't have the underlying data. In these cases, approximations are required. Fortunately, most index vendors do publish an index's yield, effective duration, and convexity, so the approximation formula above can be applied at the index level.

There are a lot of intricacies involved if you want accurate total returns, but that's why these index vendors exist. The last edition of the Lehman Brothers Global Index Guide (2008) was 336 pages...

Answer 2

Assume you have the time series of 10-year Treasury constant-maturity yield $\{y_t\}$ from FRED (here), you can calculate the total return $R_t$ from $t$ to $t+\Delta t$ as following.

Define

$$ \text{Daily: } \Delta t = 1/365 $$ $$ \text{Monthly: } \Delta t = 1/12 $$ $$ \text{Yield change: } \Delta y = y_{t+\Delta t} - y_t $$ $$ \text{Maturity: } M = 10 $$

Then the total return is

$$ R_t = \text{yield income} - \text{duration}\cdot \Delta y + \frac{1}{2} \cdot \text{convexity} \cdot (\Delta y)^2, $$

where

$$ \text{yield income} = (1+y_t)^{\Delta t}-1 \approx y_t {\Delta t} $$ $$ \text{duration} = \frac{1}{y_t} {z_t}^{2 M} $$ $$ \text{convexity} = C_1 - C_2 $$ and $$ z_t = 1+\frac{y_t}{2} $$ $$ C_1 = \frac{2}{y_t^2} (1-{z_t}^{-2 M}) $$ $$ C_2 = \frac{2 M}{y_t} {z_t}^{-2 M - 1} $$.

This formula is based on Tuckerman (2012) and Swinkels (2019).

Answer 3

I suggest the following:

$$R_c=θ\Delta t-Mdur\Delta y+\frac{1}{2}[Cvex-Mdur^2](\Delta y)^2 +(1+y)^{-1} \Delta y \Delta t)$$

where:

$R_c =$ continuously compounded return

If desired, $R_a =$ annualized simple return $\approx \exp^{R_c}-1$

$θ=\ln(1+y)$

$y$ = yield to maturity

Mdur = Modified duration, defined by $Mdur=-\frac{1}{P_0}\frac{\partial P}{\partial y}$

Cvex = Convexity, defined by $Cvex=\frac{1}{P_0}\frac{\partial^2P}{\partial y^2}$

Source: Bo Johansson, A note on approximating bond returns allowing for both yield changes and time passage, 2012. Equation 5.(link).