<p>If I understand your question correctly, you would like to know what future 1y yields will be like. These are called forward rates.</p>
<p>Because of the nature of the question and the fact that there are many ways to do forecast government yields, I'll focus on to key approaches, which are sometimes combined. For these reason the answer is only a sketch.</p>
<p><strong>1) (Augmented/Conditional/Dynamic) Factor Models</strong></p>
<p>This approach first reduces the dimensionality of the yield curve via principal component analysis, retaining the first few factors. Three factors are usually sufficient to explain the yield curve well (sometimes a fourth factor is added). In the three factor case, these are usually interpreted to the long-term level, slope, and curvature of the yield curve.</p>
<p>Note that you can then recover approximations of individual yields as
<span class="math-container">$$
r_i = \alpha_{1i} f_1 + \alpha_{2i} f_1 + \alpha_{3i} f_3
$$</span>
where the <span class="math-container">$f$</span> are the factors, and <span class="math-container">$\alpha_i$</span> the maturity specific coefficients.</p>
<p>How do you get forecasts? You estimate a VAR on the factors
<span class="math-container">$$
F_{t+1} = F_{t}\Lambda + X_t B + \epsilon_t
$$</span>
where <span class="math-container">$F$</span> are the (potentially stacked) factors, and <span class="math-container">$X_t$</span> is the &quot;augmentation&quot; (deterministic in the VAR). For example, <span class="math-container">$X$</span> could contain growth forecasts, inflation outlook, monetary policy assumptions. It could also contain forward looking information contained in financial market data, especially forward curves. It may have this information in summary form, for example factors resulting from further PCA on additional Data. It could also contain unobserved factors which then have to be estimated using a state-space model. You get your forecast from recovering the maturity yield of interest by applying the forecasted factors.</p>
<p><em>Nelson-Siegel</em>, <em>Cochrane-Piazzesi</em>, and many further developments fall in this category very broadly, that is with some minor or major deviations and refinements (Just search for &quot;forecasting the yield curve&quot; on google scholar).</p>
<p>More or less simplified versions of this approach are used by professional forecasters such as Oxford Economics, Capital Economics, IHS Markit, but these are conditional on their wider macro forecasts, and I would not necessarily trust them.</p>
<p><strong>2) Using Market Information</strong></p>
<p>You would use the first approach, or another model of your own, if you want to beat the market in that field. However, if you just want to price a product consistent with the market, it may be sufficient to use market pricing.</p>
<p>In that case you can simply use forward rates. The t-bill forward rate for a one year horizon expected at point <span class="math-container">$t$</span> will give you what market expect the actual rate to be, looking forward. So if you fund your product in line with market condition today, and then apply a markup, you should be fine.</p>
<p>Bloomberg should carry such rates directly (but be careful of how these rates are compounded and quoted, it's sometimes not obvious). Alternatively, you can compute the forward rates from spot rates.</p>
<p>Denote <span class="math-container">$rf_t^{(n)}$</span> the annualized yield for a n-year maturity expected <span class="math-container">$t$</span> periods ahead (forward rate). Denote <span class="math-container">$r_0^{(n)}$</span> the current annualized yield for an n-year maturity (spot rate). As an example, derive the market implied 1y yield in seven years time.</p>
<p>An investor should be indifferent between buying a bond that matures in seven years, and buying one that matures in six years, and then reinvest for one more year (at the forward rate). This is based on the <a href="https://en.wikipedia.org/wiki/Expectations_hypothesis" rel="nofollow noreferrer">unbiased expectations hypothesis</a>. It is generally justified by the reasoning that if it were not to hold, then there would be an arbitrage opportunity, but it's contested. Having said that, it can still be a reasonable assumption <a href="https://www.nber.org/system/files/working_papers/w2363/w2363.pdf" rel="nofollow noreferrer">for longer maturities</a>.</p>
<p>Making that assumption, we have
<span class="math-container">$$
(1+r_0^{(7)})^7 = (1+r_0^{(6)})^6 (1+rf_{6}^{(1)})
$$</span>
from which we get:
<span class="math-container">$$
rf_{6}^{(1)} = \frac{(1+r_0^{(7)})^7}{(1+r_0^{(6)})^6} -1,
$$</span>
which is the 1y yield markets currently expect to prevail in six years time.</p>
<p>With example numbers:
<span class="math-container">$$
rf_{6}^{(1)} = \frac{1.0127^7}{1.0124^6} -1 = 0.0145
$$</span>
so the implied 1y yield six years from now would be 1.45%.</p>
<p>Note that you will need a curve that is based on discount bonds (but that's usually the case) and that the raw spot rate yield curve may not cover all the maturities you need. How to estimate missing maturities is a research area in its own right. For simple applications it might be sufficient to fit a function to the available points.</p>

CC BY-SA 4.0
2021-07-30T15:45:40.660Z
2021-07-30T15:45:40.660Z
BrsG
<p>Yes you can forecast them in a various different ways. I don't think it is possible to make full literature review in a single SE post, so I will just give some examples.</p>
<ol>
<li>You can simply use random walk:</li>
</ol>
<p><span class="math-container">$$y_t = y_{t-1} + \epsilon_t$$</span></p>
<p>you would be surprised how far you can actually get with this silly little model (and in fact this is what EMH would suggest the best forecast should be). Even though it is 'quick and dirty' it was shown to be even better than 'experts forecasts' on some occasions (see <a href="https://www.sciencedirect.com/science/article/abs/pii/0304393276900143" rel="nofollow noreferrer">Bowlin and Martin,
1975</a>; <a href="https://files.stlouisfed.org/files/htdocs/publications/review/87/03/Predicting_Mar1987.pdf" rel="nofollow noreferrer">Belongia, 1987</a>; <a href="https://econpapers.repec.org/article/ucpjnlbus/v_3a65_3ay_3a1992_3ai_3a1_3ap_3a123-38.htm" rel="nofollow noreferrer">Hafer, Hein and MacDonald, 1992</a>).</p>
<ol start="2">
<li>You can use some standard time series models like ARIMA or AR model. These perform reasonably well and are very simple models models to estimate.</li>
</ol>
<p>Of course, the above are just very simple examples, because reading the question I think that is what you are interested in and complete literature review would be too cumbersome. There are of course various more fancy ways of how to do this. Something more fancy that was shown to perform quite well is Nelson-Siegel model (e.g. see example from Yu and <a href="http://Nelson-Siegel" rel="nofollow noreferrer">Zivot (2011)</a> applied to Chinese treasury rates). Widely used fancy model is also the so called Cochrane–Piazzesi regression.</p>

CC BY-SA 4.0
2021-07-27T15:23:44.717Z
2021-07-27T15:23:44.717Z
1muflon1
<p>I know, I know -- if I could predict Treasuries, I could leverage that information, various forms of the <a href="https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/efficient-markets-hypothesis/" rel="nofollow noreferrer">EMH</a>, etc. But a lot of people have mortgages (ARMs) tied to <a href="https://fred.stlouisfed.org/series/DGS1" rel="nofollow noreferrer">US1Y</a>, and so having some kind of idea of how it might perform over the next 0–360 months has practical applications even when predictive power is much less than would be needed to arbitrage a profit from the system.</p>
<p>One obvious way would be to look back over the whole history of Treasury bills and count the number of periods with a given rate and use this as a proxy for a probability table (perhaps with a small <a href="https://en.wikipedia.org/wiki/Additive_smoothing" rel="nofollow noreferrer">pseudocount</a> to handle rare and non-occurring cases). But this would give the data from 1929 the same weight as today, which seems wrong. (Perhaps a good weighting scheme would help; a few spring to mind but none of them present itself as obviously 'the right one'.)</p>
<p>Any other/better ideas? Has anyone seen literature approaching this, other than to simply say it's EMH-hard?</p>

CC BY-SA 4.0
2021-07-27T15:04:40.767Z
Welcome to applied forecasting! ;) Just looking at the history of treasury yields themselves won't help indeed. If you don't want to use forecast of t-bill yields as provided by professional forecasters, you may want to look into underlying drivers. In the short term, 1Y yields are closely tied to fed policy rates. As you go out further, inflation and gdp growth will play an increasing role. If you want to go market based, then I would look into swap rates. Incidentally, some mortgage rate pricing is based on swap rates.
BrsG
CC BY-SA 4.0
2021-07-27T15:08:08.270Z
@BrsG I'm interested in hearing more about all of those -- forecasters (what's out there?), swap rates, and how to quantify other drivers. I'm happy with quick & dirty.
Charles
CC BY-SA 4.0
2021-07-28T07:24:27.780Z
Do you want to actually want to use/produce forecasts in an applied setting? Or are you interested in theoretical modelling or simulation?
BrsG
CC BY-SA 4.0
2021-07-28T17:40:54.993Z
@BrsG Use them. If there are existing high-quality predictions that would be the best, otherwise I'm willing to construct an appropriate model. But I'd probably be more interested in the chance of different scenarios, so some kind of ensemble forecasting would seem most appropriate in that case. (With someone else's predictions, I'm willing to take point estimates.)
Charles
<reference-request><government-debt><efficient-markets><prediction>
How to weakly predict T-bill rates?
Charles
# How to weakly predict T-bill rates?

(Asked by: username_0 on 2021-07-27)

I know, I know -- if I could predict Treasuries, I could leverage that information, various forms of the [EMH](https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/efficient-markets-hypothesis/), etc. But a lot of people have mortgages (ARMs) tied to [US1Y](https://fred.stlouisfed.org/series/DGS1), and so having some kind of idea of how it might perform over the next 0–360 months has practical applications even when predictive power is much less than would be needed to arbitrage a profit from the system.

One obvious way would be to look back over the whole history of Treasury bills and count the number of periods with a given rate and use this as a proxy for a probability table (perhaps with a small [pseudocount](https://en.wikipedia.org/wiki/Additive_smoothing) to handle rare and non-occurring cases). But this would give the data from 1929 the same weight as today, which seems wrong. (Perhaps a good weighting scheme would help; a few spring to mind but none of them present itself as obviously 'the right one'.)

Any other/better ideas? Has anyone seen literature approaching this, other than to simply say it's EMH-hard?

**Tags:** reference-request, government-debt, efficient-markets, prediction

> **Comments:**
> **username_1**: Welcome to applied forecasting! ;) Just looking at the history of treasury yields themselves won't help indeed. If you don't want to use forecast of t-bill yields as provided by professional forecasters, you may want to look into underlying drivers. In the short term, 1Y yields are closely tied to fed policy rates. As you go out further, inflation and gdp growth will play an increasing role. If you want to go market based, then I would look into swap rates. Incidentally, some mortgage rate pricing is based on swap rates.  
> **username_0**: @username_1 I'm interested in hearing more about all of those -- forecasters (what's out there?), swap rates, and how to quantify other drivers. I'm happy with quick & dirty.  
> **username_1**: Do you want to actually want to use/produce forecasts in an applied setting? Or are you interested in theoretical modelling or simulation?  
> **username_0**: @username_1 Use them. If there are existing high-quality predictions that would be the best, otherwise I'm willing to construct an appropriate model. But I'd probably be more interested in the chance of different scenarios, so some kind of ensemble forecasting would seem most appropriate in that case. (With someone else's predictions, I'm willing to take point estimates.)  

---

## 2 answers

### Answer by username_1 on 2021-07-30. Score: 2 | Accepted answer

If I understand your question correctly, you would like to know what future 1y yields will be like. These are called forward rates.

Because of the nature of the question and the fact that there are many ways to do forecast government yields, I'll focus on to key approaches, which are sometimes combined. For these reason the answer is only a sketch.

**1) (Augmented/Conditional/Dynamic) Factor Models**

This approach first reduces the dimensionality of the yield curve via principal component analysis, retaining the first few factors. Three factors are usually sufficient to explain the yield curve well (sometimes a fourth factor is added). In the three factor case, these are usually interpreted to the long-term level, slope, and curvature of the yield curve.

Note that you can then recover approximations of individual yields as
$$
r\_i = \alpha\_{1i} f\_1 + \alpha\_{2i} f\_1 + \alpha\_{3i} f\_3
$$
where the $f$ are the factors, and $\alpha\_i$ the maturity specific coefficients.

How do you get forecasts? You estimate a VAR on the factors
$$
F\_{t+1} = F\_{t}\Lambda + X\_t B + \epsilon\_t
$$
where $F$ are the (potentially stacked) factors, and $X\_t$ is the "augmentation" (deterministic in the VAR). For example, $X$ could contain growth forecasts, inflation outlook, monetary policy assumptions. It could also contain forward looking information contained in financial market data, especially forward curves. It may have this information in summary form, for example factors resulting from further PCA on additional Data. It could also contain unobserved factors which then have to be estimated using a state-space model. You get your forecast from recovering the maturity yield of interest by applying the forecasted factors.

*Nelson-Siegel*, *Cochrane-Piazzesi*, and many further developments fall in this category very broadly, that is with some minor or major deviations and refinements (Just search for "forecasting the yield curve" on google scholar).

More or less simplified versions of this approach are used by professional forecasters such as Oxford Economics, Capital Economics, IHS Markit, but these are conditional on their wider macro forecasts, and I would not necessarily trust them.

**2) Using Market Information**

You would use the first approach, or another model of your own, if you want to beat the market in that field. However, if you just want to price a product consistent with the market, it may be sufficient to use market pricing.

In that case you can simply use forward rates. The t-bill forward rate for a one year horizon expected at point $t$ will give you what market expect the actual rate to be, looking forward. So if you fund your product in line with market condition today, and then apply a markup, you should be fine.

Bloomberg should carry such rates directly (but be careful of how these rates are compounded and quoted, it's sometimes not obvious). Alternatively, you can compute the forward rates from spot rates.

Denote $rf\_t^{(n)}$ the annualized yield for a n-year maturity expected $t$ periods ahead (forward rate). Denote $r\_0^{(n)}$ the current annualized yield for an n-year maturity (spot rate). As an example, derive the market implied 1y yield in seven years time.

An investor should be indifferent between buying a bond that matures in seven years, and buying one that matures in six years, and then reinvest for one more year (at the forward rate). This is based on the [unbiased expectations hypothesis](https://en.wikipedia.org/wiki/Expectations_hypothesis). It is generally justified by the reasoning that if it were not to hold, then there would be an arbitrage opportunity, but it's contested. Having said that, it can still be a reasonable assumption [for longer maturities](https://www.nber.org/system/files/working_papers/w2363/w2363.pdf).

Making that assumption, we have
$$
(1+r\_0^{(7)})^7 = (1+r\_0^{(6)})^6 (1+rf\_{6}^{(1)})
$$
from which we get:
$$
rf\_{6}^{(1)} = \frac{(1+r\_0^{(7)})^7}{(1+r\_0^{(6)})^6} -1,
$$
which is the 1y yield markets currently expect to prevail in six years time.

With example numbers:
$$
rf\_{6}^{(1)} = \frac{1.0127^7}{1.0124^6} -1 = 0.0145
$$
so the implied 1y yield six years from now would be 1.45%.

Note that you will need a curve that is based on discount bonds (but that's usually the case) and that the raw spot rate yield curve may not cover all the maturities you need. How to estimate missing maturities is a research area in its own right. For simple applications it might be sufficient to fit a function to the available points.


---

### Answer by username_2 on 2021-07-27. Score: 2

Yes you can forecast them in a various different ways. I don't think it is possible to make full literature review in a single SE post, so I will just give some examples.

1. You can simply use random walk:

$$y\_t = y\_{t-1} + \epsilon\_t$$

you would be surprised how far you can actually get with this silly little model (and in fact this is what EMH would suggest the best forecast should be). Even though it is 'quick and dirty' it was shown to be even better than 'experts forecasts' on some occasions (see [Bowlin and Martin,
1975](https://www.sciencedirect.com/science/article/abs/pii/0304393276900143); [Belongia, 1987](https://files.stlouisfed.org/files/htdocs/publications/review/87/03/Predicting_Mar1987.pdf); [Hafer, Hein and MacDonald, 1992](https://econpapers.repec.org/article/ucpjnlbus/v_3a65_3ay_3a1992_3ai_3a1_3ap_3a123-38.htm)).

2. You can use some standard time series models like ARIMA or AR model. These perform reasonably well and are very simple models models to estimate.

Of course, the above are just very simple examples, because reading the question I think that is what you are interested in and complete literature review would be too cumbersome. There are of course various more fancy ways of how to do this. Something more fancy that was shown to perform quite well is Nelson-Siegel model (e.g. see example from Yu and [Zivot (2011)](http://Nelson-Siegel) applied to Chinese treasury rates). Widely used fancy model is also the so called Cochrane–Piazzesi regression.


---

