## Expected Future Yield Curve

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.

## If today's 7 & 5 yr rates are 2.24 & 1.77, what will the 5 yr rate be in 2 yrs?

1.77
2
11%
2.24
2
11%
Between 1.77 and 2.24
5
28%
Other/Explain
9
50%

Spec7re
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### Expected Future Yield Curve

I'm coming at this from the angle of estimating returns on an intermediate treasury fund. I will present a simplified scenario below.

Say I buy a 7 year note paying coupon of 2.24 with the intention of selling it in two years. Also assume the current 5 year note yield is 1.77%. What is expected return over the next two years? There seem to be at least three views on this subject.

A. The current yield curve is the best estimate of the future yield curve. This means that we expect the 7 yr note to sell at a premium in two years since we expect the 5 year yield to remain at 1.77%. The total return would be the coupon plus the increase in bond value. This scenario implies that "riding the yield curve" is a valid strategy. This also means that the expected return on intermediate treasury fund is actually higher than the current yield (assuming the yield curve is not inverted).

B. The current yield curve predicts the future yield curve. This implies that in 2 years the 5 year yield is expected to be equal to today's 7 year yield. This implies that the expected return on the bond is the coupon. This also implies that the expected return on an intermediate treasury fund is equal to the current yield.

C. Some combination of two where part of the curve is predictive and the other part implies a term premium. This also implies that the expected return on an intermediate treasury fund is higher than the current yield (assuming the yield curve is not inverted).

AzRunner
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### Re: Expected Future Yield Curve

I'd say "A". If things play out as viewed from today, then yes, you will achieve a greater return, if you sell in two years, but then you would have to decide where to invest the money five years earlier than if you held until maturity.

The logic in "B" does not make sense. Borrowing for seven years is different than borrowing for five years. In two years the seven year bond will be a five year bond, as explained in "A".

As long as you are happy with the yield of the fixed income instrument, I would buy the bond whose maturity does not exceed your anticipated holding period. Then you have a reasonable chance of being happy with your investment for the entire holding period. For this, you can treat the bond fund as an individual bond with a maturity that is equal to the fund's duration.

acegolfer
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### Re: Expected Future Yield Curve

Kevin M
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### Re: Expected Future Yield Curve

I don't really know, and don't try to predict interest rates, but voted for the C option since it makes the most sense to me. Still, I'd rather take 2.3% for a 5-year direct CD and not take the term risk of the 7-year treasury or the intermediate-term treasury bond fund.

Kevin
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Spec7re
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### Re: Expected Future Yield Curve

Kevin M wrote:I don't really know, and don't try to predict interest rates, but voted for the C option since it makes the most sense to me. Still, I'd rather take 2.3% for a 5-year direct CD and not take the term risk of the 7-year treasury or the intermediate-term treasury bond fund.

Kevin
If you assume A, then the expected return on a 7 yr note held for 2 years is actually 3.42%.

lee1026
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### Re: Expected Future Yield Curve

The yield curve is usually curved upwards, so rolling a long termed bond for some time and then selling it as a shorter termed bond yield average out to better returns then holding it to maturity. That said, you are also taking more risk in the process.

Kevin M
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### Re: Expected Future Yield Curve

Spec7re wrote:
Kevin M wrote:I don't really know, and don't try to predict interest rates, but voted for the C option since it makes the most sense to me. Still, I'd rather take 2.3% for a 5-year direct CD and not take the term risk of the 7-year treasury or the intermediate-term treasury bond fund.

Kevin
If you assume A, then the expected return on a 7 yr note held for 2 years is actually 3.42%.
But I said C, not A, and I also said I don't really know. If riding the yield curve produced a risk-free premium then it would be arbitraged away, and the yield curve would flatten until there was no more risk-free premium. Institutional bond traders understand this phenomenon, and I I'm sure that they're much more knowledgeable about this than I am. I don't want to take the other side of the bet.

Kevin
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Angst
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### Re: Expected Future Yield Curve

I feel I have to be fairly agnostic about this sort of thing, although I willingly believe in the truthfulness of the current yield curve:
http://www.treasury.gov/resource-center ... data=yield

A), that's my scripture.

nisiprius
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Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

### Re: Expected Future Yield Curve

Inflation went from 6.8% in 1977 to 13.3% in 1979, and I think interest rates did much the same thing, so I think it is an exercise in futility to be making predictions of nominal rates to ±0.25% precision.

Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

stlutz
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### Re: Expected Future Yield Curve

We discussed in another thread last year that estimating the return of a "bullet" strategy as outlined here is really same as estimating the return on a rolling bond ladder.

If you are rolling, say, a 10 year ladder, then your expected return in determined by the yield of the bonds when you buy them. In fact, held to maturity that is your guaranteed return. So, a 10 year bond yields 2.57% right now. So, I would say that a bullet strategy with a 5 year bond should be expected to return the same annually.

For this scenario posited in the poll, we would compare it something more like a 13 year ladder, where the yield on new bonds is about 2.8%. So, I would expect the 5 year yield to end up wherever a return forecast of 2.8% per year would cause it to end up (too lazy to run the calc. now).

Of course, there are no good forecasts of the future shape of the yield curve, so the error bars here should be plenty large.

Spec7re
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Joined: Tue May 06, 2014 2:56 pm

### Re: Expected Future Yield Curve

Using data from multpl.com, it seems the average yield difference between the 5 and 7 year note is 0.32.

This represents an excess expected return above the coupon of 0.32*5/2=0.8. So expected return would be 2.24 + 0.8 = 3.04.

If you assume that 0.32 difference between 5 and 7 yr represents the term premium, then you could deduct that a greater than 0.32 difference in yield represents an expectation of rising interest rates and a less than 0.32 difference an expectation of falling interest rates.

magician
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### Re: Expected Future Yield Curve

What's the 2-year rate today?
Simplify the complicated side; don't complify the simplicated side.

market timer
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### Re: Expected Future Yield Curve

If there is no term premium, then the return from owning any Treasury bond for two years should be the same as the current return on 2-year Treasuries, i.e., 0.55%. Using your 2.24% number for the 7-year, that means after two years the 7-year would be expected to become a 5-year yielding roughly 2.92%.

acegolfer
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Joined: Tue Aug 25, 2009 9:40 am

### Re: Expected Future Yield Curve

The following is a bit technical. But I'll explain how to properly calculate the expected yields in the future.

If one want to predict the future interest rate, then the best method is to use a term structure model, which explains the evolution of short rate (a very short term interest rate such as 1-day) dynamics. BTW, these models are not just about explaining the current yield curve, it's about explaining the possible future yield curves. Specifically, the model explains the short rate movement in the future and we can also construct future rates for all terms such as 5-year zero rate. And with the zero-rates for all maturities, we can then calculate the yield of a 5-yr coupon bond.

The bad news is that I think the term structure models are beyond the scope of average BH. One of the known facts of interest rates is mean reversion, which is embedded in term structure models. A very simple term structure model looks like

dr_t = kappa (theta - r_t) * dt + sigma * dWt, where theta is the mean interest rate.

Without providing the average interest rates in OP, it's impossible to predict what the 5-yr yield will be in 2 yrs. Predicting the 5-yr yield using the current 2-yr and 7-yr rates will require too many strong assumptions, which doesn't happen in reality.

Even if understanding term structure models is hard, if anyone is interested in models that advanced fixed income experts use, try out the following term structure models.

discrete time models: Ho-Lee model, BDT model
continuous time models: Merton model, Dothan model, Vasicek model, CIR model, HJM model

Once you have these models with parameters (which can't be estimated using 5-yr and 7-yr yields alone), then you can find all the possible rate dynamics 2 years from today. With that information, one can calculate the expected 5-yr yield using either binomial model or solving the partial differential equation using Feyman-Kac formula.

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Joined: Mon May 07, 2007 12:33 pm

### Re: Expected Future Yield Curve

What are the 4 choices? Are they:

exactly 1.77
exactly 2.24
between 1.77 and 2.24
other = below 1.77 or above 2.24

skepticalobserver
Posts: 937
Joined: Tue Jul 29, 2014 11:29 am

### Re: Expected Future Yield Curve

They say there are two types of economists: those who don't know where interest rates are heading and those who don't know they don't know where interest rates are heading. Accordingly, buy your fixed income securities for the coupon and hold to maturity. As to maturities, I try to do what the banks are supposed do: lend short and borrow long. Be mindful that there is no practical way for the retail investor to hedge a fixed income investment; the institutional investors try, but as recent events have shown, they often get it wrong.

Keep in mind, treasuries are very marginable and have excellent spreads when you need principal. Corporates, agencies and other "spread" products tend to have liquidity, and in the case of corporates, credit problems. And for those of you who reside in "tax hell" states (NY, NJ, MD...), Treasuries are state tax free too!

Johno
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Joined: Sat May 24, 2014 4:14 pm

### Re: Expected Future Yield Curve

acegolfer wrote:The following is a bit technical. But I'll explain how to properly calculate the expected yields in the future.

If one want to predict the future interest rate, then the best method is to use a term structure model, which explains the evolution of short rate (a very short term interest rate such as 1-day) dynamics. BTW, these models are not just about explaining the current yield curve, it's about explaining the possible future yield curves. Specifically, the model explains the short rate movement in the future and we can also construct future rates for all terms such as 5-year zero rate. And with the zero-rates for all maturities, we can then calculate the yield of a 5-yr coupon bond.

The bad news is that I think the term structure models are beyond the scope of average BH. One of the known facts of interest rates is mean reversion, which is embedded in term structure models. A very simple term structure model looks like

dr_t = kappa (theta - r_t) * dt + sigma * dWt, where theta is the mean interest rate.

Without providing the average interest rates in OP, it's impossible to predict what the 5-yr yield will be in 2 yrs. Predicting the 5-yr yield using the current 2-yr and 7-yr rates will require too many strong assumptions, which doesn't happen in reality.

Even if understanding term structure models is hard, if anyone is interested in models that advanced fixed income experts use, try out the following term structure models.

discrete time models: Ho-Lee model, BDT model
continuous time models: Merton model, Dothan model, Vasicek model, CIR model, HJM model

Once you have these models with parameters (which can't be estimated using 5-yr and 7-yr yields alone), then you can find all the possible rate dynamics 2 years from today. With that information, one can calculate the expected 5-yr yield using either binomial model or solving the partial differential equation using Feyman-Kac formula.
Calculating the 5 yr rate two years forward requires knowing the current 2 yr and 7yr spot rates, 5yr rate not relevant. The forward rate is not necessarily the expected rate, but the type of models you mention would in the vast majority of cases in the hands of practitioners be calibrated so that the 'expected' rate was equal to the forward rate. The main use of such models, after also inputting rate volatility parameters (derived from various simple rate option prices), is to price more complex interest rate options, or price multiple simple options with a single set of inputs (rather than using a simpler model where the 'volatility of rates' input differs from one option to the next). Those prices depend on the hedging rates the trader can lock in now, the current forward rates. So the model must be calibrated to 'find' an expected rate equal to the current forward rate.

The difference between forward rate and expected rate is some function of relative risk preference of borrowers and lenders. In general lenders are believed to be more averse to rate risk than borrowers, so the market clears where forward rates are higher than expected rates, though there's no general axiom saying that's always true. But the difference can't be observed directly, nor AFAIK are the models mentioned practically useful for trying to derive what it is.

acegolfer
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### Re: Expected Future Yield Curve

Johno wrote: Calculating the 5 yr rate two years forward requires knowing the current 2 yr and 7yr spot rates, 5yr rate not relevant. The forward rate is not necessarily the expected rate, but the type of models you mention would in the vast majority of cases in the hands of practitioners be calibrated so that the 'expected' rate was equal to the forward rate. .
That statement is only true in the risk-neutral world. Yes, we calibrate the expected future rate such that it is equal to the forward rate in that risk-neutral world.

But in the risk-natural world (the world we live in), the expected rate using those model is far different from the forward rate, which is the question in this thread. In fact, each model results in different expected future rates.

Johno
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### Re: Expected Future Yield Curve

acegolfer wrote:
Johno wrote: Calculating the 5 yr rate two years forward requires knowing the current 2 yr and 7yr spot rates, 5yr rate not relevant. The forward rate is not necessarily the expected rate, but the type of models you mention would in the vast majority of cases in the hands of practitioners be calibrated so that the 'expected' rate was equal to the forward rate. .
1. That statement is only true in the risk-neutral world.
2. Yes, we calibrate the expected future rate such that it is equal to the forward rate in that risk-neutral world.
3. But in the risk-natural world (the world we live in), the expected rate using those model is far different from the forward rate, which is the question in this thread. In fact, each model results in different expected future rates.
1. I don't believe I made any statement about the real world that's only true of a risk neutral world. You'll have to be more specific.
2. To use the models you mentioned, we calibrate the *model* (if that's what you mean by 'calibrate the expected future rate') so that it returns the forward rate as the expected rate. Otherwise the model is of no use to price options, which is what the models you mention are used for.
3. The models such as you mention have no separate degree of freedom by which they determine a separate forward rate and expected rate, nor any intrinsic explanatory power to determine that difference. As in point 2, once you calibrate them for their practical use, options pricing, they return expected rate=forward rate. And if you want to calibrate them to return what you believe should be the actual expected future rates, there's no directly observable market parameter to tell you what those inputs should be anyway.

acegolfer
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### Re: Expected Future Yield Curve

Johno wrote: 1. I don't believe I made any statement about the real world that's only true of a risk neutral world. You'll have to be more specific.
2. To use the models you mentioned, we calibrate the *model* (if that's what you mean by 'calibrate the expected future rate') so that it returns the forward rate as the expected rate. Otherwise the model is of no use to price options, which is what the models you mention are used for.
3. The models such as you mention have no separate degree of freedom by which they determine a separate forward rate and expected rate, nor any intrinsic explanatory power to determine that difference. As in point 2, once you calibrate them for their practical use, options pricing, they return expected rate=forward rate. And if you want to calibrate them to return what you believe should be the actual expected future rates, there's no directly observable market parameter to tell you what those inputs should be anyway.
Ok, sorry if I misquote you. I don't want to go into details discussing how to calibrate those models.

Can we at least agree that the forward rate is the expected future rate, only under the expectations hypothesis?

If you disagree, then
Here, it is proved theoretically that the forward rate can only predict the future spot rate under the forward measure. It is therefore a biased estimator of the future spot rate under the normal situation (i.e., original probability space).
http://www.personal.psu.edu/jxh56/paper ... 02rqfa.pdf pg 264
Last edited by acegolfer on Thu Jul 31, 2014 9:06 pm, edited 2 times in total.

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### Re: Expected Future Yield Curve

tadamsmar wrote:What are the 4 choices? Are they:

exactly 1.77
exactly 2.24
between 1.77 and 2.24
other = below 1.77 or above 2.24
The most likely choice has to be either 3 or 4.

Not sure what you are getting at with this poll, but it has little to do with what you are discussing in the OP.

Johno
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### Re: Expected Future Yield Curve

acegolfer wrote:
Johno wrote: 1. I don't believe I made any statement about the real world that's only true of a risk neutral world. You'll have to be more specific.
2. To use the models you mentioned, we calibrate the *model* (if that's what you mean by 'calibrate the expected future rate') so that it returns the forward rate as the expected rate. Otherwise the model is of no use to price options, which is what the models you mention are used for.
3. The models such as you mention have no separate degree of freedom by which they determine a separate forward rate and expected rate, nor any intrinsic explanatory power to determine that difference. As in point 2, once you calibrate them for their practical use, options pricing, they return expected rate=forward rate. And if you want to calibrate them to return what you believe should be the actual expected future rates, there's no directly observable market parameter to tell you what those inputs should be anyway.
Ok, sorry if I misquote you. I don't want to go into details discussing how to calibrate those models.

Can we at least agree that the forward rate is the expected future rate, only under the expectations hypothesis?

If you disagree, then
Here, it is proved theoretically that the forward rate can only predict the future spot rate under the forward measure. It is therefore a biased estimator of the future spot rate under the normal situation (i.e., original probability space).
http://www.personal.psu.edu/jxh56/paper ... 02rqfa.pdf pg 264
I don't know much more clearly I could have stated, in both my posts, that the forward rate and the expected future spot rate are not the same thing. However the models you mentioned are not of any direct help in deriving the expected rate, which is inherently non-observable.

acegolfer
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Joined: Tue Aug 25, 2009 9:40 am

### Re: Expected Future Yield Curve

Johno wrote:I don't know much more clearly I could have stated, in both my posts, that the forward rate and the expected future spot rate are not the same thing. However the models you mentioned are not of any direct help in deriving the expected rate, which is inherently non-observable.
I disagree with your 2nd statement. You may think those models are not helpful. But I have calibrated those models and predicted the expected rates using them. IMO, it was helpful to me.