Bond Funds - Higher Rates - Point of Indifference

[Electron]

The Wiki section labeled Bonds - Advanced Topics describes the Point of Indifference in a bond fund following an increase in interest rates. See the section on Duration.

http://www.bogleheads.org/wiki/Bonds:_Advanced_Topics

I've attempted to show this concept in graphical form in the attached chart. This is a hypothetical example only.

I'd be interested in any comments and if you think this analysis is correct.

The chart shows a bond fund starting out with a yield of 2%. The yield increases by 1% in January 2016, January 2018, and January 2020. Duration of the fund was assumed to be 5.5 years, and the NAV was dropped just slightly under 5.5% following each rate increase.

The lines associated with the three higher interest rates cross the 2% line in July 2021, July 2022, and July 2023. The Point of Indifference does come out very close to 5.5 years after a single rate increase. The combined Point of Indifference for all three rate increases is about 7.5 years.

[billyt]

Electron: I think your result is approximately correct. The calculation for the NAV hit is an approximation, not an exact calculation. To check this another way, you can run a calculation with an 11 year rolling bond ladder. I think the answer will be somewhat different, and the difference will say something about the uncertainties involved in the modeling.

[garlandwhizzer]

If you subtract a conservative estimate of the compound annual rate of inflation over the next 5, 10, or 15 years, say 2% -3%, all of these scenarios result in a real return of about zero over all the time periods. Real returns are what we spend in the future so I don't find these graph at all reassuring. Historically, a long period of rising interest rates starting from a point of low inflation and very low bond interest rates is not a pretty picture when it is drawn in real dollars. One such period was 1940 to 1980, during which high quality intermediate term bonds with reinvested interest lost real purchasing power for 40 straight years, even though the graphs drawn in dollars indexed to 1940 paint a very happy picture.


Garland Whizzer

[thx1138]

As billyt points out the calculation based on duration is approximate. Getting it "exact" is very onerous and I don't think would improve your illustration in any particular way.

Personally I think this is an excellent illustration of the concept! Something like this would be a good addition to that topic on the wiki.

[linuxizer]

Nicely done. I had intended to something like this for awhile now, but have been too busy/lazy. Thanks for doing this. Note that your graph also illustrates the common refrain here that long-term investors are better off after rate increases.

[billyt]

Garland Whizzer: This seems to be a recurring theme of yours. Of course, you are absolutely right that inflation is the deadly enemy of fixed income investments. However, there are a couple of other points here. Currently inflation is closer to 1% than 2%, and it is anyone's guess what happens in the future. As long as nominal bond rates stay ahead of inflation, as they are now, then there is no problem. The other thing is the existence of inflation protected securities and ibonds. Anyone who is concerned about the effect of inflation on their fixed income portfolio should own some of these.

[z3r0c00l]

One such period was 1940 to 1980, during which high quality intermediate term bonds with reinvested interest lost real purchasing power for 40 straight years, even though the graphs drawn in dollars indexed to 1940 paint a very happy picture.

How did cash do in that time?

[garlandwhizzer]

billyt wrote:

Garland Whizzer: This seems to be a recurring theme of yours. Of course, you are absolutely right that inflation is the deadly enemy of fixed income investments. However, there are a couple of other points here. Currently inflation is closer to 1% than 2%, and it is anyone's guess what happens in the future. As long as nominal bond rates stay ahead of inflation, as they are now, then there is no problem. The other thing is the existence of inflation protected securities and ibonds. Anyone who is concerned about the effect of inflation on their fixed income portfolio should own some of these.

You're right, Billy, inflation is something that I worry about a lot even though it is currently low and has been decreasing for 30+ years. Maybe substantial inflation won't show up at all in the next decade or two and bond returns will return to historical norms. But, as a contrarian at heart, I worry more about something that has been getting better for 3 decades (inflation) than a disaster that has occurred twice in the last 13 years (stock market crashes). Most of us take steps to prepare for whatever the last disaster was, but rarely do we show the same diligence in preparing for the rare and unexpected one that is not fresh in anyone's memory. To be honest, in the present environment I fear bonds much more than stocks. The good news for bond holders is that a bear market in bonds doesn't decimate your portfolio quickly and they are necessary to cushion the volatility of stocks. My allocation to bonds is 25% at present, enough, I believe, to cushion the blows of stock downturns, but I am not counting on bonds for any significant real returns in the next decade or two. Whereas for stocks, I think real returns over that long period are overwhelmingly likely.

Garland Whizzer

[linuxizer]

billyt wrote:

Garland Whizzer: This seems to be a recurring theme of yours.

You're right, Billy, inflation is something that I worry about a lot even though it is currently low and has been decreasing for 30+ years.

The math is the same whether for real rates or nominal rates. The graph answers a very particular concern of many posters on this forum--what will happen to my bond fund when rates rise.

All of these scenarios result in a real return of about zero over all the time periods.

That's exactly the point--when you hold for D years past an interest rate change, you get the same return.

That rates are low right now (which is your concern) is commonly known and doesn't really add anything to Electron's noble effort.

[rustymutt]

Nicely done. I had intended to something like this for awhile now, but have been too busy/lazy. Thanks for doing this. Note that your graph also illustrates the common refrain here that long-term investors are better off after rate increases.

+1 I understand it. Good job explaining it visually.

[billyt]

For me, the most important point of the graph is not the point of indifference, but what happens after the breakeven point. It is crystal clear that higher rates lead to higher returns. My investment horizon is 30 or 40 years if I want my portfolio to outlive me. What do I care if a sharp rise in rates leads to a short term (duration plus) hit, if the increased returns mean I am that much closer to achieving my goals.

I don't understand that rising rates are a bond bear market. Since when are increasing returns a bear market? The prospect of trading a short term loss for bigger long term gains should appeal to the long term buy and hold investor. The only investors that are affected by these short term moves are those that are trying to time the bond market.

[Electron]

Thanks to all for your comments.

I forgot to mention that the chart reflects weekly compounding to provide a smoother curve. Bond yields typically change in smaller increments but the chart should show the basic principles involved. In reviewing past history, I see that there were six increases in the Fed Funds Rate in 1994 and eight in 2005. All increases in 2005 were 0.25%, while in 1994 they ranged from 0.25% to 0.75%. The Fed Funds Rate impacts short maturities while intermediate and long term rates are a function of supply and demand in the market.

It is also my understanding that estimates of bond price changes are most accurate for very small changes in the interest rate. Another factor not mentioned is that the duration of a bond index fund would typically decline as rates rise. As a result, the drop in price for a 1% increase in rates would be less for each higher rate.

[thx1138]

I don't understand that rising rates are a bond bear market. Since when are increasing returns a bear market? The prospect of trading a short term loss for bigger long term gains should appeal to the long term buy and hold investor. The only investors that are affected by these short term moves are those that are trying to time the bond market.

You show a 3% rise over six years with a duration of about 5 years.

Check 10 yr treasuries from 1956 to 1981. 3% to 15% in 25 years - that would be a negative nominal return despite the long holding period. And with inflation real prices increased by a factor of 3.5 over that period. So that would be a substantial real loss.

Understand now?

[billyt]

Thx1138: Ten year nominal returns for 10 year treasuries between 1956 and 1981 were uniformly positive and increasing over the interval. Of course, the effects of inflation were devastating.

[Electron]

I don't understand that rising rates are a bond bear market. Since when are increasing returns a bear market?

I think a bear market in bonds needs to be defined. The bear market exists while rates are actually rising. In the example shown in the chart, the bond investor had essentially no gain as measured from January 2014 to January 2020.

The bear market effectively ends when rates stop rising. At that point you benefit from the higher rates.

[linuxizer]

The bear market exists while rates are actually rising. In the example shown in the chart, the bond investor had essentially no gain as measured from January 2014 to January 2020.

The bear market effectively ends when rates stop rising. At that point you benefit from the higher rates.

Depends on the rate of rise vs. the rates bonds were purchased at....

[billyt]

As far as I know, bear markets involve losses. The losses here are very short lived (a few months?) and are zero in terms of a rolling return (I guess maybe 12 month rolling might be always positive in your example, you would have to check) Anyway, there are no significant losses here. What bear market?

[Jack]

I might be informative to overlay on this graph the same scenario involving a short-term bond fund, say, initially yielding 1% and a duration 2.5 years. This would give some indication of the trade-offs in short vs intermediate funds in a rising rate environment, which is a subject of debate on this forum. Of course there is no guarantee that the slope of the yield curve would remain constant, but at least it is a starting point for comparison.

[MnD]

I notice the subsequent new color lines immediately attain a steeper slope than the lines above them.
That implies that interest distributions in dollars and cents are assumed to rise immediately following a rate increase and NAV decline.
Otherwise the subsequent line would parallel the line above it at a set level below it for at least a period time.

In the case of TBM the decline in NAV and increase in distribution yield on a percentage basis is obvious, but an uptick in distribution amounts (which is needed for TBM to "regain" lost NAV) isn't jumping out at me. The June-July cumulative distribution was .04338 cents versus April-May distributions of .04333 cents, an increase of only 0.12%.
Contrast that to something like the G Fund that does respond immediately to changes in market yields - those have risen 50% April to July (from .12% monthly to .18%) or 33% if we compare two month averages.
https://personal.vanguard.com/us/funds/ ... =INT#tab=4
https://www.tsp.gov/investmentfunds/mon ... urns.shtml

Perhaps the "curves" indicating this tidy recovery of NAV via near-immediate higher distributions is an oversimplification or even erroneous depiction of what actually happens to the amount of distributions from a bond fund in dollars and cents following a market rate spike.

[billyt]

MnD: The assumption of an immediate rate increase is an approximation which accounts for the slowly increasing actual interest rate and the recovery of the NAV that is due to bonds getting closer and closer to maturity. So that is how it can be modeled in a simple way. The actual results you will see is a steady recovery in your NAV and a slowly increasing yield.

[Electron]

In the case of TBM the decline in NAV and increase in distribution yield on a percentage basis is obvious, but an uptick in distribution amounts (which is needed for TBM to "regain" lost NAV) isn't jumping out at me.

I don't believe that distribution amounts affect NAV which is related to the price of bonds. It is best to focus on Total Return which is also what is shown in the chart. Total Return will recover as income is reinvested at lower NAV which also implies a higher interest rate.

The four growth curves shown should represent the total return in a bond index fund if intermediate term rates happened to stay constant. The initial growth rates are correct since you are reinvesting essentially the same dividend at lower NAV. The dividend from a bond fund generally changes very slowly depending on the average maturity of the fund.

I agree with billyt that NAV and Dividend would have some relationship if rates remained stable after a rate increase. After an increase in rates, the dividend would slowly rise as bonds mature and are replaced. If intermediate term yields happen to remain the same, the NAV would be slowly changing as well as bonds get closer to maturity. Note also that the quoted SEC yield in a bond fund takes into account yield-to-maturity. A bond fund may have bonds trading at a discount, near par, and at a premium.

In the real world we won't see constant yields from bond index funds due to supply and demand factors in the market as well as changes in Federal Reserve policy. Even so, the chart should demonstrate the basic principle in play after an increase in rates.

[thx1138]

Thx1138: Ten year nominal returns for 10 year treasuries between 1956 and 1981 were uniformly positive and increasing over the interval. Of course, the effects of inflation were devastating.

Whoops! It looks like I used inflation adjusted data when I meant to use nominal data. Not sure how I did that, but you are right nominal returns are positive. Inflation adjusted are flat to very slightly negative depending on which intervals you use. Nothing makes inflation look worse than counting it twice

[billyt]

Jack: I agree that it would be nice to see a comparison with a short term fund. I compared Vanguards Intermediate term bond index (2.54% yield, 6.6 year duration) fund to their short-term index fund (0.69% yield, 2.7 year duration), using a model similar to Electron's rate change scenario (semi annual compounding to simplify). Here are the general characteristics of the results:

1) The total return of both funds is very similar during the period of the rate increases, but the intermediate term fund is a bit more volatile (gains more in yield, loses more from rate increases).

2) Neither fund shows significant, lasting negative returns.

3) After the rate increases are over, the intermediate term fund pulls ahead due to its higher yield.

I do not see any advantage to switching from the intermediate term index to the short term index, unless you care more about NAV change than total return. If rates do rise, you will not hurt your returns by going with the short term index, but you are losing return with every passing day if those rate increases don't materialize. Even worse, of course, if rates actually fall.

If you know in advance the exact timing, direction, and magnitude of interest rate changes, you might gain a small advantage by switching back and forth between funds. Good luck with that.

I have not figured out how to post a chart yet. Maybe Electron can run his model to check my result and post a chart for us.

[Electron]

I might be informative to overlay on this graph the same scenario involving a short-term bond fund, say, initially yielding 1% and a duration 2.5 years. This would give some indication of the trade-offs in short vs intermediate funds in a rising rate environment, which is a subject of debate on this forum. Of course there is no guarantee that the slope of the yield curve would remain constant, but at least it is a starting point for comparison.

I added a Short Term fund with Initial Yield of 1.5% and Duration of 2.5 years. Short term yields also rose by 1% on the same dates as the Intermediate Term fund. The Point of Indifference for a single rate increase came out very close to 2.5 years. The value of the fund shares was dropped by 2.5% at each rate increase.

The starting point in dollars was adjusted higher to make a cleaner looking chart. This may add a slight visual error relative to the other fund even though the percentage increases are correct.

[billyt]

Electron: Your results are somewhat different than mine. I get that the intermediate fund clearly outperforms after the rate increases are over. Before that they are neck and neck, with a little more volatility in the intermediate. I think it makes an important difference if you use actual data (interest rate and duration) for 2 comparable funds.

Today's yield curve is much steeper than in your model, with differences of about 1% between short and intermediate fund rates. It makes a difference. Comparing apples to apples for Vanguard funds:

Treasury bonds: short 0.23% yield; 2.3 year duration, intermediate 1.30% yield; 5.3 year duration

Investment grade bonds short 1.37% yield; 2.4 year duration, intermediate 2.74% yield; 5.4 year duration

Bond index: short 0.69% yield; 2.7 year duration, intermediate 2.54% yield; 6.6 year duration.

We are not the only people that are playing with bond models in the world, and I imagine if there were any clear short term difference in the outcomes of different maturities, the market would quickly (instantly?) arbitrage it away.

[Electron]

I think it makes an important difference if you use actual data (interest rate and duration) for 2 comparable funds.

Here is a chart with the two bond index funds that you suggested. The Short Term Bond Index has a large percentage in Treasury and Agency securities which results in a fairly low yield. That was one reason that I chose an actively managed short term fund in my earlier example.

In this chart I assumed that the yield curve steepens over time. The rate increase is 0.5% for the Short Term Index and 1.0% for the Intermediate Term Index.

[billyt]

Thanks for that Electron. That is more of a dramatic win for the intermediate term bonds than I got, probably because you steepened the yield curve. I think that it is more typical for the opposite to happen when the Fed is cranking up the rates.

[Kevin M]

I think the OP makes the most important point, so nice job! Each subsequent rate increase pushes the point of indifference further out in time relative to the first rate increase. I have seen many posts indicating that some folks don't understand this.

I agree that something like this would make a good addition to any Wiki article on duration as it applies to bond funds.

Kevin

[billyt]

No, the most important point is that higher rates lead to higher returns.

[Chris M]

I agree these graphs would be good addition to the wiki. But I'm trying to reconcile the graphs with an example in a paper by Vanguard that was highlighted by another recent thread. The paper is here

https://advisors.vanguard.com/iwe/pdf/ICRROL.pdf

All of the graphs in this thread, for all the various examples, show that the point of indifference equals, or comes close to equalling, the duration. But in Figure 1 of the Vanguard paper the authors give an example of a 3% rise in yield, from 2.1% to 5.1%, that occurs in the course of a single year. Vanguard takes the average of 2.1% and 5.1% as the yield in the first year, then 5.1% thereafter. The duration is 5.5 years. Although Figure 1 only provides Vanguard's calculations out to 5 years, when I extend the calculations I don't reach the point of indifference until around 6.5 years. At 6.5 years I calculate cumulative returns of about 14.5% regardless of whether or not the 3% rate rise occurs.

I'm hoping someone could check my arithmetic, and explain the cause of the discrepancy. Am I making a mistake? Is Vanguard making an assumption different from electron's?

Thanks in advance for your help.

Chris

[thx1138]

I'm hoping someone could check my arithmetic, and explain the cause of the discrepancy. Am I making a mistake? Is Vanguard making an assumption different from electron's?

Yes the assumptions are different. Remember duration is only an approximation and the larger the interest rate change the less accurate it is. So Vanguard's instantaneous 3% change is huge (as they point out in the paper almost un-historical in magnitude) and will "break" the duration rule of thumb more than electron's 1% spread out changes. There may also be differences in Vangaurd's model and electron's model that are more subtle.

[Kevin M]

Remember duration is only an approximation and the larger the interest rate change the less accurate it is. So Vanguard's instantaneous 3% change is huge (as they point out in the paper almost un-historical in magnitude) and will "break" the duration rule of thumb more than electron's 1% spread out changes.

Yes, but it should "break it" in the other direction for the aggregate bond fund, and not by that much.

The rule of thumb does not account for convexity; i.e., that the relationship between price change and rate change is non-linear. For a non-callable bond, the curve is such that rate increases at higher rates result in less price reduction (a good thing for bond investors). Vanguard does not account for convexity, but simply assumes a linear duration model; hence the -16.5% drop for a 3 percentage point increase in rates for duration of 5.5 years.

Using actual bond calculations, a bond with a yield and coupon of 2% with a term to maturity of six years has a modified duration of 5.63. With a coupon of 2% and yield of 5% (representing a 3 pp instantaneous rate increase), price is 84.61, so a drop of -15.4%--about 1.5 pp less than the -16.9% predicted by the linear duration model. Incidentally, modified duration drops to 5.51 with the yield increase to 5% (note that the Vanguard paper mentions that duration changes as yield changes, but doesn't give any examples).

Note that the Vanguard paper never mentions the point of indifference concept, although it does discuss time to recover the loss.

Kevin

[Chris M]

Thanks much for the clear explanations Kevin and thx1138.

[Electron]

If anyone would like to model a bond fund following a rise in rates, you can do quite a bit with just a calculator. You can also see how the Point of Indifference works.

Here is an example for a bond fund with duration of 5 years and current SEC yield of 2%. Assume dividends are reinvested.

If rates remain unchanged for 5 years, a $1000 investment would increase in value to $1000 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 = $1104.08.

Alternately, if the yield of the fund suddenly increased to 3%, the value of the fund shares would be expected to drop 5%. That is based on the duration of the fund times the percentage change in rates (5 x 1% = 5% = 0.05). Create a multiplier using (1-.05) = 0.95. The new fund value is $1000 x 0.95 = $950.00.

If rates remained at 3% for 5 years, the fund value would become $1000 x 0.95 x 1.03 x 1.03 x 1.03 x 1.03 x 1.03 = $1101.31.

Note that the value of $1101.31 after 5 years is very close to the first case where rates remained at 2%. That is the Point of Indifference and is very close to the duration of the fund which is 5 years. After this point the investment earning 3% would pull ahead of the earlier case earning 2%.

With a little effort, you can model multiple rate changes. Just make sure you calculate the correct multiplier for each rate change and then compound at the new rate until rates change again.

[CWRadio]

Electron
Have you done any back testing of your chart?
Between 1953-1963, when interest rates were last this low, and 10 year bonds went from just under 3% to just over 4%.
Thanks

[Electron]

Have you done any back testing of your chart? Between 1953-1963, when interest rates were last this low, and 10 year bonds went from just under 3% to just over 4%.

It would be difficult to apply the chart technique to actual periods such as 1953-63. There would be many small rate changes along the way.

You might look at Putnam Income A (PINCX) on Morningstar. That fund goes back to 1954. Click "Maximum" on the chart to see the full period.

http://quote.morningstar.com/fund/chart ... ture=en-US

We don't know how the fund was being managed but the chart may provide some useful information.