Positive Bond Convexity and You

[craigr]

Strangely I've had this blog post in a draft form for a long time and finally got around to finishing it after some discussions on this topic came up here. Hope you enjoy it:

https://web.archive.org/web/20160324133 ... y-and-you/

The short of it is that positive convexity makes long term treasuries more powerful as interest rates fall. With German and Japanese interest rates well under 2% at this point, the US bonds could follow so there is still some room for big gains in this asset class in the Permanent Portfolio.

[Alanw]

Thanks Craig.  Very informative.  Exciting time for long bonds.  We'll see how it all plays out.

[sophie]

This is great.  I'd been wondering if I shouldn't pull out of LT bonds at some point and keep the money in cash, but now I can quit worrying.  Thank you so much!

[MediumTex]

It's hard to accept, but it really doesn't matter where interest rates are at any point in time so long as they are somewhere north of zero.

I understand that in theory zero is not a barrier because interest can go negative, but I would be comfortable saying that if 30 year yields got close to zero it would be a good idea to shift to all cash and move back into bonds when 30 year yields "spiked" up to .50% or so.

Until we get to sub-1% yields on 30 year bonds, though, I don't think this will be anything anyone needs to worry about.

The bottom line is that in the current multi-decade bull market for treasuries, the biggest gains have occurred with yields under about 4.3%, which is not at all the message you would get from watching TV and reading the financial press.  The man on the street seems to believe that gains in long term bonds are mostly about coupon payments and the idea of capital gains due to interest rate moves is poorly understood by most people  The concept of convexity is something even fewer people probably grasp, though once you "get it", it makes perfect intuitive sense that bond values would be more dramatically influenced by interest rate moves that are closer to zero. 

For example, if I said to someone "Would it surprise you to learn that the value of a bond would be more significantly affected by a move in interest rates from 4% to 2% than it would be by an interest rate move from 16% to 14%?", I would expect that most people who understood the question would say "No, that wouldn't surprise me."

[Alanw]

This is great.  I'd been wondering if I shouldn't pull out of LT bonds at some point and keep the money in cash, but now I can quit worrying.  Thank you so much!

Sophie,
I entered the PP world in the 2nd Quarter of 2011 after much study and deliberation.  I literally had to hold my nose and buy LTT's at a yield of around 4.25%.  I initially set my rebalance bands at 30/20 and have already had to rebalance once because of the increase in value of LTT's.  I have now moved my rebalance bands to 35/15.

[AdamA]

It's hard to accept, but it really doesn't matter where interest rates are at any point in time so long as they are somewhere north of zero.

I understand that in theory zero is not a barrier because interest can go negative, but I would be comfortable saying that if 30 year yields got close to zero it would be a good idea to shift to all cash and move back into bonds when 30 year yields "spiked" up to .50% or so.

But even if they did become negative, don't you think the convexity idea still applies?  In other words, an investor who hung on to LTT's when rates were -1% could make a lot of money if they dropped to -2%, right?

I realize this very unlikely, but if they (LTT's) ever have negative rates, who's to say how negative they could actually become.

[Alanw]

It's hard to accept, but it really doesn't matter where interest rates are at any point in time so long as they are somewhere north of zero.

I understand that in theory zero is not a barrier because interest can go negative, but I would be comfortable saying that if 30 year yields got close to zero it would be a good idea to shift to all cash and move back into bonds when 30 year yields "spiked" up to .50% or so.

But even if they did become negative, don't you think the convexity idea still applies?  In other words, an investor who hung on to LTT's when rates were -1% could make a lot of money if they dropped to -2%, right?

I realize this very unlikely, but if they (LTT's) ever have negative rates, who's to say how negative they could actually become.

Wouldn't we have to be in an extremely deflationary environment to have LTT's become negative with investors fleeing to the safety of US Treasuries to obtain real positive yields?  And at that point, wouldn't cash under the mattress or in a safe deposit box yielding 0% interest be more favorable than negative yields?  I realize that we would lose any convexity with cash but with negative yields on LTT's we would be gambling that rates would fall even further.  It seems we might want to consider going to cash if LTT's became negative and not gamble.  In reality I don't believe we will see negative LTT's but who knows.

[Xan]

But even if they did become negative, don't you think the convexity idea still applies?  In other words, an investor who hung on to LTT's when rates were -1% could make a lot of money if they dropped to -2%, right?

I realize this very unlikely, but if they (LTT's) ever have negative rates, who's to say how negative they could actually become.

If convexity did apply in such a hypothetical situation, the price would have hit infinity when the yields were zero, and then be climbing back towards zero from negative infinity.

In other words, negative nominal rates are impossible.  (Or were we talking about negative real rates?)

[MediumTex]

But even if they did become negative, don't you think the convexity idea still applies?  In other words, an investor who hung on to LTT's when rates were -1% could make a lot of money if they dropped to -2%, right?

I realize this very unlikely, but if they (LTT's) ever have negative rates, who's to say how negative they could actually become.

If convexity did apply in such a hypothetical situation, the price would have hit infinity when the yields were zero, and then be climbing back towards zero from negative infinity.

In other words, negative nominal rates are impossible.  (Or were we talking about negative real rates?)

In a negative interest rate environment, wouldn't the value of the bonds still just rise or fall to reflect current interest rates?

Think about zero coupon bonds as an example because there is no need to include the coupon payments in the thought process.  At 0% interest rates, the value of a zero coupon bond would be exactly the same for the whole life of the bond.  If I bought the bond at $100, it would pay me $100 when the bond matured.  If rates went to 1%, the value of my $100 bond would probably go WAY down, but if interest rates fell to -1%, where people were having to pay perhaps $120 to get $100 back when the bond matured, wouldn't my 0% bond now be worth at least $120?

Of course, this is kind of a silly exercise, since presumably people would herd into the short end of the yield curve in such a case (assuming that short term yields weren't even MORE negative than long term yields), but it is interesting to think about.

[Tortoise]

AdamA and MT are right. To convince yourself, you can use an online bond price calculator like this one or this one to see exactly what happens to the bond price as yields go negative.

For example, if you buy a $1000 30-year Treasury bond at a 3% coupon rate, you get:

Yield (%)  Price ($)
---------  ---------
    5        691
    4        826
    3        1000
    2        1225
    1        1517
    0        1900
  -1        2404
  -2        3069
  -3        3953
  ...        ...

So bond prices don't shoot up to infinity as yield approaches 0. At a yield of 0, the bond price is simply equal to its face value plus the sum of its remaining coupon payments. At negative yields it just starts to require very large price increases to keep steadily pushing that yield more negative.

I suppose it's possible to have a bizarre type of "inflation" scenario where tons of new money is being created, but instead of using that money to bid up commodities, stocks, housing, or the general cost of living, investors decide instead they just really want Treasury securities. In that strange scenario, bond prices could keep going up, up, up, pushing negative bond yields down, down, down.

This LTT low-yield situation is one of the most fascinating parts of the PP strategy that I've been grappling with for some time. My opinion on the matter keeps changing as I continue to think about it. Part of me really wants to say that I'd bail out and go into cash at sub-1% or negative LTT yields. But the peaks of speculative bubbles are notoriously hard to predict accurately. Bail out too early and you miss some impressive gains. Bail out too late and you suffer painful losses. So why bail out at all?

[BearBones]

AdamA and MT are right. To convince yourself, you can use an online bond price calculator like this one or this one to see exactly what happens to the bond price as yields go negative.

For example, if you buy a $1000 30-year Treasury bond at a 3% coupon rate, you get:

Yield (%)  Price ($)
---------  ---------
    5         691
    4         826
    3        1000
    2        1225
    1        1517
    0        1900
   -1        2404
   -2        3069
   -3        3953
   ...        ...

So bond prices don't shoot up to infinity as yield approaches 0. At a yield of 0, the bond price is simply equal to its face value plus the sum of its remaining coupon payments. At negative yields it just starts to require very large price increases to keep steadily pushing that yield more negative.

I suppose it's possible to have a bizarre type of "inflation" scenario where tons of new money is being created, but instead of using that money to bid up commodities, stocks, housing, or the general cost of living, investors decide instead they just really want Treasury securities. In that strange scenario, bond prices could keep going up, up, up, pushing negative bond yields down, down, down.

This LTT low-yield situation is one of the most fascinating parts of the PP strategy that I've been grappling with for some time. My opinion on the matter keeps changing as I continue to think about it. Part of me really wants to say that I'd bail out and go into cash at sub-1% or negative LTT yields. But the peaks of speculative bubbles are notoriously hard to predict accurately. Bail out too early and you miss some impressive gains. Bail out too late and you suffer painful losses. So why bail out at all?

I am going to bring this up again because it is very important, and the bottom line passed over my feeble head.

In the related discussion (http://gyroscopicinvesting.com/forum/in ... ic=2585.30), you make a good point, Tortoise, that LTTs might theoretically reach a yield whereby they are incapable of reaching a rebalancing band (so you do not capture gains during a low interest rate "LTT ping-pong match."

In order for a 25% LTT allocation to hit a 35% rebalancing band, LTTs have to rise in price by 40%. The lowest LTT yield that supports a 40% increase in bond price is 1.3%. At yields below that, your max upside drops below 40%, so your LTTs would be increasingly unlikely to hit that rebalancing band to take profits.

Is this still true, given the discussion on positive bond convexity? I don't care about volatility if I am unable to capture it!

A related question: I wonder if market forces have to become increasingly severe to move LTT yields in a negative direction as the yield approaches 0. Therefore, at some point, the probability of moving in a positive direction greatly outweighs that of a negative (kind of like a tight rubber band getting increasingly difficult to stretch and increasingly likely to snap back).

[Tortoise]

In order for a 25% LTT allocation to hit a 35% rebalancing band, LTTs have to rise in price by 40%. The lowest LTT yield that supports a 40% increase in bond price is 1.3%. At yields below that, your max upside drops below 40%, so your LTTs would be increasingly unlikely to hit that rebalancing band to take profits.

Is this still true, given the discussion on positive bond convexity? I don't care about volatility if I am unable to capture it!

It sounds like we share a similar level of concern about this interesting issue, BearBones.

Whether it is true or not depends entirely on whether you believe it's possible for LTT yields to go negative.

If it's impossible for LTT yields to drop below zero, then there's definitely a yield below which LTTs no longer have the power to push a 25% allocation up to a 35% rebalancing band. But if negative LTT yields are possible, then who's to say exactly how negative they could possibly go?

Positive bond convexity doesn't change this situation. It's just a complicated-sounding name for a concept that can be stated in plain English: the bond price vs. yield curve increases faster than a straight line as yields decrease. All it means is that as yields decrease, the bond price's sensitivity to yield increases. But for non-negative yields, the sensitivity only increases up to a point. We know this is true because the bond price vs. yield curve crosses the y-axis; it doesn't shoot up towards infinity at 0% yield.

A related question: I wonder if market forces have to become increasingly severe to move LTT yields in a negative direction as the yield approaches 0. Therefore, at some point, the probability of moving in a positive direction greatly outweighs that of a negative (kind of like a tight rubber band getting increasingly difficult to stretch and increasingly likely to snap back).

I agree completely. I just wish I knew how in the world to calculate such probabilities. If we could do that, we could probably make a lot of money timing the stock market.

Here's what I think you need to ask yourself: If LTT yields were to reach 0% and I were to convert all of my LTTs to STTs to protect myself from an increasingly likely LTT crash, and if LTT yields then mocked me by proceeding to go negative, would I be okay with missing out on those gains? And if LTT yields were then to bounce back and forth between -1% and +1% for the next 10 years, would I be okay with missing out on the opportunity to harvest that volatility?

Those are not easy questions to answer. I still haven't answered them for myself.