Thursday, January 22, 2015

First Derivatives and Second Moments

A common argument that bond funds and bond ladders are identical involves their duration. Duration, though precisely defined mathematically, is roughly the number of years it would take a bond or fund to recover the capital loss from a 1% increase in interest rates (yield).

An increase in interest rates would lower the price of the bond but the bond's future interest payments could then be reinvested at the higher yield and would eventually make up for the capital loss. A bond with a duration of 5 would recover a 1% capital loss in about 5 years.

Another way to look at duration is that is the percentage capital loss one would expect from a 1% increase in interest rate. So, that same bond with a duration of 5 would lose about 5% of its value if interest rates rose 1%. (The opposite would happen if rates fell 1%.)

The argument goes like this. If the duration of the TIPS bond ladder and the average duration of the bonds in the fund are the same, then the risk of the ladder and the fund are identical. Therefore, it doesn't matter which you buy.

There are a couple of problems with this argument. First, because the ladder locks in yields and the fund doesn't, their returns won't be the same unless interest rates happen to remain unchanged over time. And second, two investments with the same duration don't necessarily have the same risk.

I recently had a brief chat with Professor Moshe Milevsky at York University in Toronto. I told him that I had read opinions that a TIPS bond fund with the same duration as a TIPS ladder provides equal risk for retirees. In other words, it was suggested that owning a 5-year ladder of individual TIPS bonds with an average duration of about 2.5 years has the same risk as owning a TIPS bond fund with a duration of about 2.5 years.

Dr. Milevsky responded, "Duration is just one moment. [I] would like to match [the] second moment (convexity) and perhaps higher, before I agree.”

That response didn’t help me a lot, because he seemed to be saying that he would agree that there is no significant risk difference between the two so long as I could match several risk factors. But, I could only match several risk factors by holding nearly identical bonds in both the ladder and the fund, and of course those would have equal risk.

Notice what Dr. Milevsky didn't say – that it makes no difference because ladders and funds are identical.

"To match all those moments, don't you effectively need to hold the same bonds in the ladder as in the fund?” I then asked.

"Good point,” he replied. "Match all moments and you get the same portfolio (of strips.) To get "reasonably" close, give me two moments.”

Now, that was something I could work with.

I have often written that the duration of a bond is the percentage loss of a bond's value that would result from a 1% increase in interest rates, but that is only precisely correct for small interest rate movements. The real amount of loss (or gain) a bond will experience also depends on how much rates change. Duration is just an estimate of bond price sensitivity when the interest rate change is very small.

In order to compare the risk of a bond fund to a bond ladder, or to a different bond fund for that matter, simply knowing their durations isn’t enough information. We also need to understand at least their convexities. Higher moments would allow us to make an even better comparison of risks, but duration and convexity get us “reasonably close.”

I try to avoid the weeds on this blog, but please bear with me and I promise to bring us back out of them and onto smoothly-mowed lawn as quickly as possible.

The following chart from shows how much a change in interest rates (yield) along the x-axis changes the price of the bond along the y-axis. Duration and convexity can be calculated for both bonds and bonds fund.

The red line shows bond duration and illustrates the fact that bond prices move in the opposite direction of yields. Duration is one estimate of interest rate risk. A bond with a duration of 5 will decline in value about 5% for every 1% decline in interest rates. But duration is just a first-order estimate of the impact of an interest rate change on a bond’s price.

(Duration is the first derivative of the price/yield curve, the blue curve, for anyone who remembers first semester calculus. And because it is the first derivative, it is the calculation of duration at a single point along that curve. At any other point on the curve this is an estimate of the curve's slope. Convexity is the second derivative.)

The precise change in the bond’s price as yields change is shown by the blue curve. The yellow area shows the estimation error of the bond’s duration. The larger the yield change, the greater the error.

The next chart is similar, but adds a second bond. Bond A has greater convexity (a sharper curve) than Bond B.

Notice the red arrow. Near the current yield and price at point (*Y,*P), the duration and convexity of both bonds are identical. But, as the yield moves farther to the right or left along the x-axis, Bond B’s price changes differently than its duration predicts.

Duration predicts that the price change will be linear, but it will not be. In fact, though this simplified chart shows the curves as symmetrical, there is more error when bond yields decline than when they rise.

In other words, duration is a good estimate of expected price change if yields increase or decrease just a little, but the difference (estimation error) becomes substantial if yields change a lot in either direction.

Also notice that Bond A’s price (**P) changes less than Bond B’s price (**P) for the same change of yield. Bond A has greater convexity than Bond B.

So, if your bond fund looks like Bond B and a ladder looks like Bond A, they both have the same duration but your fund has more interest rate risk. A rate increase from *Y to **Y will cause Bond A (the ladder) to fall from price *P to price **P, but Bond B (your fund) will fall farther, to price **P.

Of course, you might be comparing a ladder with greater convexity than your fund. Your fund could look like Bond A and the ladder like Bond B, in which case the opposite would be true, but the point here is only that they are different.

There are a lot of ways to build bond portfolios (funds or ladders) with the same duration. It is far more challenging to build two bond portfolios with both the same duration and the same convexity and, therefore, the same risk. (More challenging unless, as I suggested to Dr. Milevsky, we put the same bonds in both the ladder and the fund, but that isn't an interesting scenario.)

What does this have to do with the ladder-versus-fund debate? It throws a monkey wrench into the argument that a TIPS bond fund has the same risk as a TIPS bond ladder if the duration of the two is the same. The risk is “reasonably close”, according to Dr. Milevsky, only if the convexity of the fund also matches that of the ladder.

Some advisers suggest mixing funds of different durations to achieve the duration you need. Mix a fund with a duration of 4 years with an equal amount of a fund with a duration of two years, for example, to create a fund of funds with a 3-year duration. This might work for duration, depending on your needs and whether workable funds exist, but that math doesn’t work with convexity. (An excellent Powerpoint lecture explaining why can be found here.) It will be impractical to combine funds to generate both the duration and convexity of the ladder you seek to replace. And at some point, buying the individual bonds is just a lot less work.

This all assumes, of course, that you can learn the convexity of a fund and that the manager will hold it steady while you own it. Bond durations are fairly easy to find online, even if they aren’t guaranteed over time, at places like, but funds don’t typically even report their convexity.

Here is where I honor my promise to return from the weeds. A TIPS bond fund and a TIPS ladder won’t have the same risk unless they both hold about the same bonds in the same proportion. Good luck finding a bond fund that needs the same bonds as you. A fund and a ladder can have the same price, yield and duration but if one has lower convexity, their risk is different.

The intent of my post today is to dispel the notion that a TIPS bond fund and a TIPS ladder are no different so long as they have the same duration. Duration is a first-order estimate of interest rate risk. A ladder and a fund can have the same duration but different amounts of risk. Even if the risk is similar enough, they won't have the same expected return.

A bond fund is not a bond ladder unless they effectively hold the same bonds. A bond fund is not even another bond fund.

Intermediate TIPS bond fund IPE has a duration of 6.84, a 5-year return of 4.08% and a 5-year standard deviation of 5.54%. Intermediate TIPS bond ETF TIP has a duration of 7.62, a return of 3.97% and standard deviation 5.08. Though they are both “intermediate term TIPS bond funds”, TIP has a longer duration, similar return and 8% less volatility. We don't know the convexity of either.

Because ladders lock in current interest rates and bond funds continue to track rate changes after shares are purchased, a TIPS ladder will have a different expected return than a TIPS bond fund. A fund could have the same risk as a ladder if duration and convexity match, but achieving this with mutual funds is not practical. A fund's convexity is not generally available information and, even if it were, the fund manager makes no commitment to maintain it over time.

This shows that ladders and funds are not identical unless they hold very similar bonds in the same proportion, but it doesn't show which is better, under what conditions it is better, and when it is superior enough to matter.

It largely depends on how you will use them. More on that next time. (See Funds and Ladders: What Matters?) And, if you don't enjoy the math, it should be more interesting.


  1. Very interesting Dirk ... and quite correct for the level of detail on what is essentially for a short period of time for the near term income generation needs (unless one were to construct the ladder for the entire retirement period - an unknown in itself - ref: period life tables change that remaining time period as you age).

    I've always argued that, in reality, the whole picture is one of total return where gains and dividends as well as principal, are also important sources of income. Secondly, once the short term money (bonds) are spent on living expenses, those need to be replaced and it becomes a rolling replacement of spent money.

    Thus, it is important to understand the dynamics of both the near term bucket (bonds or funds) as well as the dynamics of the long term bucket (its' overall portfolio mixture math) used to replace spent money from the short term bucket. What is important is to measure and monitor how spending may be prudent over the entire time period you have remaining, where that end age keeps shifting out ahead of you as you age. Over reliance on either funds or a ladder may lead to spending too much if markets misbehave badly.

    And, as Kitces recently posted, essentially saying we all live in an interconnected world investing in the same stuff at the end of the day so risk retention or risk transfer may result in the same thing because everything possible is contained on the same blue marble we all live on.

    Excellent post on forgotten math Dirk! I, for one, will be interested in part 2 you've alluded to.

    1. Larry, as I sit at my favorite coffee shop – this blog's namesake, in fact – on this beautiful, warm Carolina winter day and start to write that next post, I find your comments an excellent segue. Thanks!

      I hadn't read the Kitces comment, but it reminds me of a message from a Bernstein explanation from long ago that the key to how much we save for retirment isn't an absolute number but that we save more than everyone else.

      Thanks for writing!

  2. To make a true apples-to-apples comparison of a bond fund to a bond ladder, don't you also need to look at the dispersion (standard deviation) of durations/maturities of the set of bonds in each? For instance, a bond fund and a bond ladder could each have an average duration of 10 years. But the ladder might have bonds that mature every year for the next 20 years whereas maturities in the fund might be all in the 8-12 year range.

    Would differences in dispersion of maturities be reflected directly in convexity differences? Or is dispersion a separate factor to employ in fund-to-ladder comparisons? At any rate, it doesn't make a lot of sense to speak about the effects of a *singular* interest rate change when talking about the behavior of a bond ladder with a wide range of maturities (unless you're convinced that rates along the entire maturity spectrum can be counted on to move uniformly up or down).

  3. Great post for lay non-bond guru people! It reinforces my current thoughts regarding using rolling TIPS ladders for guaranteed income requirements for the next 5-10 years where you spend the current maturing TIPS while buying another one for the far end. The TIPS fund would not provide guaranteed income but could fluctuate in value and yield significantly with the market.

    The TIPS bond fund would have its place in the asset allocation portfolio that would be rebalanced periodically and would be the source of a variable income with a safe withdrawal rate approach of some sort. The purchase of the new TIPS bond for the far end of the guaranteed income ladder would come from that portfolio.

  4. Good questions.

    Convexity does reflect the dispersion of durations in a portfolio of bonds.

    Convexity of a single bond is primarily determined by the coupon rate and the number of those coupons remaining until maturity (time), assuming the bond has no call features. (With call features, convexity can even be negative.) A bond with a high coupon rate has less convexity for the same maturity date. Zero-coupon bonds have the highest convexity.

    The dispersion of durations you describe impacts the convexity of a portfolio of bonds (this is explained in the link above). A barbell-like portfolio will have greater convexity than a bullet-like portfolio, all other factors held constant. This is a key reason you can't rely on the average duration for comparison.

    As the reference above explains, "If there is a parallel shift in rates, the barbell will outperform. If the shift is not parallel, anything could happen."

    And you are correct, these comparisons usually assume a parallel shift of rates and not a change in the yield curve and that is precisely because anything can happen if the shift is not parallel.

    Excellent questions. Thanks!

  5. I don't understand why all the original and concluding statements restricted the argument to TIPS. I cannot see why any of the explanations would not apply to all bonds/bondfunds.

    1. You are correct. The duration and convexity explanations apply to all types of bonds and bond funds. You can even calculate the duration for stocks.

      Other types of bonds, however, introduce credit risk (the U.S. government is the safest bond investment in the world) and inflation risk. Restricting my discussion to TIPS bonds allows me to ignore those two risks and simplify the comparison. TIPS bonds are safe enough to consider a "floor" investment, while many other bond types are inappropriate.

      Duration and convexity, however, work the same with all bonds.

    2. A bit more explanation on the credit risk issue. This post is a comparison of funds and ladders for retirees. TIPS and other Treasuries have no credit risk, so there is no need for diversification. You could make the same duration/convexity argument for say, corporate bonds, but you do need to diversify corporates. Funds provide better diversification than ladders. For that reason, the diversification advantage of owning a fund of many corporate bonds will, in most cases, outweigh any advantages of holding a ladder, so the duration/convexity issue is far less important.

      I don't recommend that retirees fund the risk-free portion of their portfolio with corporate bonds because of their risk and lack of inflation protection. If I did, I would recommend they be held in a fund and not a ladder for greater diversification. Corporate bonds may well have a place in your risky portfolio for diversification purposes, however, and I recommend they be owned in a fund.

  6. "the U.S. government is the safest bond investment in the world": how do you know?

    1. I suppose no one can really know that with certainty, but it's where everyone else in the world invests in troubled times, so it's at least a consensus.

    2. We don't know. But if it becomes unsafe, then you can throw out pretty much every other investing guideline as well because we will be undergoing a financial and fiscal crisis bigger than the 2008-9 Financial Crisis, the 1974-1982 inflation period, and the 1929-1940 Great Depression.

      There aren't many investment strategies that can survive Russia in 1917 or the Weimar Republic.