# The Perplexing Convexity Of Rates, Bonds And Bond Funds

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by: Jari Ulmer

It should go without saying that Bond Exchange Trade Funds (ETFs) are not bonds. Even Maturity-Targeted Bond ETFs, which actually hold bonds until they expire, are not bonds. In a world where rates never changed the differences between these might be negligible, but they do change. When they do, and how that affects the price and yield of the \$100B+ bond fund market is more complex than most investors are probably aware.

There are many published differences between holding an actual bond and holding shares of a bond fund. That main difference I want to discuss in this article however, actually goes back to the most fundamental aspect of bonds themselves.

It's frequently rehearsed that bond prices move inversely to going Treasury rates. When rates move, the market prices existing bonds so that new buyers can get the going rate while current holders are stuck with their original coupon. The image below illustrates this concept:

If you haven't guessed it, this relationship isn't 1-to-1. In fact to calculate the price discount/premium of our bonds when rates move we need to use the equation below, plugging in the final value of the original coupon bond, and the new rate:

A good example for this concept are Zero-Coupon Bonds which rely solely on their face value discount to generate yield.

Understanding the Bond Convexity Chart

Running the math for every possible movement in rates and for every possible bond coupon is time consuming, obviously. This information is much better portrayed by a reference chart. The Bond Convexity chart shows the expected movement in a particular bond's price given a change in the underlying rate and the bond's starting coupon. The following is for a 10-Year Treasury Bond:

The red line shows how to utilize these charts. If an investor purchased a 10-year bond yielding 2%, and rates rose to 2.6% (an increase of 30%), then their bond should have lost ~5% in price.

Keep in mind this relationship is for a single, actual bond. If the holder never sells this bond it will continue to behave like a 2% bond and move accordingly. Note: these charts are built off a given maturity, every day that passes slowly changes a bond's price sensitivity.

Nonetheless, understanding this phenomenon is critical for investors looking to play movements in interest rates. The most direct introduction into this world is to become a broker and buy and sell actual bonds on the market. For the rest of us, there are maturity-targeted funds. Not accounting for liquidity or expenses these funds will behave most like their underlying yield-to-maturity when rates move. Matthew Patterson explains how even in the ETF construct these funds enable individualized yields.

The trouble with index-tracking bond funds, on the other hand, like the \$7.7B iShares Barclays' 1-3 Year Treasury Bond Fund (NYSEARCA:SHY), is they continually turnover their holdings once they reach a minimum maturity. As a result these funds are constantly adjusting their duration and resulting yield.

If we plot the returns for a high yielding Treasury ETF like iShares Barclays' 20+ Year Treasury Bond Fund (NYSEARCA:TLT),on a Bond Convexity chart for an appropriately weighted maturity (27.6 Year) we see an interesting phenomenon. When rates fall, bond funds behave only somewhat accordingly with their underlying "coupon" rate. As money revolves through the fund, it purchases more and more shares of higher yielding bonds, which pushes the behavior of the fund to that of a higher original coupon. The opposite occurs when rates rise:

Not impressed yet? Well not shown on the chart above is that this relationship for TLT actually has a 0.99 correlation since the fund's inception. That's profound. In no time since TLT began trading has it behaved like anything outside of a 1.5% to 3% coupon bond. Let me remind you the 20-Year Treasury rate hasn't dropped below 2% in the last 20 years, so what could TLT possibly be holding that behaves like a 2% yielding bond?

This correlation is extremely important. Many investors likely hold the misconception that because funds like TLT hold bonds of a specific maturity, they're somehow gaining access to "real" bonds with similar behavior. The implications of this restricted price movement plays significant games with our understanding of rates and price.

For example, if investors were clairvoyant enough to purchase TLT on December 18, 2008, when rates were 5.37% and sell on June 12, 2013, after they fell to a then low of 2.62% they would have taken advantage of a 51% drop in the going rate. According to our textbook bond-price relationship, and believing that buying TLT when rates were 5.37% somehow made those shares equate to that yield, TLT should have gained almost 95%. In reality, investors only saw 48%:

What holds these bond funds back is two-fold. The ETF structure of TLT necessitates constant creation and destruction of shares against a revolving holding of discounted/premium assets, which averages the underlying yields of the fund's bonds, flattening their growth. Additionally, I hypothesize these movements take into account that TLT's holdings will be turned over long before they mature. By cutting short the potential future cash flows from the bonds, the discount/premium subsequently experienced by the bonds is less.

Turnover itself even presents increased losses, Patterson again has a fantastic two part series (1, 2) on the often unquoted costs of "indexing."

So now you can begin to imagine the mathematical complexity involved in calculating these funds' indexes. With money flowing in and out of the funds and shares being created and destroyed constantly the overall price position against the original money flow curve of the underlying issues is constantly in flux.

The point of all this isn't to scare you away from bond funds. Rather, its that each specific bond fund will always have its own unique price convexity curve to an underlying rate and it often has nothing to do with any published rate or yield you'll find in its prospectus. Before investors considers a fund, they need to do the math and prepare their own Bond Covexity chart to fully gauge what rate changes will actually mean for their fund.

At the same time many of these funds still translate their assets' coupons into dividends. While changing rates affect bonds and bond funds' return, their dividend yield is often similar to their underlying holdings' coupon.

Closing Remarks

Understanding bond convexity is critical to projecting price movements of a bond, especially in an era of super low rates. Because of the way bond funds are structured, they do not act like individual bonds. Even funds that purport to track a specific segment of the Treasury curve do not behave anything like an actual bond from that segment.

The first step in considering a specific fund is always to build its convexity chart. If the price movements of the fund aren't what you were expecting given a swing in rates you might have to consider a leveraged approach.

If you're not comfortable with the complexities of rates and asset turnover within an indexed fund, maturity-targeted funds offer a more traditional buy-and-hold approach to bond investing.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.