Expected short-term return of a longer-term bond fund

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grabiner
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Expected short-term return of a longer-term bond fund

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Summary: Assuming an efficient market, the best estimate of the 1-year return of a fund with an N-year duration (or a bond) is the SEC yield of a 1-year bond, plus N times the historical average slope of the yield curve from N years to N-1, plus N-2 times the historical average slope from N-1 years to 1 year. This is a correction to a previous discussion in Expected return of a bond fund

The SEC yield of the bond or fund itself is less relevant over a short time, because it also reflects expected changes in yields. (If yields do not change, the fund will return more than the SEC yield.) However, the SEC yield is still important for investors because the SEC yield of an N-year bond is the N-year return on that bond, and investors with a time horizon less than N years should not be holding N-year bonds.

In particular, the SEC yield is the right yield to use for comparison to paying down a loan. If you make an extra payment on your 10-year mortgage, your return is the same as if you bought a 10-year bond with a yield equal to the mortgage rate, since you will get the money back with interest in 10 years.

The reason that the irrelevance of the SEC yield to 1-year returns follows from an efficient market is that the expected return of an N-year bond should be the same as the expected return of a 1-year bond, plus a risk premium. The yield curve may change, or even invert, reflecting investors' expectations of future interest rates, but this does not change the risk premium.

The SEC yield of a bond fund is defined as the average yield to maturity of the bonds in the fund, less expenses. (The official definition is somewhat confusing; it is based on the "net investment income" of the bonds over the last 30 days, which includes not only accrued dividends, but also amortized premiums and discounts.) Therefore, it makes sense to do calculations with yield to maturity, as the expenses on any bond fund are subtracted from the expected return.

Bond yields are quoted as double the six-month yield, rather than as the annual yield, because most bonds pay coupons every six months. Thus a $1000 bond which pays a $20 coupon every six months has a 4% yield, even though the investor's one-year return will be 4.04% if the coupon is reinvested in a similar bond and the bond yield doesn't change. The calculations below will use annual yields because this is simpler, but will mention the SEC yield in the numbers.

The simplest computation is for zero-coupon bonds. If the bond is held for one year and rates do not change, the return is the product of the duration and the one-year slope of the yield curve.

Consider some examples with numbers. Say a one-year zero-coupon bond worth $1000 at maturity is currently worth $975, and a two-year bond is worth $945, and bond investors do not expect rates to change. The two-year bond has a 2.85% yield to maturity (computed as 2*[(1000/945)^{-1/4}]; the annual yield is 2.87%), and the one-year bond has a 2.54% yield to maturity (annual yield of 2.56%). If rates do not change over the next year, the two-year bond will be worth $975 next year, for a 3.17% one-year return. The estimate above gives 3.18%.

Now, suppose that the two-year bond is worth $955, yielding 2.32% (annual yield 2.33%). Investors would only buy such a bond if they expect rates to fall; otherwise, they could buy a one-year bond, earn 2.56%, and then earn more over two years by buying a new one-year bond. If the risk premium is the same (based on the possibility that rates might fall more or less than expected), investors should still expect a 3.17% return from the two-year bond, so that they expect a value of $985.27 next year for the one-year bond which will earn 1.49% in the second year.

So, if this is the correct risk premium, investors have a guaranteed one-year return of 2.56% on a one-year bond, and demand a risk premium of 0.61% for holding a two-year bond instead. This is twice the SEC yield difference when rates are not expected to change.

Now do the same calculation with a three-year bond worth $912 and an expectation that rates will not change. The three-year bond has an SEC yield of 3.09% (annual yield 3.12%). If rates do not change, the three-year bond will be worth $945 as a two-year bond next year, a return of 3.62%. This gives a one-year risk premium of 1.06% over the one-year bond, and 0.45% over the two-year bond.

I'll do some algebra to show how the 1.06% can be calculated. Let Y_1, Y_2, and Y_3 be the annual yields of the three bonds when the yield curve projects no change in yields. If yields don't change, the two-year bond returns Y_2 + (Y_2-Y_1). The three-year bond returns Y_2 + 3(Y_3-Y_2). Thus the risk premium for three years over two years is 3(Y_3-Y_2) - (Y_2-Y_1), which is triple the slope from three years to two, minus the slope from two years to one, and the risk premium for three years over one year is 3(Y_3-Y_2) + (Y_2-Y_1), which is triple the slope from three years to two, plus the slope from two years to one. In our example, Y_3=3.12%. Y_2=2.87%, Y_1=2.56%, which does give the 1.06% premium.

Continuing the same calculation to four years, if the four-year bond yields Y_4, its one-year return is Y_3+4(Y_4-Y_3). The three-year bond has a one-year return of Y_3 + 2(Y_3-Y_2). Thus the risk premium for four years over three years is 4(Y_4-Y_3) - 2(Y_3-Y_2), and the risk premium for four years over one year is 4(Y_4-Y_3) + (Y_3-Y_2) + (Y_2-Y_1).

Repeating this calculation over N years says that the risk premium for N years over one year is N(Y_N-Y_{N-1}) + (Y_{N-1}-Y_1). Again, this assumes that the Y_i yields represent a neutral yield curve. The N-2 in the introduction is the result of using the slope; the difference Y_{N-1}-Y_1 is N-2 times the slope since it covers an N-year period.

A bond fund holds multiple bonds, and each bond is a set of payments, but the math works out the same for each one. The fund expects to keep a constant duration, and gets its excess return if yields do not change by doing so. For example, if the fund held zero-coupon bonds maturing in 6-10 years (for an 8-year duration), then after one year, it could sell a 5-year bond and replace it with a 10-year bond. If rates did not change, the fund benefited from the 10-year bond becoming a 9-year bond, ..., and the 6-year bond becoming a 5-year bond; it then sold the 5-year bond at the end of the year to buy a 10-year bond, which did not change the returns for the previous year. Thus, if the current yield curve indicates that rates are not expected to change, and they actually do not change, the return of the fund should be close to the return of a 1-year bond plus the 8-year-over-1-year risk premium.

Note that the coupon payments do not directly enter into this calculation. They are relevant because they affect the duration of a bond fund. A 10-year bond worth $1000 paying a $20 coupon every six months is effectively twenty $20 zero-coupon bonds with durations from half a year to 10 years, plus one $1000 zero-coupon bond with a ten-year duration. The overall duration of this bond is less than 10 years.
Wiki David Grabiner
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