# Subsequent returns of bond higher than the initial bond yield

I am currently reading Bogle's common sense investing. In one chapter he states that the initial yield of a 10 year U.S. Treasury Note has a high correlation with the subsequent 10 year return. That makes sense of course.

In the corresponding graph one can see two lines. One for the initial yield of the 10 year U.S. Treasury bond and one for the corresponding 10 year return in percent.

There are different occurrences when the subsequent 10 year return is higher than the initial yield. How can this be?

Per my understanding the subsequent 10 year return for a 10 year note means holding it to maturity. Doesn't that also mean the subsequent return must be the same as the initial yield?

The bond yield to maturity (YTM) assumes all coupons are reinvested at the YTM, which is the internal rate of return (IRR) of the bond. In practice, coupons will be reinvested at different rates as the yield curve changes over time. The result is that the 10-year buy and hold return can be different - higher or lower - than the initial yield on the bond.

To illustrate, consider a 10-year T-note with a 5% coupon priced at par, implying a 5% YTM. Assume the yield curve is flat at 5%. The instant after purchasing the bond, assume the yield curve shifts down to 4% where it stays until maturity.

To compute the buy and hold return over the 10-year period, we need to do the following.

1. Compute the future value of each coupon at the 4% reinvestment rate and sum.
2. Divide this sum by the initial investment, \$100, and subtract 1.

This gives us a 10-year buy and hold return of 60.74% in which we reinvested every coupon at a 4% APR. Annualizing this 10-year return produces 4.86% (4.80% APR), less than the initial 5% yield because of the lower returns on the reinvested coupons.

Repeating the exercise with an assumed increase in interest rates to 6% and you'll find the 10-year buy and hold return is 67.18% or an annualized 5.27% (5.21% APR).

(The APRs are a result of the semi-annual compounding inherent to Treasury securities.)

Note the intuition here holds regardless of whether we purchase a bond at auction or some time during its life. The buy and hold to maturity return will almost surely be different from the yield to maturity at purchase if there are any coupons because of reinvestment risk. For T-notes and STRIPs, there is no reinvestment risk so the yield at purchase will equal with the buy and hold to maturity return (in APR terms).

Assuming the bond is held to maturity fluctuations in the yield will primarily be down to variations in the rate at which the coupon payments are reinvested e.g.

1. You buy a 10-year US Treasury Note with a face value of \$1,000 and an annual coupon rate of 2% (= yield to maturity). You'll receive \$20 per year in interest payments if the rate stays at 2%
2. At the end of year 1 the \$20 is paid and reinvested but the rate now shifts to 3%
3. At the end of year 2 the previous coupon grows to \$20.60, and another coupon is reinvested
4. At the end of year 3 the first coupon grows to \$21.22, the second coupon to \$20.60
5. This continues through the life of the bond

This is a simple example because in reality rates will fluctuate more within a 10-year period and travel in both directions. This is termed reinvestment risk.

The above also clearly illustrates why there can be a difference between the internal rate of return (IRR) of a bond, and its initial yield to maturity (YTM).