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## 4.6 Calculating Returns

### Learning Objective

1. What is the rate of return and how does it differ from yield to maturity?

This is not all you need to know about bonds if you were to become a bond trader because the bond market, which in the United States is over 200 years old, has some odd conventions that do not make much economic sense. Most students will not become professional bond traders, so in the interest of sanity, yours and ours, we will not delve into the intricacies here. (If you do become a bond trader, you will quickly and easily pick up on the conventions anyway.) Our goal here is to understand the basics of PV, FV, yield to maturity (YTM), and, finally, rate of returnA measure of the profitability of an investment that takes into account changes in the value of the bond or other asset.. Students sometimes conflate the last two concepts. The yield to maturity is merely a measure of the interest rate. The rate of return is more a measure of how lucrative an investment is because it accounts for changes in the price of the bond (or other asset, financial or otherwise). More formally,

$R = ( C + P t 1 - P t 0 ) / P t 0$

where:

R = return from holding the asset for some time period, t0 to t1

Pt0 = the price at time t0 (this can also be thought of as the purchase price)

Pt1 = the price at time t1 (this can also be thought of as the sale or going market price)

C = coupon (or other) payment

### Key Takeaways

• The rate of return accounts for changes in the market price of a bond or other asset while the yield to maturity does not.
• Yield to maturity (YTM) is almost always positive but returns are often negative due to interest rate risk, the risk that interest rates will rise, depressing bond prices.
• When the market interest rate increases, bond prices decrease because the opportunity cost of lending money has increased, making bonds less attractive investments unless their price falls.
• Algebraically, PV = FV/(1 + i)n. The interest rate is in the denominator, so as i gets bigger, PV must get smaller.
• Bonds with longer periods to maturity have more volatile prices, ceteris paribus, because the PV of their distant FV shrinks more, to very small sums.