Bond Yield : Formula & Calculation

What Is Bond Yield?

Bond yield is the return an investor realizes on a bond. The bond yield can be defined in different ways. Setting the bond yield equal to its coupon rate is the simplest definition. The current yield is a function of the bond's price and its coupon or interest payment, which will be more accurate than the coupon yield if the price of the bond is different than its face value.

More complex calculations of a bond's yield will account for the time value of money and compounding interest payments. These calculations include yield to maturity (YTM), bond equivalent yield (BEY), and effective annual yield (EAY).

Overview of Bond Yield

When investors buy bonds, they essentially lend bond issuers money. In return, bond issuers agree to pay investors interest on bonds through the life of the bond and to repay the face value of bonds upon maturity. The simplest way to calculate a bond yield is to divide its coupon payment by the face value of the bond. This is called the coupon rate.

Coupon Rate = Annual Coupon Payment Bond Face Value \text{Coupon Rate}=\frac{\text{Annual Coupon Payment}}{\text{Bond Face Value}} Coupon Rate=Bond Face ValueAnnual Coupon Payment

If a bond has a face value of $1,000 and made interest or coupon payments of $100 per year, then its coupon rate is 10% ($100 / $1,000 = 10%). However, sometimes a bond is purchased for more than its face value (premium) or less than its face value (discount), which will change the yield an investor earns on the bond.

Bond Yield Vs. Price

As bond prices increase, bond yields fall. For example, assume an investor purchases a bond that matures in five years with a 10% annual coupon rate and a face value of $1,000. Each year, the bond pays 10%, or $100, in interest. Its coupon rate is the interest divided by its par value.

If interest rates rise above 10%, the bond's price will fall if the investor decides to sell it. For example, imagine interest rates for similar investments rise to 12.5%. The original bond still only makes a coupon payment of $100, which would be unattractive to investors who can buy bonds that pay $125 now that interest rates are higher.

If the original bond owner wants to sell the bond, the price can be lowered so that the coupon payments and maturity value equal a yield of 12%. In this case, that means the investor would drop the price of the bond to $927.90. In order to fully understand why that is the value of the bond, you need to understand a little more about how the time value of money is used in bond pricing, which is discussed later in this article.

If interest rates were to fall in value, the bond's price would rise because its coupon payment is more attractive. For example, if interest rates fell to 7.5% for similar investments, the bond seller could sell the bond for $1,101.15. The further rates fall, the higher the bond's price will rise, and the same is true in reverse when interest rates rise.

In either scenario, the coupon rate no longer has any meaning for a new investor. However, if the annual coupon payment is divided by the bond's price, the investor can calculate the current yield and get a rough estimate of the bond's true yield.

Current Yield = Annual Coupon Payment Bond Price \text{Current Yield}=\frac{\text{Annual Coupon Payment}}{\text{Bond Price}} Current Yield=Bond PriceAnnual Coupon Payment

The current yield and the coupon rate are incomplete calculations for a bond's yield because they do not account for the time value of money, maturity value, or payment frequency. More complex calculations are needed to see the full picture of a bond's yield.

Yield to Maturity

A bond's yield to maturity (YTM) is equal to the interest rate that makes the present value of all a bond's future cash flows equal to its current price. These cash flows include all the coupon payments and its maturity value. Solving for YTM is a trial and error process that can be done on a financial calculator, but the formula is as follows:

Price = ∑ t − 1 T Cash Flows t ( 1 + YTM ) t where: YTM =  Yield to maturity \begin{aligned} &\text{Price}=\sum^T_{t-1}\frac{\text{Cash Flows}_t}{(1+\text{YTM})^t}\\ &\textbf{where:}\\ &\text{YTM}=\text{ Yield to maturity} \end{aligned} Price=t−1∑T(1+YTM)tCash Flowstwhere:YTM= Yield to maturity

In the previous example, a bond with $1,000 face value, five years to maturity, and $100 annual coupon payments was worth $927.90 in order to match a YTM of 12%. In that case, the five coupon payments and the $1,000 maturity value were the bond's cash flows. Finding the present value of each of those six cash flows with a discount or interest rate of 12% will determine what the bond's current price should be.

Bond Equivalent Yield – BEY

Bond yields are normally quoted as a bond equivalent yield (BEY), which makes an adjustment for the fact that most bonds pay their annual coupon in two semi-annual payments. In the previous examples, the bonds' cash flows were annual, so the YTM is equal to the BEY. However, if the coupon payments were made every six months, the semi-annual YTM would be 5.979%.

The BEY is a simple annualized version of the semi-annual YTM and is calculated by multiplying the YTM by two. In this example, the BEY of a bond that pays semi-annual coupon payments of $50 would be 11.958% (5.979% X 2 = 11.958%). The BEY does not account for the time value of money for the adjustment from a semi-annual YTM to an annual rate.

Effective Annual Yield – EAY

Investors can find a more precise annual yield once they know the BEY for a bond if they account for the time value of money in the calculation. In the case of a semi-annual coupon payment, the effective annual yield (EAY) would be calculated as follows:

EAY = ( 1 + YTM 2 ) 2 − 1 where: EAY = Effective annual yield \begin{aligned} &\text{EAY} = \left ( 1 + \frac { \text{YTM} }{ 2 } \right ) ^ 2 - 1 \\ &\textbf{where:}\\ &\text{EAY} = \text{Effective annual yield} \\ \end{aligned} EAY=(1+2YTM)2−1where:EAY=Effective annual yield

If an investor knows that the semi-annual YTM was 5.979%, they could use the previous formula to find the EAY of 12.32%. Because the extra compounding period is included, the EAY will be higher than the BEY.

Complications Finding a Bond's Yield

There are a few factors that can make finding a bond's yield more complicated. For instance, in the previous examples, it was assumed that the bond had exactly five years left to maturity when it was sold, which would rarely be the case.

When calculating a bond's yield, the fractional periods can be dealt with simply; the accrued interest is more difficult. For example, imagine a bond that has four years and eight months left to maturity. The exponent in the yield calculations can be turned into a decimal to adjust for the partial year. However, this means that four months in the current coupon period have elapsed and there are two more to go, which requires an adjustment for accrued interest. A new bond buyer will be paid the full coupon, so the bond's price will be inflated slightly to compensate the seller for the four months in the current coupon period that have elapsed.

Bonds can be quoted with a "clean price" that excludes the accrued interest or the "dirty price" that includes the amount owed to reconcile the accrued interest. When bonds are quoted in a system like a Bloomberg or Reuters terminal, the clean price is used.

What does a bond's yield tell investors?

A bond's yield is the return to an investor from the bond's coupon (interest) payments. It can be calculated as a simple coupon yield, which ignores the time value of money and any changes in the bond's price or using a more complex method like yield to maturity. Higher yields mean that bond investors are owed larger interest payments, but may also be a sign of greater risk. The riskier a borrower is, the more yield investors demand to hold their debts. Higher yields are also associated with longer maturity bonds.

Are high-yield bonds better investments than low-yield bonds?

Like any investment, it depends on one's individual circumstances, goals, and risk tolerance. Low-yield bonds may be better for investors who want a virtually risk-free asset, or one who is hedging a mixed portfolio by keeping a portion of it in a low-risk asset. High-yield bonds may instead be better-suited for investors who are willing to accept a degree of risk in return for a higher return. The risk is that the company or government issuing the bond will default on its debts. Diversification can help lower portfolio risk while boosting expected returns.

What are some common yield calculations?

The yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures. Yield to maturity is considered a long-term bond yield but is expressed as an annual rate. YTM is usually quoted as a bond equivalent yield (BEY), which makes bonds with coupon payment periods less than a year easy to compare. The annual percentage yield (APY) is the real rate of return earned on a savings deposit or investment taking into account the effect of compounding interest. The annual percentage rate (APR) includes any fees or additional costs associated with the transaction, but it does not take into account the compounding of interest within a specific year. An investor in a callable bond also wants to estimate the yield to call (YTC), or the total return that will be received if the bond purchased is held only until its call date instead of full maturity.

How do investors utilize bond yields?

In addition to evaluating the expected cash flows from individual bonds, yields are used for more sophisticated analyses. Traders may buy and sell bonds of different maturities to take advantage of the yield curve, which plots the interest rates of bonds having equal credit quality but differing maturity dates. The slope of the yield curve gives an idea of future interest rate changes and economic activity. They may also look to the difference in interest rates between different categories of bonds, holding some characteristics constant. A yield spread is the difference between yields on differing debt instruments of varying maturities, credit ratings, issuer, or risk level, calculated by deducting the yield of one instrument from the other -- for example the spread between AAA corporate bonds and U.S. Treasuries. This difference is most often expressed in basis points (bps) or percentage points.