# Bond Yield Explained

Suppose you are considering investing in one one of two bonds.

 Bond A Bond B Coupon Rate 5% 6% Maturity (years) 10 7 Price 92 97

Both bonds pay coupons semi-annually starting six months from today, and ending in 10 and 7 years. If neither bond defaults, which will be the more profitable investment? Since these two bonds have different coupon rates, maturities and trade at different prices, the answer to this question is not immediately clear. To answer this question, it is important to look at the yield of the bond.

The Yield to Maturity (YTM) of a bond is the discount rate at which the present value of the bond’s cashflows is equal to its market price. It is also the annual rate of return on your bond investment if the bond doesn’t default and all coupon payments are reinvested. (We will revisit this later.) Concretely, the YTM is the rate $y$ that satisfies

$\displaystyle P=\frac{c/2}{(1+y/2)^{.5\times 2}}+\frac{c/2}{(1+y/2)^{1\times 2}} + \cdots + \frac{100+c/2}{(1+y/2)^{T\times 2}}$

where $c$ is the annual coupon payment on $100 of notional,$T$is the number of years to maturity, and $P$ is the market price of the bond. Note that bond yields are almost always communicated as a semi-annually compounded rate since the vast majority of U.S. corporate bonds pay coupons semi-annually. The easiest way to compute the yield of a bond is by using the YIELD function in Excel. From here we can see that Bond A has a yield of 6.08% and Bond B has a yield of 6.54%. We can check these numbers by confirming that they satisfy the equation above. $\displaystyle \frac{\2.50}{1+3.04\%}+\frac{\2.50}{(1+3.04\%)^2}+\cdots+\frac{\102.50}{(1+3.04\%)^{20}}=92.00$ $\displaystyle \frac{\3.00}{1+3.27\%}+\frac{\3.00}{(1+3.27\%)^2}+\cdots+\frac{\103.00}{(1+3.27\%)^{14}}=97.00$ Since Bond B has the higher yield, Bond B is the more profitable investment (assuming your earnings are always reinvested). The fact that Bond B trades at a higher yield than Bond A also implies that the market considers Bond B to have greater default risk and thereby demands a higher yield. ## Important Properties of Yield Yield is a very important part of bond analysis and investing. Here are a few properties of yield that are important to understand. Yield is another, better way of saying price There is a one-to-one relationship between the price and the yield of the bond: any price implies exactly one yield, and any yield implies exactly one bond price. They are simply two different ways of communicating the same information. In practice, yield is usually the better indicator of how risky and profitable the bond is. Yield and price are inversely related At any given point in time, the lower the price of the bond, the higher the yield, and the higher the price the lower the yield. Below is a plot of the yield vs. price of a 10-year bond with a 5%, 10% and 15% coupon rate. Notice that at par, the yield of each bond is exactly equal to its coupon. This, it turns out, is always true for all bonds. Bond prices pull to par over time The conversion between yield and price depends on the time to maturity of the bond. As time passes and the bond approaches its maturity date, either the yield or the price must change to reflect the shortened bond. If the yield stays constant throughout the life of the bond, the price of the bond must drift to par over time. This effect is called pulling to par. Consider a bond that pays a 9% coupon and matures in 10 years. Under various fixed yields, the price of this bond over time looks like this: Yield is an annual return Suppose you invest D dollars in a bond that always trades at a fixed yield y (expressed as a semi-annually compounded rate as usual). Additionally, when the bond makes coupon payments, you reinvest the payments by buying more notional of the bond. After t years, the market value of your investment will be exactly $D(1+y/2)^{t\times 2}$. I.e. your annualized rate of return will be exactly y. This is the basis for our earlier assertion that the yield of a bond like an annual rate of return on the bond. A proof of this is provided at the end of this article. ## The Nitty Gritty of Yield Computation Let’s consider this bond issued by Rent-A-Center Inc. On 03/08/2017, this bond traded at 85.00, which implied a yield of 11.765%. Let’s see if we can verify that number by discounting the bond’s cashflows by its yield and arriving at its price. In order to do that, however, we first need to discuss a few more nuances in bond yield calculation. Settlement Date vs. Trade Date When a bond is traded between two parties, it doesn’t officially exchange hands typically until three business days later. This is becuase it takes time to verify that the seller actually owns the bond and to update central records to reflect the new owner of the bond. Yields are generally computed by discounting cashflows to the settlement date, not the trade date, which in this case is 03/13/17. Discounting and Day Counts This bond makes coupon payments on the 15th of May and November every year until its maturity in May 2023. The next coupon will be paid on 05/15/17, which is sooner than six months from the settlement date of 03/13/17. Therefore we can no longer assume that the next coupon will be paid exactly six months from now, and must discount each cashflow $CF_i$ by $(1+y/2)^{2\times t_i}$ where $t_i$ is the remaining time to the next coupon date (as a fraction of a year). Specifically, $t_1$ is the number of years between 03/13/17 and 05/15/17, $t_2$ is the years between 03/13/17 and 11/15/17, and so on. Computing each $t_i$ exactly can be complicated, especially when dealing with leap years, so the calculation is conventionally simplified using a Day Count Convention. The vast majority of all U.S. corporate bonds use a “30/360” day count convention, which assumes that every year is comprised of twelve equal 30-day months. In this case, the number of days between 03/13/17 and 5/15/17 is $(5-3)\times 30 + (15-13)=62$, and $t_1=\frac{63}{360}=0.173$. The 30/360 approximation does produce an error in the yield computation, but it’s fairly small as you’ll see below.  Date Payment Actual Days Actual Years Discounted Value 30/360 Days 30/360 Years Discounted Value 05/15/17 3.3125 63 0.173 3.25 62 0.172 3.25 11/15/17 3.3125 247 0.677 3.07 242 0.672 3.07 05/15/18 3.3125 428 1.173 2.90 422 1.172 2.90 11/15/18 3.3125 612 1.677 2.73 602 1.672 2.74 05/15/19 3.3125 793 2.173 2.58 782 2.172 2.58 11/15/19 3.3125 977 2.677 2.44 962 2.672 2.44 05/15/20 3.3125 1159 3.174 2.30 1142 3.172 2.30 11/15/20 103.3125 1343 3.677 67.86 1322 3.672 67.89 Total 87.13 87.17 Summing up the 30/360 discounted cashflows, we get 87.17 as the price of the bond, which doesn’t match 85.00. That’s because we’re missing one final piece of the story. Accrued Interest, Clean Price and Dirty Price The seller of this bond received the last coupon payment on 11/15/16, and has already waited almost three months without any coupon payment. Therefore, the seller should receive around half of the coupon that will be paid out three months from now. To adjust for this, the seller simply charges the buyer of the bond the fractional coupon that is owed in addition to the$85 price. On a 30/360 basis, $\frac{118}{360}$ will have passed between the last coupon date and the settlement date, so the seller is owed $\frac{118}{360}\times \6.625=\2.17$. Therefore, even though the bond traded at a price of $85, the buyer actually paid$87.17 for the bond, which matches our result above. In general, the actual amount that is paid for the bond ($87.17) is called the dirty price while the paid amount less accrued interest ($85.00) is called the clean price.

Dirty Price = Clean Price + Accrued Interest

Why is it called dirty price? Since Accrued Interest drops down to zero right after coupon payments are made, the dirty price also suffers from the same sudden jumps. If the yield on this bond never changed, the clean and dirty price of the bond would look like this:

To avoid these jumps, bonds are always quoted with their clean price, and it is on the buyer to know that he or she will pay accrued interest on top of the quoted price.

## Proofs on Yield

Here we will provide rough proofs for two of the assertions made above. For each we assume coupons are paid annually, and therefore yield will be expressed as an annually compounded rate. Generalizing to the k-compounded case requires minimal extensions.

The yield of a bond trading at par is equal to the coupon rate.
If a bond with an annual coupon rate of $c$, maturing in $T$ years is trading at a yield of $c$, its price (per $1 notional) must be: $\displaystyle P = c\sum_{k=1}^T\frac{1}{(1+c)^k}+\frac{1}{(1+c)^T}$ The sum here is simply a geometric sum which can be simplified as follows. $\displaystyle P = c\times \frac{1}{1+c}\times \frac{1-\left(\frac{1}{1+c}\right)^T}{1-\frac{1}{1+c}}+\frac{1}{(1+c)^T} = 1$ Thus, the price of this bond is exactly par, independent of $c$ and $T$. An investment in a bond with a fixed yield y will produce annual returns of y. Consider a bond that pays a coupon c with payments made annually, and matures in T years. Additionally, assume that the bond trades at a fixed yield y at all times. The price of$1 of notional of the bond at the end of year $t$ must then be

$\displaystyle P_t = \sum_{p = 1}^{T-t}\frac{c}{(1+y)^p} + \frac{1}{(1+y)^T}$

Note that we can write this as the recursive formula

$\displaystyle P_t = P_{t-1}(1+y)-c$

Now, suppose you spend $D$ dollars and buy $\frac{D}{P_0}$ in notional of this bond. As we recieve coupon payements on this investment, we will reinvest those payments back into the bond so the notional of our investment will change over time. Let $B_t$ represent the notional balance of our investment in the bond at the end of year $t$, where $B_0=\frac{D}{P_0}$. We can write $B_t$ recursively in the following way:

$\displaystyle B_t = B_{t-1}+\frac{B_{t-1}c}{P_t} = B_{t-1}\left(\frac{P_t+c}{P_t}\right)=B_{t-1}\frac{P_{t-1}(1+y)}{P_t}$

Expanding the recursion, we get:

$\displaystyle B_t = \frac{D}{P_0}\times \frac{P_0(1+y)}{P_1}\times \frac{P_1(1+y)}{P_2}\times \cdots \times \frac{P_{t-1}(1+y)}{P_t} = \frac{D}{P_t}(1+y)^t$

The market value of our investment at time $t$ is simply the notional balance times the price at time $t$. Therefore

$\displaystyle MV_t = B_tP_t=D(1+y)^t$