A Guide to Bond Basics
Ombretta Pettinato and William L. Silber
Table 1: Relationship Between Price and Yield to Maturity
The Set-Up and Some Insights:
1)YTMis defined as thediscount ratethat equates the present value of cash flows received from a debt instrument with its value (price) today.It is the internal rate of return of an investment in an annual-pay bond if the investor holds the bond until maturity and if all payments are made as scheduled.For a semi-annual pay bond the yield to maturity is the internal rate of return times 2.Table 1 reports the Yield to Maturity (YTM) calculated for a 10% annual-pay coupon bond at different prices.
2)Column Bliststhe different prices for the bond with a face value of $1,000 (Cell C4). In particular, in Cell B8 and Cell B9pricesare above their face value ($1,200 and $1,100 respectively) in Cell B10the price is equal toits face value ($1,000)and in Cell B11 and Cell B12 prices are below the face value ($900 and $800 respectively).The prices in column B have negative signs because they are cash outflows while the annual coupons and face value listed under columns C through L in lines 8 through 12 are all positive because they are cash inflows.
3)The Yield to Maturityfor each bond appears in column M (rows 8 to 12). These values equate the lefthand side and the righthand side of the equation below and are derived by trial and error as with all IRR calculations (or simply by using the IRR(values) function as describedin the Excel worksheet).
Lessons from Table 1:
4)It is easy to see that the value of a bond’s price and its Yield to Maturity are inversely related. Lower prices in column B are associated with higher YTMs in column M;5)When a bond sells at par (face value) its Yield to Maturity(Cell M10) equals the coupon rate (C/F) which is 10% in this case;
6)The YTM is greater than the coupon rate (Cells M11 and M12 are greater than 10%) when the bond price is below its face value (it is selling at a discount) and the YTM is lower than the coupon rate (Cells M8 and M9 are lower than 10%) when the bond price is above its face value (it is selling at a premium).
Extensions:
7) You can change the coupon rate (Cell C3) and/or the prices of the bonds (Column B, lines 8 through 12) to see the effect on the YTM (Column M, lines 8 through 12).
Table 2: Yield to MaturityvsAnnual Rate of Return
The Set-Up and Some Insights:
How is the yield to maturity (YTM) related to the annual rate of return on the bond when yields change and the bond issold before maturity, in particular, after one year?
The following example shows that in general the YTMand the annual return are not necessarily the same.The example assumes that you buy the bond at par (face value), hold the bond for one year (a 1-year holding period) and the coupon is paid at the end of the year.
1)TheAnnual rate of return of the bond can be computed as:
In our case, since you bought the bond at par, Pt is equal to F so you have the following:
2)Table 2 reports one-year returns (Column F, rows 10-15) on different maturity 10% coupon rate bonds when yield to maturity rises from 10% (Cell B6) to 20% (Cell B7).
3)When the yield to maturity rises from 10% to 20%, the price at which the bond is sold (Pt+1) is computed by discouting back the future cash flows at the new YTM for the remaining period. An example is the computation of the price in year 2 (Pt+1) of the 10 year coupon bond which appears in Cell D12.
Lessons from Table 2:
4)The YTM depends on the current price, the bond’s coupon payment, and its face value. All of these values are observable today. Conversely, the rate of return over a particular investment period (one year in our example) depends on the market price of the bond at the end of the holding period. Of course, this price is not known today.
5)As a consequence, if you sell the bond before maturity, changes in theYTM can generate capital gains or losses.Look at the 10-Year Coupon bond: if you sell the bond after one year (when the YTM has increased from 10% to 20%), you suffer a capital loss of 40.3% (Cell E12) hence the return on the bond drops to -30.3% (Cell F12). This value is given by the sum of the coupon rate of 10% (Cell B12) and the capital loss of -40.3% (Cell E12). Even though a bond has a substantial initial coupon rate, its return can turn out to be negative if the YTM rises.
6)For a given coupon rate, prices and returns for long-term bonds are more volatile than those for shorter-term bonds. This is seen by comparing Cell F10 through Cell F15 which records larger price declines associated with the increase in YTM the longer the maturity of the bond.
7)There is no risk of capital gain or loss when YTM changes (interest rate risk) for a bond whose time to maturity matches the holding period (see line 15).Thishappens becasue the price at the end of the holding period is fixed at the face value.
Extensions:
8) You can change the value of the YTM in Cell B7 to see the impact on the capital gain/loss (Column E, lines 10 through 15) and the annual return (Column F, lines 10 through 15).
Table 3: Calculating Duration
The Set-Up and Some Insights:
The definition of (Macaulay) duration(which assumes a fixed cash flow bond)is a weighted average of the time periods when payments are made. It can be used to measure the riskiness of a bond because longer duration bonds have greater price sensitivity per unit change in YTM.
The formula is the following:
where CPt= Cash Payments for each period t
1)For a 3% coupon (annual pay) 10-year bond with a yield to maturity of 5% (Cell B4) - as reported in Table 3- we have:
Lessons from Table 3:
2)All else being equal, when YTM increases, the duration of a coupon bond falls. For example, if you replacethe 5% YTM with a 7% YTM (Cell B4) the duration of the bond falls from 8.66 to 8.52.Try it yourself!
3)All else being equal, the higher the coupon rate, the shorter the bond's duration. Starting again with the 5% YTM, an increase of the coupon rate (Cell B6) from 3% to 4%, produces a reduction in the duration of the bond from 8.66 to 8.36. Try it again!
4)For a zero coupon bond, with 10 years to maturity, the duration is exactly equal to 10.You can easily get this result by replacing the 3% coupon rate with 0% in Cell B6.
5)The table under column J through column M shows the sensitivity of duration to the YTM and the coupon rate. You can run your own sensitivity analysis by changing the YTM and/or the coupon rate respectively in Cells B4 and B6.
Extensions:
Using the information provided in the following press release, set up a table (like Table 1) and check if they report the right YTM for both maturities. We do this for the 7 year bond in the excel document so you can check (but don’t peek first).