Pricing a fixed rate bond is made easy with the calculator in this page. It calculates not only the proceed, clean price and accrued interest, but also the various risks such as present value of one basis point (pvbp01), macaulay duration, modified duration, the first derivative of bond value with respect to yield to maturity and the second derivative (convexity) as well.
The coupon structure of the bond is also presented in a table with individual valuation of each cash flow. User have the flexibility to change the individual coupon in the table to create coupon structure such as step-up/down or just different coupon for each coupon period. Changes made will also be reflected in its proceed, accrued interest, clean price and all the risk measurements mentioned earlier.
Value date of the bond. The field must have a value.
Issue date of the bond. If issue date is not a valid value e.g. blank, the following describes the effect:
Maturity date of the bond. The field must have a value and must fall on a regular coupon date if 'Day Count' is set to 'Actual'Actual'. The calculator still works but the day count calculation is not accurate as the last coupon period needs to be identified as an odd coupon period. For other conventions, the day count factor for the odd period adjusted properly.
Day count convention for the purpose of calculating coupon interest and discounting the cash flows.
Business day convention used to adjust the coupon date.
Coupon rate of the bond.
Coupon frequency of the bond.
Yield to maturity of the bond
This is a button to calculate the proceed, accrued interest, clean price and other risk measurements based on the input made in the calculator.
Present value of one basis point. This the difference between the value of the bond at YTM + 0.01% and bond value at YTM.
The standard Macaulay and modified duration. Futher information can be found in Wikipedia.
This is the first derivative of the bond with respect to YTM assuming a face value of 100.
This is the second derivative of the bond with respect to YTM , or also known as convexity assuming a face value of 100.
The calculator generated a coupon structure which is used to revalue and calculate the risk statistics of the bond. Some of the information in the coupon structure are tabulated below. The following are description of some of the columns:
The coupon rate for the coupon period. It is a user-editable field. User may change the coupon rate on each row to see the effect on values in 'Price and Risks Measurements' table. The field must have a numeric value. An empty cell will not affect anything and the shown value may be incorrect/invalid.
Day count factor used to calculate the coupon interest based on the day count convention selected.
The discount factor for the coupon period
Coupon interest for the period.
For bonds, the face value flows only occurs on maturity of the bond. Values should be zero except for the last coupon date.
Total cash flow on the date. Basically its the sum of 'Coupon Interest' and 'Face Value Flows'.
Discount factor for the date, calculated based on YTM.
Present value of the cash flows occuring on the date. It basically 'Cash Flows' multiplied by 'Discount Factor'.
The first three charts are using information in 'Coupon Structures' table. Hence changing any coupon will change the values of the vertical axis.
The following chart illustrate the relationship between the two effectively.A higher yield to maturity will lead to lower price of the bond. Note that the relationship is not linear, hence the need for convexity in addition to duration to estimate the change in price.
The duration referred to is not modified or macaulay duration. Instead, it refers to the 1st derivative of the bond price to with respect to yield. The relation is is positive i.e upward sloping
The second derivatives of the bond's price with respect to yield.
Subsequent 3D charts are using the basic bond information provided in the calculators. Small icons will appear once the cursor hover a chart. User can rotate, zoom, pan and even save the chart in png format via the menu to analyse the charts in detail.
The chart shows the behaviour of price given yield and time to maturity. At a given coupon rate, price is highest at the lowest yield and the longest maturity. It is at its lowest at the highest yield and the shortest maturity. Prices of the bond do not have linear relationships with neither yield or time to maturity.
It is however, best to note that for a given yield, price tend to move to 100 or par. This convergence to par has an accounting implication. Since we charted the price as clean price, it is clearer and easier to explain. A convergence to par for a premium bond, is called amortisation, and accretion for a discount bond. The accretion/ amortisation amount is interest earned/ deduction from the coupon interest. True capital gain, therefore occurs only when the bond are sold at higher yield.
Duration of the bond is always negative i.e. it is always inversely related to yield and at its highest (in terms of absolute value) at the lowest yield and the longest time to maturity i.e. price is at those ranges.
At high yield, time to maturity seems to have little impact on duration. At lower yield, however, the impact is more pronounced.
Convexity is hardly looks at by many. It can be considered as adjustment to duration when it comes to estimating prices changes for a move in yield. The proper way of estimation changes using duration and convexity would be Taylor series approximation, although current technology simply allows us to put in the new yield and revalue a bond or portfolio of bond.
There are other analysis/ charts that can be look at. For instance, coupon and time to maturity against price, duration and convexity. We can also look at coupon and yield against price, duration and convexity, for a total of additional 6 3D charts. Inclusion of these charts would be made at a later date.