Bond pricing

Will eventually lead to an update of Bonds: Advanced Topics and to reorganize the definitions of duration as shown in Duration - Definitions Refined.

Required Yield
The interest rate, or discount rate, that the investor wants from investing in a bond is called the required yield. Required yield is often specified as an annual interest rate. For purposes of calculating cash flows, a bond's semiannual interest rate is conventionally set to one-half the annual interest rate.

However, note that using a periodic rate as one-half the annual interest rate will result in an effective annual yield that is greater than the annual interest rate (an error in the actual interest rate).

Pricing A Bond
Bond price is the sum of the present value of the cash flows, which is determined by adding:
 * 1) The present value of the (semiannual) coupon payments
 * 2) The present value of the par or maturity value

This is shown mathematically as:
 * {| style="border:1px solid darkgray;"


 * align="left" |p = c/(1 + i)1 + c/(1 + i)2 + c/(1 + i)3 + ... + c/(1 + i)n + M/(1 + i)n
 * }

where
 * p = Price
 * c = Semiannual coupon payment
 * i = Periodic interest rate (required yield/2)
 * n = Number of periods (number of years x 2)
 * M = Maturity value

As the semiannual coupon payments are equivalent to an ordinary annuity, the same present value formula used for Comparing Investments applies. Use the financial variable PMT to represent the coupon payment.

Price/Yield Curve
A basic property of present value is that as the interest rate rises, the price drops and vice versa.


 * {| style="margin:1em auto 1em auto;" cellpadding= "3" border="1" style="border:1px solid black; border-collapse: collapse;"


 * +Increasing the interest rate lowers the price.
 * Increases in the interest rate lowers the price.png
 * }

In a bond, the number of period payments (NPER) and the coupon rate (PMT) is fixed. Note how the price of a bond changes (sum of PV) as the required yield (I) varies.


 * {| style="margin:1em auto 1em auto;" cellpadding= "3" border="1" style="border:1px solid black; border-collapse: collapse;"


 * +Bond Price/Yield Relationship
 * Bond_Price-Yield_Curve.png
 * }

The curve of this graph has a convex shape, which can have significant implications for the investment properties of a bond.

The above graph was created in Excel (axis labels removed) as an example to show how the price of a 20 year, 9% coupon bond changes when the required yield varies from 1% to 20%.

Displayed values (first, last row only):
 * {| style="margin:1em auto 1em auto;" cellpadding= "3" border="1" style="border:1px solid black; border-collapse: collapse;"


 * align = "center"|
 * align = "center"|Required Yield (Annual) Column C
 * align = "center"|Present Value of Coupon payments Column D
 * align = "center"|Present Value of Par Value Column E
 * align = "center"|Price of Bond
 * align = "center"|Row 8
 * align = "right"|1%
 * align = "right"|$1,627.75
 * align = "right"|$819.14
 * align = "right"|$2,446.89
 * align = "center"|Row 46
 * align = "right"|20.00%
 * align = "right"|$440.06
 * align = "right"|$22.09
 * align = "right"|$462.15
 * }
 * align = "right"|$462.15
 * }

Cell formulas (first, last row only): Cell D4 =2*20 (20 years, semiannual, NPER), D5 =-1000 (par value), D6 =-45 (coupon payment, PMT)
 * {| style="margin:1em auto 1em auto;" cellpadding= "3" border="1" style="border:1px solid black; border-collapse: collapse;"


 * align = "center"|
 * align = "center"|Required Yield (Annual) Column C
 * align = "center"|Present Value of Coupon payments Column D
 * align = "center"|Present Value of Par Value Column E
 * align = "center"|Price of Bond
 * align = "center"|Row 8
 * align = "right"|0.01
 * align = "right"|=PV(C8/2,$D$4,$D$6)
 * align = "right"|=PV(C8/2,$D$4,0,$D$5)
 * align = "right"|=SUM(D8:E8)
 * align = "center"|Row 46
 * align = "right"|0.20
 * align = "right"|=PV(C46/2,$D$4,$D$6)
 * align = "right"|=PV(C46/2,$D$4,0,$D$5)
 * align = "right"|=SUM(D46:E46)
 * }
 * align = "right"|=SUM(D46:E46)
 * }

Yield to Maturity
p. 74 p. 94 the yield curve

Distribution Yield vs Yield to Maturity

Duration
p. 187 (graph using center of mass concept for weighted average)