Government of Canada Securities - Technical Guide

Determining Bond and Treasury Bill Prices and Yields

Bonds

In general, there are two parameters that are needed to describe fully the cash flows on a bond. The first is the maturity date of the bond, at which time the principal, or face amount of the bond is paid and the bond retired. The second parameter need to describe a bond is the coupon rate. A Government of Canada bond issued in the domestic market pays one-half of its coupon rate times its principal value every six months up to and including the maturity date. Thus, a bond with an 8 per cent coupon maturing on December 1, 2005 will make future coupon payments of 4 per cent of principal value on each June 1 and December 1 between the purchase date and the maturity date.

The price of the bond is found by discounting future cash flows back to their present value as indicated in the following formula:

In other words, the price equals the sum of each cash flow at a period N, that is divided by a discount rate, which is the yield to maturity divided by 200, plus 1, and to the power N.

where

P = current price

N = number of semi-annual periods

y = yield to maturity (expressed in percentage points). The yield is divided by 200 to convert the yield to a percentage on a semi-annual basis.

CF = cash flow in a given semi-annual period (coupon⁄2) and at maturity (coupon⁄2) + 100

This may be better understood by considering the following example, which shows the price per $100 that would equate a 2-year bond with an 8 per cent coupon to a 6 per cent yield to maturity.

N = 4 (two semi-annual payment in each of the two years)

coupon = 8% ( CF = 8⁄2 = 4 and at maturity CF = 4 +100 =104)

y = 6% (yield to maturity)

The price equals each cash flow, 4, 4, 4, and 104 at maturity, divided by 1.03, the discount rate, to the power N, 1, 2, 3, and 4, corresponding to each period.

If the bond is purchased between coupon payment dates, the price must be adjusted for accrued interest that is owed to the seller of the bond.

Treasury Bills

Treasury bills are priced at a discount. The return to the investor is the difference between the purchase price and the par value. The rate of return is calculated by dividing this difference by the purchase price and expressing the result as an annual percentage rate, using a 365-day year. For example, a price of $990.13 per $1,000 of face amount for a 91-day bill would produce an annualised rate of return equal to 4.00 per cent, computed as follows:

For details see preceding paragraph

Market Convention for Quoting Bond and Treasury Bill Prices and Yields in the Secondary Market

Bonds

Bond prices are quoted as a percentage of the bond's par or face value and exclude accrued interest; e.g. if a nominal fixed coupon bond is quoted as 101.59, then the price of that bond is 101.59% or 1.0159 times the value of the bond at maturity.

When an investor buys a bond between coupon payments, the investor must compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date of the bond. This amount is called accrued interest. Accrued interest for Government of Canada bonds are calculated as follows:

C/2   actual number of days from the last coupon payment to settlement date
actual number of days in coupon period

where C = coupon payment

The market convention for calculating accrued interest on Government of Canada bonds is known as actual over 365 basis, which considers a year to have 365 days.

Bond yields are quoted as the yield to maturity; i.e. the quoted yield is the yield necessary to make the present value of the bond's cash flow equal to the current market price. However, as previously mentioned, the quoted price and, consequently, the quoted yield-to-maturity excludes accrued interest. Therefore, the yield-to-maturity is determined by solving for y in the following equation:

In other words, the price equals the sum of each cash flow at a period N, that is divided by a discount rate, which is the yield to maturity divided by 200, plus 1, and to the power N.

where

P = current price

N = number of semi-annual periods

y = yield to maturity (expressed in percentage points). The yield is divided by 200 to convert the yield to a percentage on a semi-annual basis.

CF = cash flow in a given semi-annual period (coupon⁄2) and at maturity (coupon⁄2) + 100

Treasury Bills

The convention in Canada is to quote Treasury bills in yield terms. . The yield is calculated as follows:

the difference between the face value and the Treasury Bill price is multiplied by 365, which is then divided by the product of the price and the actual number of days remaining to maturity

where

Y = quoted value

F = face value of the Treasury Bill

P = price

t = actual number of days remaining to maturity

The price of the Treasury bill can be determined by rearranging the above equation and solving for (F-P)⁄P and then given F, solving for P.

Resources

Investment Industry Association of Canada: Canadian Conventions in Fixed Income Markets [PDF 295.99 Kb]

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