The simplest form of classification is accrued versus actual interest. The accrued interest is the standard policy as it considers the interest the moment it becomes due. Actual interest is only considered when it is actually received.

Interest rates can also be expressed in several different equivalent ways:

§ Discount factors

§ Spot rates

§ Forward rates

§ Yields

The prices of Treasury securities may be used to compute discount factors, spot rates, forward rates and yields. Discount factors can be computed directly from the prices of Treasuries. Let us understand these different approaches to interest rates.

Discount Factors

A discount factor represents the present value of a sum. For example, if the one-year discount factor is 0.942900, this indicates that the present value of Rs.1 received in one year is Rs.0.942900. Discount factors cannot exceed 1 and will fall continuously as their maturity increases.

Discount factors may be computed from the prices and coupons of Treasury securities. As an example, the following table shows the prices and coupon rates of off-the-run Treasury securities with maturities of 6 months, 12 months, 18 months and 24 months. Off-the-run securities are those that have been issued in the past; the most recently issued securities are said to be on-the-run. For example, if a one-year Treasury bill was issued six months ago, it is now considered to be an off-the-run six month Treasury bill. A newly-issued six-month Treasury bill is considered to be on-the-run.

Maturity

Coupon Rate

Price

6 months

0.80%

Rs.990.00

12 months

1.40%

Rs.975.00

18 months

2.20%

Rs.960.00

24 months

2.80%

Rs.935.00

The above table looks at the interest rate as a discount rate and uses the same to calculate the present value or the price of the bond using this methodology. Each bond has a face value of Rs.1,000 and makes semi-annual coupon payments.

When the six month bond matures, it will pay one final semi-annual coupon along with the bond’s principal or face value of Rs.1,000. The annual coupon rate is 0.8%, so the annual coupon payment is (0.008)(Rs.1,000) = Rs.8. Each semi-annual coupon payment will therefore be Rs.8/2 = Rs.4. As a result, the bond will provide a cash flow of Rs.1,004 in six months.

Since the price of a bond equals the present value of its promised future cash flows, the following relationship holds for the six-month bond:

990 = 1004d(0.5)

where:

d(t) = the discount factor with a maturity of t years

According to this equation, the price of the bond (Rs.990) equals the present value of the remaining cash flow of Rs.1,004. Since this cash flow occurs in six months, its present value is obtained by multiplying it by the six month discount factor:

1004d(0.5)

Solving for d(0.5) gives:

990 = 1004d(0.5)

d(0.5) = 990 / 1004 = 0.986056

For the 12 month bond, the following relationship holds:

975 = 7d(0.5) + 1007d(1)

This is because the bond’s price is Rs.975. With a 1.4% annual coupon, the bond pays two semi-annual coupons each year of (1.4%/2)(Rs.1,000) = Rs.7. This bond will make a coupon payment in six months, and then a final coupon payment and repayment of principal when the bond matures in twelve months.

Since d(0.5) = 0.986056,

975 = 7d(0.5) + 1007d(1)

975 = 7(0.986056) + 1007d(1)

975 = 6.9024 + 1007d(1)

968.0976 = 1007d(1)

d(1) = 968.0976/1007

= 0.961368

Using this same approach:

d(1.5) = 0.928366

d(2) = 0.882386

These results can be confirmed by pricing the 24 month bond using these discount factors, as follows. The 24 month bond has four remaining cash flows. The bond’s annual coupon rate is 2.8%, so that its semi-annual coupon payments equal (2.8%/2)(Rs.1,000) = Rs.14. Therefore, the price of the bond equals:

P = 14d(0.5) + 14d(1) + 14d(1.5) + 1,014d(2)

P = 14(0.986056) + 14(0.961368) + 14(0.928366) + 1,014(0.882386)

P = 13.8048 + 13.4592 + 12.9971 + 894.7394

P = Rs.935.00005

Except for a small rounding error, this matches the bond’s market price of Rs.935.00.

Spot Rates

A spot rate of interest is the yield to maturity of a zero-coupon bond. Spot rates may be derived directly from discount factors implicit. These spot rates are used frequently in bond trading by experts.

The simplest form of classification is accrued versus actual interest. The accrued interest is the standard policy as it considers the interest the moment it becomes due. Actual interest is only considered when it is actually received.

Interest rates can also be expressed in several different equivalent ways:

§ Discount factors

§ Spot rates

§ Forward rates

§ Yields

The prices of Treasury securities may be used to compute discount factors, spot rates, forward rates and yields. Discount factors can be computed directly from the prices of Treasuries. Let us understand these different approaches to interest rates.

Discount FactorsA discount factor represents the present value of a sum. For example, if the one-year discount factor is 0.942900, this indicates that the present value of Rs.1 received in one year is Rs.0.942900. Discount factors cannot exceed 1 and will fall continuously as their maturity increases.

Discount factors may be computed from the prices and coupons of Treasury securities. As an example, the following table shows the prices and coupon rates of

off-the-runTreasury securities with maturities of 6 months, 12 months, 18 months and 24 months. Off-the-run securities are those that have been issued in the past; the most recently issued securities are said to beon-the-run. For example, if a one-year Treasury bill was issued six months ago, it is now considered to be an off-the-run six month Treasury bill. A newly-issued six-month Treasury bill is considered to be on-the-run.MaturityCoupon RatePrice6 months

0.80%

Rs.990.00

12 months

1.40%

Rs.975.00

18 months

2.20%

Rs.960.00

24 months

2.80%

Rs.935.00

The above table looks at the interest rate as a discount rate and uses the same to calculate the present value or the price of the bond using this methodology. Each bond has a face value of Rs.1,000 and makes semi-annual coupon payments.

When the six month bond matures, it will pay one final semi-annual coupon along with the bond’s principal or face value of Rs.1,000. The annual coupon rate is 0.8%, so the annual coupon payment is (0.008)(Rs.1,000) = Rs.8. Each semi-annual coupon payment will therefore be Rs.8/2 = Rs.4. As a result, the bond will provide a cash flow of Rs.1,004 in six months.

Since the price of a bond equals the present value of its promised future cash flows, the following relationship holds for the six-month bond:

990 = 1004d(0.5)

where:

d(t) = the discount factor with a maturity of t years

According to this equation, the price of the bond (Rs.990) equals the present value of the remaining cash flow of Rs.1,004. Since this cash flow occurs in six months, its present value is obtained by multiplying it by the six month discount factor:

1004d(0.5)

Solving for d(0.5) gives:

990 = 1004d(0.5)

d(0.5) = 990 / 1004 = 0.986056

For the 12 month bond, the following relationship holds:

975 = 7d(0.5) + 1007d(1)

This is because the bond’s price is Rs.975. With a 1.4% annual coupon, the bond pays two semi-annual coupons each year of (1.4%/2)(Rs.1,000) = Rs.7. This bond will make a coupon payment in six months, and then a final coupon payment and repayment of principal when the bond matures in twelve months.

Since d(0.5) = 0.986056,

975 = 7d(0.5) + 1007d(1)

975 = 7(0.986056) + 1007d(1)

975 = 6.9024 + 1007d(1)

968.0976 = 1007d(1)

d(1) = 968.0976/1007

= 0.961368

Using this same approach:

d(1.5) = 0.928366

d(2) = 0.882386

These results can be confirmed by pricing the 24 month bond using these discount factors, as follows. The 24 month bond has four remaining cash flows. The bond’s annual coupon rate is 2.8%, so that its semi-annual coupon payments equal (2.8%/2)(Rs.1,000) = Rs.14. Therefore, the price of the bond equals:

P = 14d(0.5) + 14d(1) + 14d(1.5) + 1,014d(2)

P = 14(0.986056) + 14(0.961368) + 14(0.928366) + 1,014(0.882386)

P = 13.8048 + 13.4592 + 12.9971 + 894.7394

P = Rs.935.00005

Except for a small rounding error, this matches the bond’s market price of Rs.935.00.

A spot rate of interest is the yield to maturity of aSpot Rateszero-couponbond. Spot rates may be derived directly from discount factors implicit. These spot rates are used frequently in bond trading by experts.