Bond Pricing Fundamentals[/b]
The cash inflow for an investor in a bond includes the coupon payments and the payment on
maturity (which is the face value) of the bond. Thus the price of the bond should represent the sum total of the discounted value of each of these cash flows (such a total is called the present value of the bond). The discount rate used for valuing the bond is generally higher than the risk-free rate to cover additional risks such as default risk, liquidity risks, etc.
Bond Price = PV (Coupons and Face Value)
Note that the coupon payments are at different points of time in the future, usually twice each year. The face value is paid at the maturity date.
Clean and dirty prices and accrued interest
Bonds are not traded only on coupon dates but are traded throughout the year. The market price of the bonds also includes the accrued interest on the bond since the most recent coupon payment date. The price of the bond including the accrued interest since issue or the most recent coupon payment date is called the ‘dirty price’ and the price of the bond excluding the accrued interest is called the ‘clean price’. Clean price is the price of the bond on the most recent coupon payment date, when the accrued interest is zero.
Dirty Price = Clean price + Accrued interest
For reporting purpose (in press or on trading screens), bonds are quoted at ‘clean price’ for ease of comparison across bonds with differing interest payment dates (dirty prices ‘jump’ on interest payment dates). Changes in the more stable clean prices are reflective of macroeconomic conditions, usually of more interest to the bond market.
Bond Yields
Bond yield are measured using the following measures:
Coupon yield
Current Yield
The main drawback of coupon yield and current yield is that they consider only the interest
payment (coupon payments) and ignore the capital gains or losses from the bonds. Since they consider only coupon payments, they are not measurable for bonds that do not pay any interest, such as zero coupon bonds. The other measures of yields are yield to maturity and yield to call. These measures consider interest payments as well as capital gains (or losses) during the life of the bond.
Yield to maturity
Yield to maturity (also called YTM) is the most popular concept used to compare bonds. It
refers to the internal rate of return earned from holding the bond till maturity. Assuming a
constant interest rate for various maturities, there will be only one rate that equalizes the
present value of the cash flows to the observed market price in equation given earlier. That
rate is referred to as the yield to maturity.
Yield and Bond Price:
There is a negative relationship between yields and bond price. The bond price falls when yield increases and vice versa.
Further, for a long-term bond, the cash flows are more distant in the future and hence the
impact of change in interest rate is higher for such cash flows. Alternatively, for short-term
bonds, the cash flows are not far and discounting does not have much effect on the bond price. Thus, price of long-term bonds are more sensitive to interest rate changes. Bond equivalent yield and Effective annual yield: This is another important concept that is of
importance in case of bonds and notes that pay coupons at time interval which is less than a
year (for example, semi-annually or quarterly). In such cases, the yield to maturity is the discount rate solved using the following formula, wherein we assume that the annual discount rate is the product of the interest rate for interval between two coupon payments and the number of coupon payments in a year:
YTM calculated using the above formula is called bond equivalent yield. However, if we assume that one can reinvest the coupon payments at the bond equivalent yield (YTM), the effective interest rate will be different. Yield rate calculated using the above formula is called effective annual yield. Yield to call is calculated for callable bond. A callable bond is a bond where the issuer has a right (but not the obligation) to call/redeem the bond before the actual maturity. Generally the callable date or the date when the company can exercise the right, is pre-specified at the time of issue. Further, in the case of callable bonds, the callable price (redemption price) may be different from the face value. Yield to call is calculated with the same formula used for calculating YTM, with an assumption that the issuer will exercise the call option on the exercise date.
Interest Rates
While computing the bond prices and YTM, we assumed that the interest rate is constant
across different maturities. However, this may not be true for different reasons. For example, investors may perceive longer maturity periods to be riskier and hence may demand higher interest rate for cash flow occurring at distant time intervals than those occurring at short time intervals.
The cash inflow for an investor in a bond includes the coupon payments and the payment on
maturity (which is the face value) of the bond. Thus the price of the bond should represent the sum total of the discounted value of each of these cash flows (such a total is called the present value of the bond). The discount rate used for valuing the bond is generally higher than the risk-free rate to cover additional risks such as default risk, liquidity risks, etc.
Bond Price = PV (Coupons and Face Value)
Note that the coupon payments are at different points of time in the future, usually twice each year. The face value is paid at the maturity date.
Clean and dirty prices and accrued interest
Bonds are not traded only on coupon dates but are traded throughout the year. The market price of the bonds also includes the accrued interest on the bond since the most recent coupon payment date. The price of the bond including the accrued interest since issue or the most recent coupon payment date is called the ‘dirty price’ and the price of the bond excluding the accrued interest is called the ‘clean price’. Clean price is the price of the bond on the most recent coupon payment date, when the accrued interest is zero.
Dirty Price = Clean price + Accrued interest
For reporting purpose (in press or on trading screens), bonds are quoted at ‘clean price’ for ease of comparison across bonds with differing interest payment dates (dirty prices ‘jump’ on interest payment dates). Changes in the more stable clean prices are reflective of macroeconomic conditions, usually of more interest to the bond market.
Bond Yields
Bond yield are measured using the following measures:
Coupon yield
Current Yield
The main drawback of coupon yield and current yield is that they consider only the interest
payment (coupon payments) and ignore the capital gains or losses from the bonds. Since they consider only coupon payments, they are not measurable for bonds that do not pay any interest, such as zero coupon bonds. The other measures of yields are yield to maturity and yield to call. These measures consider interest payments as well as capital gains (or losses) during the life of the bond.
Yield to maturity
Yield to maturity (also called YTM) is the most popular concept used to compare bonds. It
refers to the internal rate of return earned from holding the bond till maturity. Assuming a
constant interest rate for various maturities, there will be only one rate that equalizes the
present value of the cash flows to the observed market price in equation given earlier. That
rate is referred to as the yield to maturity.
Yield and Bond Price:
There is a negative relationship between yields and bond price. The bond price falls when yield increases and vice versa.
Further, for a long-term bond, the cash flows are more distant in the future and hence the
impact of change in interest rate is higher for such cash flows. Alternatively, for short-term
bonds, the cash flows are not far and discounting does not have much effect on the bond price. Thus, price of long-term bonds are more sensitive to interest rate changes. Bond equivalent yield and Effective annual yield: This is another important concept that is of
importance in case of bonds and notes that pay coupons at time interval which is less than a
year (for example, semi-annually or quarterly). In such cases, the yield to maturity is the discount rate solved using the following formula, wherein we assume that the annual discount rate is the product of the interest rate for interval between two coupon payments and the number of coupon payments in a year:
YTM calculated using the above formula is called bond equivalent yield. However, if we assume that one can reinvest the coupon payments at the bond equivalent yield (YTM), the effective interest rate will be different. Yield rate calculated using the above formula is called effective annual yield. Yield to call is calculated for callable bond. A callable bond is a bond where the issuer has a right (but not the obligation) to call/redeem the bond before the actual maturity. Generally the callable date or the date when the company can exercise the right, is pre-specified at the time of issue. Further, in the case of callable bonds, the callable price (redemption price) may be different from the face value. Yield to call is calculated with the same formula used for calculating YTM, with an assumption that the issuer will exercise the call option on the exercise date.
Interest Rates
While computing the bond prices and YTM, we assumed that the interest rate is constant
across different maturities. However, this may not be true for different reasons. For example, investors may perceive longer maturity periods to be riskier and hence may demand higher interest rate for cash flow occurring at distant time intervals than those occurring at short time intervals.