Advanced Bond Pricing Techniques: NISM Series XXII Study Notes (Part Two)

This guide provides the second part of high-quality short notes for Chapter 4: Pricing of Bonds from the NISM Series XXII: Fixed Income Securities workbook. This section explores the pricing of semi-annual and short-term instruments, the calculation of accrued interest, price-yield dynamics, and the mechanics of floating rate bonds.

4.4 Pricing of Different Bonds

Bonds are priced by summing the present values of their respective cash flows, which include periodic coupons and the final principal repayment.

Semi-Annual Coupon Pricing

In the Indian market, most Government Bonds pay coupons twice a year. When interest is paid semi-annually, the frequency "n" is 2, and the discount rate and time periods must be adjusted accordingly.

• The Power of Frequency: A semi-annual coupon-paying bond is valued higher than an annual coupon-paying bond with identical maturity and yield because interest is received and reinvested sooner.
• Valuation Formula (Semi-Annual): Value = (Coupon * PVIF) + (Face Value * Discount Factor for maturity year).
• General Valuation Formula: V = ∑ (C / n) / (1 + R% / n)^(n * t) + Face Value / (1 + R% / n)^(n * t).

Continuous Compounding

Continuous compounding occurs when "n" (the frequency of payments) becomes extremely large, assuming interest is earned and reinvested at every infinitesimal point in time.

• Market Practice: While continuous compounding yields a marginally higher interest rate than daily compounding, it is rarely used in standard bond market practice due to its infinite nature.

Valuing Short-Term Instruments (< 1 Year)

For instruments like Treasury Bills that mature within one coupon cycle, the pricing formula is simplified to account for the exact days to maturity.

• T-Bill Formula: V = Face Value / (1 + R% * n / 365), where "n" is the number of days between the purchase and maturity.
• Zero Coupon Bond (> 1 Year): These are valued using a bond equivalent yield formula: V = Face Value / (1 + R% / 2)^(2 * time).

Day Count Conventions

Precise valuation requires standardized day count conventions to calculate the time between dates.

• Actual/365: Used in the Indian Treasury Bill market, counting the actual calendar days between two dates while assuming a 365-day year.
• 30/360 European: Standard for the Indian Government Bond market, where every month is treated as 30 days and the year as 360 days.

Valuing Bonds at Non-Coupon Dates

Most bond trades occur between coupon payment dates, requiring the separation of the bond's value into Clean and Dirty prices.

Clean Price vs. Dirty Price

• Dirty Price: This is the actual Present Value of the bond, representing the total sum the buyer pays the seller. It includes both the asset's value and the interest earned but not yet paid.
  • Formula: Dirty Price = Clean Price + Accrued Interest.
• Clean Price: This is the bond's price excluding the interest that has accrued since the last coupon payment.
• Market Convention: In India, trading quotes and yield computations are based on the Clean Price, while the actual settlement of the trade is conducted at the Dirty Price.

Accrued Interest (AI)

Accrued interest is a static calculation of interest for the "broken period" since the last coupon date. Because there is no reinvestment opportunity between the last payment and the trade date, this interest is not discounted.

• AI Formula: AI = Coupon * (Days from last coupon / Days between coupon dates).

Specialized Pricing Models

Perpetual Bond Pricing

Perpetual bonds continuously pay coupons indefinitely and never return the face value.

• Formula: PV = Coupon / Yield.
• Example: If a perpetuity pays an 8% coupon and the market yield is 6%, its value is (100 * 8%) / 6% = ₹133.33.

Discount Factors and Bootstrapping

A Discount Factor (DF) is the ratio of Present Value to Future Value for a specific time and rate.

• Formula: DF = 1 / (1 + r)^t.
• Bootstrapping: This is the process of deriving a set of discount factors (the discount function) from current market bond prices.
• Interpolation: Traders use straight-line interpolation to calculate rates for non-standard maturities: Rn = R1 + (R2 - R1) / (T2 - T1) * (Tn - T1).

4.5 Price-Yield Relationship

The relationship between a bond’s price and its yield is inverse and non-linear.

Core Principles

1. Inverse Movement: When the required rate of return (yield) increases, the bond's price falls.
2. Par, Premium, and Discount:
   • If Coupon = Yield: Bond trades at Par.
   • If Coupon < Yield: Bond trades at a Discount.
   • If Coupon > Yield: Bond trades at a Premium.
3. Sensitivity: Bonds with longer maturities or lower coupon rates are more sensitive to interest rate changes.

4.6 Price Time Path of a Bond

As a bond approaches its maturity date, its price will inevitably converge toward its par value, a phenomenon known as being "Pulled to Par".

• Discount Bonds: Increase in value over time toward par.
• Premium Bonds: Decrease in value over time toward par.
• Visualizing the Path: Regardless of the yield environment during its life, the value at "Year 0" (maturity) is always the face value (₹100).

4.7 Pricing of a Floating Rate Bond (FRB)

A floating rate bond features a coupon that is not fixed but instead indexed to a benchmark, such as the 182-day T-Bill rate in India.

• Reset Mechanism: The coupon rate is periodically reset on payment dates.
• Price Certainty: Because the coupon rate adjusts to current market levels, an FRB is generally worth par on every coupon refixation day.
• Valuation Logic: Pricing involves discounting only the principal and the next known coupon payment, making these bonds less sensitive to interest rate risk.

Important Terms in Advanced Pricing

Key Takeaways for Chapter 4

• Standardization: Clean price is for quoting; Dirty price is for paying.
• Maturity Impact: Longer-term bonds exhibit higher price volatility.
• Compounding Impact: Increasing the frequency of interest payments raises the bond's present value.
• Convergence: All standard bonds eventually trade at par value at the moment of maturity.

This concludes the summary of Chapter 4: Pricing of Bonds.