Pricing of Bond
This sub‑topic explains how bonds are priced in the Indian market, covering the present‑value method, accrued interest, clean vs dirty price, and yield to maturity. Understanding bond pricing is essential for the NISM Series X‑A exam because questions often test the ability to compute price, yield and related concepts. The content links directly to the Fixed Income chapter and prepares you for scenario‑based calculations.
Learning Objectives
- 1Explain the cash‑flow structure of a bond and the concept of discounting.
- 2Calculate the clean and dirty price of a bond using accrued interest.
- 3Derive and approximate the Yield to Maturity (YTM) for coupon bonds.
- 4Price zero‑coupon bonds and interpret price‑yield relationships.
Bond Cash‑Flow Structure
A bond is a debt instrument that promises periodic coupon payments and the return of face value (also called principal or maturity value) at the end of its term. In India, most corporate and government bonds pay coupons semi‑annually, but the NISM syllabus treats the coupon as a regular cash flow occurring each period.
The coupon amount is calculated as a percentage of the face value. For example, a ₹1,000 face‑value bond with a 5% annual coupon pays ₹50 each year (or ₹25 every six months if semi‑annual). The total cash‑flows therefore consist of a series of equal coupons followed by the final redemption of the face value.
Exam questions frequently present a bond’s face value, coupon rate, frequency and years to maturity, then ask you to compute its price or yield. Remember that the cash‑flow pattern is the foundation for all pricing formulas.
- Coupon – periodic interest paid to bondholder.
- Face value – amount repaid at maturity.
- Maturity – number of periods until redemption.
Present Value Concept
Bond pricing relies on the principle of present value (PV): a cash flow received in the future is worth less today because of the opportunity cost of capital. The discount rate used for bonds is the yield per period, which reflects the market’s required return for a bond of similar risk and maturity.
To obtain the price, each coupon and the face value are discounted back to the valuation date using the yield per period. The sum of these discounted cash flows is called the dirty price. If the bond is traded between coupon dates, the accrued interest must be subtracted to arrive at the clean price quoted in most Indian market reports.
Understanding the PV framework helps you answer both direct calculation questions and conceptual ones that ask why bond prices move inversely to yields.
Where:
C= Coupon payment per period in rupeesy= Yield per period (decimal, e.g., 0.06 for 6%)t= Period index (1,2,...,n)n= Total number of periods until maturityM= Face (par) value of the bond in rupeesWorked Example
Given a 5‑year annual coupon bond with face value ₹1,000, coupon C = ₹50, and market yield y = 6% (0.06): Step 1: Compute PV of coupons = 50 \times \frac{1-(1+0.06)^{-5}}{0.06} = 50 \times 4.21237 = 210.62 Step 2: Compute PV of face value = 1,000 \times (1+0.06)^{-5} = 1,000 \times 0.747258 = 747.26 Step 3: Add both components: Price = 210.62 + 747.26 = 957.88 Verification: \sum_{t=1}^{5} \frac{50}{(1.06)^{t}} + \frac{1000}{(1.06)^{5}} = 957.88.
Students often forget to adjust the yield and coupon when the bond pays semi‑annually. Convert the annual yield to a per‑period rate (divide by 2) and double the number of periods; otherwise the price will be significantly off.
Accrued Interest and Clean vs Dirty Price
When a bond is bought between coupon dates, the seller is entitled to the interest that has accrued up to the settlement date. This amount is called accrued interest (AI) and is added to the clean price to obtain the dirty price that the buyer actually pays.
Accrued interest is calculated on a day‑count basis prescribed by the bond’s terms, commonly Actual/365 or 30/360 in India. The formula is AI = C \times (days accrued / days in coupon period). The clean price is the quoted market price without AI, while the dirty price = clean price + AI.
Exam questions may give the settlement date and ask you to compute the dirty price. Remember to first find the clean price using the PV formula, then add AI using the appropriate day count.
Where:
C= Coupon payment for the full period in rupeesD_{accrued}= Number of days from last coupon date to settlementD_{period}= Total number of days in the coupon period (e.g., 180 for semi‑annual)Worked Example
A semi‑annual coupon of ₹50, 45 days have passed since the last coupon, and the period is 180 days: Step 1: AI = 50 \times (45/180) Step 2: AI = 50 \times 0.25 = 12.50 Verification: 50 \times 45/180 = 12.50.
Clean Price vs Dirty Price
| Attribute | Clean Price | Dirty Price |
|---|---|---|
| Definition | Quoted market price excluding accrued interest | Actual amount paid including accrued interest |
| Use in | Quoting on exchanges and financial news | Settlement calculations |
| Effect of time | Remains constant between coupons | Increases as settlement date moves away from last coupon |
Yield to Maturity (YTM) Basics
Yield to Maturity is the internal rate of return (IRR) that equates the present value of all future cash flows to the bond’s current market price. In other words, it is the discount rate that makes the PV formula equal to the observed price.
YTM reflects the total expected return if the bond is held to maturity, assuming all coupons are reinvested at the same rate. It is the most exam‑relevant yield measure because NISM questions often ask you to compare bonds based on YTM.
Exact YTM requires solving the PV equation iteratively, but the syllabus provides a widely accepted approximation that is sufficient for most multiple‑choice items.
Where:
C= Annual coupon payment in rupeesM= Face (par) value in rupeesP= Current market price (clean) in rupeesn= Years to maturityWorked Example
Bond details: P = ₹950, C = ₹50, M = ₹1,000, n = 5 years. Step 1: Numerator = 50 + (1,000 - 950)/5 = 50 + 10 = 60 Step 2: Denominator = (1,000 + 950)/2 = 975 Step 3: YTM ≈ 60 / 975 = 0.0615 = 6.15% Verification: (50 + (1000-950)/5) / ((1000+950)/2) = 0.0615.
Current yield = C / P, which ignores capital gains/losses and time value. The exam often includes both; choose YTM when the question mentions total return or holding to maturity.
Zero‑Coupon Bond Pricing
A zero‑coupon bond does not make periodic interest payments. Instead, it is issued at a deep discount and pays only its face value at maturity. Because there are no coupons, its price is simply the present value of the face amount.
The pricing formula is straightforward: P = M / (1 + y)^n, where y is the yield per period and n is the total number of periods. This simplicity makes zero‑coupon bonds useful for illustrating the pure effect of discounting.
In the Indian market, Treasury bills are a common example of zero‑coupon securities. NISM questions may ask you to compute the discount rate given the issue price and maturity value, or vice‑versa.
Where:
P= Price of the zero‑coupon bond in rupeesM= Face (par) value in rupeesy= Yield per period (decimal)n= Number of periods to maturityWorked Example
M = ₹1,000, y = 8% (0.08), n = 3 years: Step 1: (1+0.08)^3 = 1.259712 Step 2: P = 1,000 / 1.259712 = 794.0 Verification: 1000 / (1.08)^3 = 794.0.
Bond Price Sensitivity to Yield (5‑Year, 5% Coupon)
Scenario
Rohan wants to buy a corporate bond with a face value of ₹1,000, annual coupon 5%, and 4 years remaining to maturity. The bond trades at a clean price of ₹970. Settlement occurs 60 days after the last coupon date. The bond follows a 30/360 day‑count convention.
Solution
Step 1: Compute accrued interest. For a semi‑annual coupon, each period is 180 days. Days accrued = 60, so AI = (₹25) × (60/180) = ₹8.33. Step 2: Add AI to clean price: Dirty price = ₹970 + ₹8.33 = ₹978.33. Step 3: Verify using the PV formula (optional for exam). The dirty price of ₹978.33 is the amount Rohan must pay at settlement.
Conclusion
The key is to separate clean price calculation from accrued interest, then combine them. Remember the day‑count convention; using the wrong basis leads to a common exam error.
Practical Tips for the Exam
Always identify whether the question asks for a clean price, dirty price, or yield. Write down the given cash‑flow details before plugging numbers into formulas – this prevents mixing up coupon frequency or period counts.
When the yield is not given, use the approximation formula for YTM; it is accepted in the NISM exam and saves time compared to trial‑and‑error. For zero‑coupon bonds, remember there is no coupon term in the price equation.
Finally, double‑check your day‑count convention. The Indian market commonly uses 30/360 for corporate bonds and Actual/365 for government securities. A quick mental check of the denominator (180 vs 365) can avoid costly mistakes.
⭐Exam Takeaways
- Bond price = present value of all coupons plus discounted face value; use the correct per‑period yield.
- Accrued interest = Coupon × (days accrued ÷ days in period); add it to clean price to obtain dirty price.
- Clean price excludes AI, while dirty price is the actual cash outflow at settlement.
- Approximate YTM = (C + (M‑P)/n) ÷ ((M+P)/2); use it when the exact IRR is not required.
- Zero‑coupon bond price = M ÷ (1+y)^n; no coupon term appears.
- Adjust coupon and yield for semi‑annual bonds: divide annual rate by 2 and double the number of periods.
- Remember the day‑count convention (30/360 or Actual/365) as it directly affects accrued interest.
- In multiple‑choice questions, eliminate options that ignore capital gains/losses – those are usually current‑yield values, not YTM.
Practice Questions
8 questions on Pricing of Bond
What term describes the amount repaid to the bondholder at maturity?
Which formula correctly calculates accrued interest (AI) for a bond?
How does a change in market yield affect the price of a bond?
A 5‑year annual coupon bond has a face value of ₹1,000, a coupon of ₹50 per year, and a market yield of 6% per year. What is its clean price?
Using the approximate YTM formula, what is the yield to maturity for a bond with P=₹950, C=₹50, M=₹1,000 and 5 years to maturity?
Which statement about clean and dirty prices is correct?
Rohan buys a bond with face value ₹1,000, annual coupon 5%, clean price ₹970. Settlement is 60 days after the last coupon date, using a 30/360 day‑count. What is the dirty price he must pay?
A zero‑coupon bond has a face value of ₹1,000, a yield of 8% per period, and 3 periods to maturity. What is its price?