Valuation of Bonds
This sub‑topic covers how to value fixed‑income securities, specifically bonds, using the cash‑flow approach. Understanding bond valuation is essential for PMS distributors because it directly impacts portfolio construction and client advisory. The concepts link to yield calculations, price sensitivity, and regulatory disclosures required by SEBI.
Learning Objectives
- 1Explain the components of a bond and why valuation is required.
- 2Apply the present value formula to compute a bond's price.
- 3Calculate approximate Yield to Maturity (YTM) and understand its exam relevance.
- 4Interpret duration and the effect of interest‑rate changes on bond prices.
Understanding Bond Basics
A bond is a debt instrument issued by governments, corporations, or other entities to raise capital. The issuer promises to pay a fixed coupon periodically and to return the face value (also called par value) at maturity.
Key terms include: coupon rate (percentage of face value), coupon frequency (annual, semi‑annual), maturity period (in years), and credit quality. In the Indian context, SEBI classifies bonds as government securities, corporate bonds, and tax‑free bonds, each having distinct tax treatment.
Valuation matters because the market price of a bond fluctuates with changes in prevailing interest rates. For the NISM exam, you must know how to derive the fair price, compare it with market price, and explain any premium or discount.
- Coupon – periodic interest paid to the bondholder.
- Face Value – amount repaid at maturity, typically ₹1,000 in India.
Cash‑Flow Approach to Bond Valuation
The most widely accepted method is the present value (PV) of all future cash flows. Each coupon payment and the final principal repayment are discounted back to today using the investor’s required rate of return, often called the yield or discount rate.
Mathematically, the price (P) equals the sum of discounted coupons plus the discounted face value. The discounting reflects the time value of money – a rupee today is worth more than a rupee next year.
In the NISM exam, you will be given the coupon, face value, maturity, and market yield. The task is to compute the bond price and decide whether it trades at a premium, discount, or at par.
Where:
P= Current market price of the bond (rupees)C= Annual coupon payment (rupees)y= Market yield or required rate of return (decimal, e.g., 0.06 for 6%)FV= Face (par) value of the bond (rupees)n= Number of years to maturityWorked Example
Given C = 80, FV = 1000, y = 0.06, n = 10: Step 1: Compute PV of coupons = 80 \times \frac{1-(1+0.06)^{-10}}{0.06} = 588.82 Step 2: Compute PV of principal = 1000 \times (1+0.06)^{-10} = 558.39 Step 3: Add both = 588.82 + 558.39 = 1147.21 Verification: 80 \times \frac{1-(1+0.06)^{-10}}{0.06} + 1000 \times (1+0.06)^{-10} = 1147.21.
Students often substitute the coupon rate for the market yield in the price formula. Remember: the coupon rate is fixed, while the yield reflects current market conditions and is the discount rate.
Yield to Maturity (YTM) Concept
Yield to Maturity is the internal rate of return (IRR) earned by an investor who buys the bond at its current price and holds it until maturity, receiving all coupons and the face value.
YTM equates the present value of cash flows to the market price. Because the equation cannot be solved algebraically for y, exam questions either provide the price and ask for an approximate YTM, or give the yield and ask for price.
For quick calculations, NISM permits the use of an approximation formula that yields a close estimate, especially for bonds with moderate coupons and maturities.
Where:
YTM= Approximate yield to maturity (decimal)C= Annual coupon payment (rupees)FV= Face value (rupees)P= Current bond price (rupees)n= Years to maturityWorked Example
Using the bond from the previous example: C = 80, FV = 1000, P = 1147.21, n = 10. Step 1: (FV - P)/n = (1000 - 1147.21)/10 = -14.721 Step 2: Numerator = 80 + (-14.721) = 65.279 Step 3: Denominator = (1000 + 1147.21)/2 = 1073.605 Step 4: YTM ≈ 65.279 / 1073.605 = 0.0608 = 6.08% Verification: 65.279 ÷ 1073.605 = 0.0608 (≈6.08%).
Current yield = C / P. It ignores capital gains/losses and therefore is not the correct measure for bond valuation questions that ask for YTM.
Types of Bonds and Valuation Nuances
Key bond categories relevant for PMS distributors and their valuation considerations
| Bond Type | Typical Coupon Structure | Tax Treatment (India) | Valuation Note |
|---|---|---|---|
| Government Securities (G‑Sec) | Fixed coupon, often semi‑annual | Exempt from tax for resident individuals | Yield curve published by RBI; use same price formula |
| Corporate Bonds | Fixed or floating coupon | Taxable as per investor’s slab | Credit spread added to risk‑free rate in discounting |
| Tax‑Free Bonds (e.g., NHAI, NHAI‑IR) | Fixed coupon, usually lower than G‑Sec | Interest exempt from tax | Yield comparison should be after‑tax for other bonds |
| Zero‑Coupon Bonds | No periodic coupons; sold at deep discount | Taxed on accrued interest (deemed), not on receipt | Price = FV / (1+y)^n directly |
Impact of Interest‑Rate Changes
Bond Price Sensitivity to Yield Changes (10‑Year, 8% Coupon, ₹1,000 FV)
Scenario
An investor approaches a PMS distributor to invest ₹1,00,000 in a 10‑year government bond with a coupon of 8% payable annually. The current market yield for similar securities is 6%. The distributor must compute the fair price per bond and determine how many bonds can be purchased.
Solution
Step 1: Compute price of one bond using the formula. With C = 80, FV = 1000, y = 0.06, n = 10, price = 1147.21 rupees (as shown earlier). Step 2: Number of bonds = ₹1,00,000 ÷ 1147.21 ≈ 87.2, so the investor can buy 87 bonds (round down to avoid fractional bonds). Step 3: Total investment used = 87 × 1147.21 = 99,807.27 rupees, leaving a small cash remainder. The distributor reports a premium‑priced bond (price > face value) and explains that the lower market yield relative to the coupon creates the premium.
Conclusion
The example demonstrates the cash‑flow valuation, the effect of a lower market yield creating a premium, and the practical step of converting a lump‑sum amount into discrete bond units – a typical NISM question.
Duration and Convexity – Quick Insight
Duration measures the weighted average time to receive a bond’s cash flows and is a proxy for price sensitivity to interest‑rate changes. The most common version for exam purposes is Macaulay duration.
Macaulay duration (D) is calculated as D = \frac{\sum t \times PV(CF_t)}{\sum PV(CF_t)} where CF_t denotes each cash flow at time t. A higher duration means the bond price will move more for a given change in yield.
Convexity refines the estimate by accounting for the curvature of the price‑yield relationship, but the NISM syllabus only expects you to recognise its purpose, not to compute it.
Where:
D= Macaulay duration in yearsCF_{t}= Cash flow at time t (coupon or principal) in rupeesy= Yield to maturity (decimal)t= Time period in yearsn= Number of periods to maturityWorked Example
Using the bond from earlier (C=80, FV=1000, y=0.06, n=10): Step 1: Compute PV of each cash flow (already done). Step 2: Multiply each PV by its time t and sum = 8548.19. Step 3: Sum of PVs = 1147.21 (bond price). Step 4: D = 8548.19 / 1147.21 = 7.45 years. Verification: 8548.19 ÷ 1147.21 = 7.45.
Modified duration = Macaulay duration ÷ (1 + y). The exam may ask for price change using modified duration; ensure you convert correctly.
Practical Tips for PMS Distributors
When recommending fixed‑income products, always start with the client’s required return and match it against the bond’s YTM. Use a spreadsheet or a certified bond calculator to avoid manual errors.
Check the bond’s credit rating (CRISIL, ICRA) and ensure it complies with the client’s risk profile as per SEBI’s suitability norms. Remember that tax‑free bonds may appear attractive but their after‑tax yield should be compared with taxable bonds after adjusting for the investor’s tax slab.
Document the valuation steps, the assumptions about market yield, and any sensitivity analysis (e.g., price change for a 1% move in yield). This documentation satisfies SEBI’s record‑keeping requirements for PMS distributors.
⭐Exam Takeaways
- Bond price = PV of coupons + PV of face value; discount each cash flow using the market yield.
- Yield to Maturity is the discount rate that equates price with cash flows; use the approximation formula for quick exam calculations.
- A bond trades at a premium when its coupon > market yield, and at a discount when coupon < market yield.
- Macaulay duration measures weighted average time to cash‑flow receipt; higher duration = greater price sensitivity.
- Modified duration = Macaulay duration ÷ (1 + y) and is used to estimate price change for a 1% move in yield.
Practice Questions
8 questions on Valuation of Bonds
What is the term used for the periodic interest paid to the bondholder?
Which formula correctly represents the price of a bond using the cash‑flow approach?
Using a coupon of ₹80, face value ₹1,000, market yield 6% and 10 years to maturity, what is the bond's price (rounded to two decimals)?
Applying the approximate YTM formula to the bond in the previous question (price 1,147.21), what is the estimated YTM (in % to two decimal places)?
A 10‑year government bond has an 8% annual coupon and the current market yield for similar securities is 6%. How should the bond be classified in terms of price relative to par?
For the bond described earlier, the Macaulay duration is calculated as 7.45 years. Which statement is true?
Which type of bond is sold at a deep discount and has its price calculated directly as FV/(1+y)^n?
If a bond’s Macaulay duration is 7.45 years and its yield to maturity is 6%, what is the approximate modified duration?