Fixed income: bond pricing, yields and duration

In short

Fixed income is the study of debt instruments - bonds, debentures, government securities, treasury bills - whose cash flows (coupons and principal) are contractually fixed, so their price is simply the present value of those cash flows at a market discount rate. Because the cash flows are fixed, price moves inversely with yield, and duration measures how much. The topic matters because India's bond market is large and institutional: the RBI issues G-Secs and T-bills, SEBI regulates corporate debt, and banks, insurers, mutual funds and pension funds must value and hedge huge portfolios daily. It is examined in MBA/PGDM Security Analysis, Investment Management and Fixed Income papers, in CFA/NISM/FRM certifications, and in placement tests at ICICI Bank, HDFC Bank, SBI, Axis, Kotak, Nippon India MF, HDFC AMC, CRISIL, ICRA, Deloitte and the treasury/risk desks of every bank.

Explained

Instruments, features and classification

• Bond / debenture: a contract to pay periodic coupons and repay face value (par) at maturity. In India "debenture" is the Companies Act, 2013 term for corporate debt; economically the same as a bond.
• Key features: face value (usually ₹100 for G-Secs, ₹1,000 or ₹1,00,000 for corporate bonds), coupon rate (annual, paid half-yearly on Indian G-Secs), maturity date, redemption terms, embedded options.
• Government securities (G-Secs): dated securities issued by RBI on behalf of the Centre, tenors from 2 up to 40 years (a 50-year security has also been issued), sovereign-risk-free in rupees, traded on NDS-OM; State Development Loans (SDLs) are state issues at a small spread over central G-Secs.
• Treasury bills: 91, 182, 364-day Central Government paper, zero-coupon, issued at a discount to ₹100 face value. Cash Management Bills are shorter and ad hoc.
• Corporate bonds / NCDs: issued under SEBI (Issue and Listing of Non-Convertible Securities) Regulations, 2021, rated by CRISIL/ICRA/CARE/India Ratings; mostly privately placed and traded OTC, reported on the NSE/BSE reporting platforms.
• Other Indian instruments: commercial paper and certificates of deposit (money-market, discount instruments), floating rate bonds (coupon reset off a benchmark), inflation-indexed bonds, capital-indexed, STRIPS (coupon-stripped G-Secs), Masala bonds (rupee-denominated, issued offshore), AT-1 and Tier-2 bank bonds (loss-absorbing, perpetual in the case of AT-1), tax-free PSU bonds (NHAI, REC, PFC - older issues), sovereign gold bonds (gold-linked, not plain fixed income).
• By embedded option: callable (issuer can redeem early - bad for investor when rates fall), puttable (investor can sell back - valuable when rates rise), convertible (into equity).
• By coupon structure: plain vanilla fixed coupon, zero-coupon (deep discount), step-up/step-down, floating rate, perpetual.

Pricing: present value of the cash flows

• Core principle: price = PV of all future coupons + PV of redemption value, discounted at the market-required yield. Nothing else; everything in fixed income is a restatement of this.
• Semi-annual convention: for a bond paying twice a year, use half the annual coupon per period and half the annual yield per period, over 2n periods - the Indian G-Sec market convention.
• Price-yield relationship: inverse and convex (curved, not straight). Price falls less for a yield rise than it gains for an equal yield fall.
• Premium, par, discount: if coupon > yield the bond trades above par (premium); coupon = yield → at par; coupon < yield → at a discount. A premium bond's price drifts down towards par as maturity approaches (pull to par), and a discount bond's drifts up.
• Clean vs dirty price: quoted (clean) prices exclude accrued interest; the buyer pays the dirty (settlement) price = clean + accrued interest. Indian G-Secs use a 30/360 day count for accrued interest; SEBI requires actual/actual for corporate bonds.
• Between coupon dates: price with the fractional-period formula or discount each cash flow by its exact time to receipt.
• Why price changes: the level of interest rates (repo/policy), inflation expectations, credit spread changes, liquidity, demand-supply at auction, and the passage of time.

Yield measures - and what each one leaves out

1. Nominal/coupon yield = coupon / face value. Fixed for life; tells you nothing about return at today's price.
2. Current yield = annual coupon / market price. Ignores capital gain/loss at redemption and the time value of money.
3. Yield to maturity (YTM): the single discount rate that equates the PV of all cash flows to the current market price - i.e. the bond's internal rate of return, assuming it is held to maturity and every coupon is reinvested at the YTM.
4. Yield to call (YTC) / yield to put: same calculation to the call/put date and call price; for a callable bond quote the yield to worst (lowest of YTM and all YTCs).
5. Realised (horizon) yield: actual compound return including the real reinvestment rate and the sale price - the only honest measure ex post.
6. Spot (zero) rate: yield on a zero-coupon cash flow of a given maturity; derived by bootstrapping from coupon bond prices. Forward rate: rate implied today for a future period.
7. Bond equivalent yield / discount yield for T-bills; the money market also uses simple-interest conventions.
8. YTM's limitations: assumes a flat yield curve, no default, reinvestment at YTM, and holding to maturity. Reinvestment risk makes realised yield differ from YTM when rates move.
9. Yield curve (term structure): plot of yield against maturity. Shapes - normal/upward sloping, flat, inverted, humped. Theories: expectations theory (long rates = average of expected future short rates), liquidity preference (a term premium for longer paper), market segmentation / preferred habitat (distinct investor pools by maturity). In India the 10-year G-Sec yield is the benchmark reference rate.

Duration: measuring interest rate risk

• Macaulay duration (D): the weighted-average time (in years) to receive the bond's cash flows, weights being each cash flow's PV as a share of price. Interpretation: the bond's effective maturity, and the horizon at which price risk and reinvestment risk offset.
• Modified duration (MD) = D / (1 + y/m): the percentage price change for a 1% (100 bp) change in yield. This is the practical risk number on a trading desk.
• Dollar/rupee duration and PVBP: PVBP (price value of a basis point) = MD × Price × 0.0001 - the rupee loss per ₹100 (or per crore) of face value for a 1 bp yield move. Used for hedging and VaR.
• Determinants of duration: longer maturity → higher duration; lower coupon → higher duration; lower yield → higher duration; a zero-coupon bond's Macaulay duration equals its maturity (the only such case).
• Portfolio duration = weighted average of component durations (weights = market values) - the basis of fund factsheets quoting "modified duration 4.2 years".
• Effective duration: computed by repricing for small up/down yield shifts; the correct measure for bonds with embedded options, where cash flows themselves change with yield.
• Convexity: the curvature of the price-yield curve - the second-order term. Duration alone always underestimates the price (overstates loss, understates gain), so adding convexity improves the estimate. Positive convexity is valuable to the holder; callable bonds and MBS show negative convexity at low yields.
• Immunisation: match portfolio duration to the investment horizon (or to liability duration) so that a parallel yield shift leaves the terminal value unchanged; must be rebalanced as time passes and yields move. Limitations: works for parallel shifts only, assumes no default, needs frequent rebalancing.
• Asset-liability management: insurers and banks match the duration of assets and liabilities; duration gap drives the sensitivity of net worth to rates.

Risks, credit and the Indian regulatory frame

• Interest rate (price) risk - captured by duration and convexity; reinvestment risk - moves opposite to price risk; credit/default risk - rating migration and loss given default; liquidity risk - wide bid-ask in Indian corporate bonds; call/prepayment risk; inflation risk; currency risk for foreign-currency bonds; event and sovereign risk.
• Credit spread = corporate yield − G-Sec yield of similar maturity; widens in stress. Ratings: AAA to D; investment grade is BBB− and above (Indian scale uses the "IND"/"CRISIL" prefix).
• Regulators and infrastructure: RBI (G-Sec issuance, NDS-OM trading, CCIL clearing, Retail Direct portal for individuals), SEBI (corporate bonds, debt MFs, Request for Quote platform), IRDAI and PFRDA (investment norms). Banks classify G-Sec holdings under RBI's 2024 investment directions as HTM / AFS / FVTPL (effective 1 April 2024), replacing the older HTM/AFS/HFT split; mark-to-market rules differ by bucket.
• Taxation (indicative - verify current provisions): under the Income-tax Act, 2025 (in force from 1 April 2026, with all sections renumbered), interest on bonds is taxable as income from other sources at slab rates; since the Finance Act, 2023 amendment, gains on specified mutual fund units and on market-linked debentures are treated as short-term regardless of holding period. Treatment of listed bond capital gains and indexation has changed more than once - check the current year's provisions rather than relying on a remembered section number.
• Debt mutual funds are categorised by SEBI by duration band: overnight, liquid, ultra-short, low, short, medium, long duration, gilt, corporate bond, credit risk.

Key points

• Bond price - PV of coupons plus PV of face value at the market yield; every other result follows from this.
• Inverse relation - price and yield always move in opposite directions.
• Convexity of price-yield curve - price gains from a yield fall exceed losses from an equal yield rise.
• Coupon vs yield - coupon > YTM means premium, coupon < YTM means discount, equal means par.
• Current yield - coupon/price; ignores redemption gain and time value.
• YTM - the IRR of the bond, assuming hold-to-maturity and reinvestment at YTM itself.
• Yield to worst - the lowest of YTM and all yields-to-call; the honest quote for callable paper.
• Macaulay duration - PV-weighted average time to cash flows, in years; equals maturity for a zero-coupon bond.
• Modified duration - % price change per 100 bp yield change = Macaulay / (1 + y/m).
• PVBP - rupee price change per basis point = MD × price × 0.0001; the desk's hedging unit.
• Duration drivers - rises with maturity, falls with higher coupon, falls with higher yield.
• Convexity correction - adds ½ × convexity × (Δy)² × price; always positive for plain bonds.
• Immunisation - set portfolio duration equal to the horizon or liability duration; rebalance periodically.
• Clean vs dirty price - quoted price excludes accrued interest; settlement price includes it.
• T-bills - 91/182/364-day zero-coupon Central paper issued at a discount to ₹100.
• Yield curve shapes - normal, flat, inverted, humped; explained by expectations, liquidity preference and segmentation theories.
• Credit spread - corporate yield minus matched-maturity G-Sec yield; compensation for default and liquidity risk.
• Negative convexity - callable bonds and mortgage-backed securities, because falling yields trigger redemption.

Formulas and examples

Example 1 - Pricing an annual-coupon bond at premium and discount. A corporate bond: face ₹1,000, coupon 9% annual (₹90), 5 years to maturity.

• At a required yield of 8%: annuity factor = [1 − 1.08⁻⁵]/0.08. 1.08⁵ = 1.46933, so 1.08⁻⁵ = 0.680583; (1 − 0.680583)/0.08 = 3.99271.
• PV coupons = 90 × 3.99271 = ₹359.34; PV principal = 1,000 × 0.680583 = ₹680.58. Price = ₹1,039.93 (premium, since 9% > 8%).
• At a required yield of 11%: 1.11⁵ = 1.685058 → 1.11⁻⁵ = 0.593451; annuity factor = (1 − 0.593451)/0.11 = 3.69590.
• PV coupons = 90 × 3.69590 = ₹332.63; PV principal = ₹593.45. Price = ₹926.08 (discount).
• Note the asymmetry: from the 8% price of ₹1,039.93, a 300 bp rise (to 11%) costs ₹113.85, whereas a 300 bp fall (to 5%, price = 90 × 4.32948 + 1,000 × 0.783526 = ₹1,173.18) gains ₹133.25 - convexity means the gain from a yield fall exceeds the loss from an equal yield rise.

Example 2 - Semi-annual G-Sec and its yield measures. A ₹100 face G-Sec, coupon 7.26% paid half-yearly (₹3.63 per period), 3 years (6 periods) left, required annual yield 7.00% (3.50% per period).

• 1.035⁶ = 1.229255 → 1.035⁻⁶ = 0.813501; annuity factor = (1 − 0.813501)/0.035 = 5.32854.
• PV coupons = 3.63 × 5.32854 = ₹19.34; PV face = 100 × 0.813501 = ₹81.35. Clean price = ₹100.69.
• Current yield = 7.26/100.69 = 7.21%. Nominal yield = 7.26%. YTM = 7.00% (given). Order is coupon > current yield > YTM for a premium bond - a standard exam check.
• If the last coupon was paid 73 days ago, on the G-Sec 30/360 basis (180-day half-year): accrued = 3.63 × 73/180 = ₹1.472; dirty price = 100.69 + 1.47 = ₹102.16 per ₹100 face.

Example 3 - Macaulay duration, modified duration and price prediction. A ₹1,000 face bond, 10% annual coupon, 3 years, YTM 10% (so it trades at par ₹1,000).

• Macaulay duration = 2,735.54 / 1,000 = 2.7355 years.
• Modified duration = 2.7355 / 1.10 = 2.4869 years (annual coupon, m = 1).
• Predicted price change for a 100 bp yield rise: −2.4869 × 0.01 = −2.487% → price ≈ ₹975.13.
• Actual price at 11%: 100/1.11 + 100/1.11² + 1,100/1.11³ = 90.09 + 81.16 + 804.31 = ₹975.56. Duration under-predicts the price by ₹0.43 - the convexity gap.
• Convexity: Σ t(t+1)PV = (1×2×90.91) + (2×3×82.64) + (3×4×826.45) = 181.82 + 495.87 + 9,917.40 = 10,595.09; C_x = 10,595.09 / (1,000 × 1.21) = 8.757 years².
• With convexity correction: ΔP/P = −2.4869(0.01) + 0.5(8.757)(0.0001) = −0.024869 + 0.000438 = −2.4431% → price ≈ ₹975.57, far closer to the true ₹975.56.
• PVBP = 2.4869 × 1,000 × 0.0001 = ₹0.2487 per bond; on ₹10 crore face value, about ₹24,900 per basis point (₹2.49 lakh on ₹100 crore).

Example 4 - T-bill yield and approximate YTM.

• A 91-day T-bill is bought at ₹98.35 per ₹100 face. Discount = ₹1.65. Annualised yield = (1.65/98.35) × (365/91) × 100 = 0.016777 × 4.01099 × 100 = 6.73% p.a.
• Approximate YTM of the Example 1 bond priced at ₹926.08: C = 90, F = 1,000, P = 926.08, n = 5. Numerator = 90 + (1,000 − 926.08)/5 = 90 + 14.784 = 104.784. Denominator = (1,000 + 2×926.08)/3 = (1,000 + 1,852.16)/3 = 950.72. Approx YTM = 104.784/950.72 = 11.02% - close to the true 11.00%, showing the shortcut is adequate for a quick check but not for settlement.

Example 5 - Duration and the coupon/maturity effect. Three ₹100 bonds, all yielding 8% annually:

• Takeaway: the zero-coupon bond has the highest duration for the same maturity (5.000 = its maturity exactly), but because its price is lower its rupee loss is smaller - duration measures percentage risk, PVBP measures rupee risk. Longer maturity (Bond C) raises both.
• Bond B check: 100/1.08⁵ = 100/1.469328 = ₹68.06; D = 5 years by definition; MD = 5/1.08 = 4.630.

Where it is used

• Bank and primary dealer treasuries: G-Sec and SDL trading on NDS-OM, SLR portfolio management, HTM/AFS/FVTPL classification and mark-to-market under RBI's investment directions - roles like dealer, treasury analyst, ALM officer at SBI, ICICI, HDFC Bank, Axis, PNB, ICICI Securities PD.
• Debt mutual funds and insurance: fund managers and credit analysts at HDFC AMC, SBI MF, Nippon India, ICICI Pru AMC, LIC and HDFC Life run duration calls, publish modified duration and average maturity on factsheets, and immunise against policyholder liabilities.
• Credit rating and research: CRISIL, ICRA, CARE, India Ratings analysts price spreads, build yield curves and assess default risk; CRISIL and FBIL publish the valuation curves Indian mutual funds use.
• Corporate treasury and fund-raising: issuing NCDs and commercial paper, choosing tenor and coupon structure, managing interest-rate exposure with OIS and interest-rate futures at NTPC, REC, PFC, L&T Finance, Bajaj Finance.
• Risk management and regulatory reporting: VaR and PVBP limits, interest-rate-risk-in-the-banking-book (IRRBB) and duration-gap statements, Basel capital charges for market risk.
• Retail and advisory: RBI Retail Direct and bond platforms (SEBI-registered OBPPs) sell G-Secs, SGBs and corporate bonds to individuals; wealth advisers compare YTM against FD rates after tax.

Interview questions

• Q: Why do bond prices fall when interest rates rise? A: The coupon is fixed, so when market yields rise, the old bond's cash flows must be discounted at a higher rate; its price falls until its yield matches the market.
• Q: What exactly does YTM assume, and when does it mislead? A: It assumes the bond is held to maturity, no default, and every coupon is reinvested at the YTM itself; if reinvestment rates differ or the bond is sold early, realised yield differs from YTM.
• Q: Rank coupon rate, current yield and YTM for a bond trading at a discount. A: YTM > current yield > coupon rate, because the investor also earns the capital gain from price to par at redemption.
• Q: What is Macaulay duration in words? A: The present-value-weighted average time until the bond's cash flows are received, which also measures the bond's effective maturity and its price sensitivity to yield.
• Q: Difference between Macaulay and modified duration? A: Macaulay is in years; modified is Macaulay divided by (1 + y/m) and gives the approximate percentage price change for a 100 basis point yield move.
• Q: Which has higher duration - a 5-year 12% coupon bond or a 5-year 6% coupon bond? A: The 6% bond, because more of its value sits in the distant principal payment, so its PV-weighted average time is longer.
• Q: Why add convexity to a duration-based price estimate? A: Duration is a straight-line approximation to a curved price-yield relation, so it always understates the price; the convexity term corrects most of that error, and matters most for large yield moves.
• Q: What is negative convexity and which instruments show it? A: Price appreciation slows or reverses as yields fall, because the issuer calls or borrowers prepay - seen in callable bonds, mortgage-backed securities and Indian AT-1 bonds with call dates.
• Q: How would you hedge a ₹100 crore G-Sec portfolio with modified duration 6? A: Compute PVBP = 6 × price × 0.0001 per ₹100 of face, then short interest-rate futures or pay fixed in an OIS/IRS with an equal and opposite PVBP, and rebalance as duration drifts.
• Q: Clean price or dirty price - what does a buyer actually pay? A: The dirty price: the quoted clean price plus accrued interest from the last coupon date, computed on a 30/360 basis for Indian G-Secs (actual/actual for corporate bonds).
• Q: What does an inverted yield curve suggest? A: That the market expects short rates - and usually growth and inflation - to fall; historically it has preceded slowdowns, though it is a signal, not a certainty.
• Q: Explain immunisation and its weaknesses. A: Setting portfolio duration equal to the liability horizon so price and reinvestment effects offset; it assumes parallel yield-curve shifts, ignores default, and needs periodic rebalancing as duration and the horizon both shorten at different rates.

Common mistakes

• Using the annual coupon and annual yield with 2n periods, or half-rates with n periods, in semi-annual bonds - halve both the coupon and the yield, and double the periods.
• Confusing coupon rate with yield: the coupon is fixed on face value, the yield is computed on market price.
• Quoting Macaulay duration where modified duration is wanted (and vice versa) - a 1% yield move prediction needs modified duration.
• Forgetting accrued interest, then being unable to reconcile a quoted price with the settlement amount.
• Treating duration as a reliable predictor for large yield moves without the convexity term, and applying plain duration to callable bonds instead of effective duration.
• Averaging durations using face value or number of bonds instead of market value weights when computing portfolio duration.
• Assuming a zero-coupon bond has no interest-rate risk - it has none from reinvestment, but the highest price risk for its maturity.

Related topics

Time value of money · Financial markets and instruments · Security analysis and portfolio management · Derivatives: futures and options basics · Bank management: Basel norms, NPAs and asset-liability management · Financial risk management: market, credit and operational risk