Learn Fixed Income,
from first principles

A bond is a loan you make to a government or company, packaged as a tradable security. You hand over money today; the issuer promises fixed payments on fixed dates and returns your principal at maturity.

The key mental shift from equity: your upside is contractually capped at the promised cash flows. You are not a part-owner hoping for growth — you are a lender whose main concerns are getting paid and what rates do in the meantime.

The amount repaid at maturity, and the base on which coupons are calculated. A 7% coupon on ₹100 face pays ₹7 a year regardless of what you paid for the bond.

Face value is fixed by the issuer at issuance. Market price moves; face value does not.

The periodic interest payment, quoted as an annual percentage of face value. A 7.26% semiannual bond on ₹100 face pays ₹3.63 twice a year, not ₹7.26 twice.

Coupon rate is fixed at issuance and never changes (for a fixed-rate bond). It tells you the cash flows — it does not tell you your return, because you may buy above or below par.

The date the issuer repays face value and the bond ceases to exist. Longer maturity generally means greater price sensitivity to interest-rate changes.

Maturity is the crude measure of "how long"; duration is the precise one. Two bonds maturing the same day can behave very differently if one pays high coupons and the other pays none.

A bond's price is the present value of its remaining cash flows, discounted at the yield the market demands. Nothing more mysterious than that.

Because the cash flows are fixed, price and yield move in opposite directions by construction. If demanded yield rises, the same fixed payments are worth less today.

If the coupon rate exceeds the market yield, the bond's fixed payments are better than what's currently on offer, so it trades above par (premium). If the coupon is below market yield, it trades below par (discount).

Useful sanity check when using any calculator: coupon > yield should give price > 100. If it doesn't, an input is wrong.

A bond with no periodic coupons. You buy at a discount and receive face value at maturity; the entire return comes from the price appreciation.

Zeros matter far beyond their own market: because each pays exactly one cash flow on one date, they are the natural building block for spot rates and for decomposing any coupon bond into a portfolio of zeros.

The rule for converting calendar days into the year-fractions used in discounting and accrual. The same bond priced under Actual/Actual and 30/360 will give slightly different numbers — and both are "correct" under their own convention.

This is a frequent source of small mismatches when reconciling against a counterparty. Always confirm the convention before disputing a price.

Interest earned by the seller since the last coupon date but not yet paid out. The buyer compensates the seller for it at settlement, because the buyer will receive the whole next coupon.

It accrues roughly linearly between coupon dates, then resets to zero on each payment date — producing the familiar sawtooth pattern.

Clean price is what gets quoted in the market. Dirty price (or full/invoice price) is what you actually pay: clean price plus accrued interest.

Quoting clean is a convention that hides the sawtooth of accrual, making price series look smooth and comparable across dates. But discounting always happens on the dirty price — that is the real cash amount changing hands.

Annual coupon divided by clean price — a quick measure of the income you earn relative to what you paid.

It deliberately ignores the capital gain or loss you'll realise as the price pulls toward par at maturity, which makes it a poor measure of total return. Useful as a rough income screen, misleading as a comparison tool.

The single discount rate that sets the present value of all remaining cash flows equal to the dirty price you pay. It is, precisely, the bond's internal rate of return if held to maturity.

YTM carries an assumption worth stating plainly: it assumes every coupon is reinvested at the YTM itself. In practice rates move, so realised return usually differs — see reinvestment risk.

The risk that rising market yields reduce your bond's price. It is the dominant risk for government bonds, where default is not a realistic concern.

Note the asymmetry of perspective: if you truly hold to maturity, price swings are noise and you still receive par. Interest-rate risk bites when you must sell early, or must mark to market.

The risk that coupons are reinvested at rates lower than assumed. It is the mirror image of interest-rate risk: falling rates help your price but hurt your reinvestment.

This is the honest caveat to YTM. A bond quoted at 7% YTM only delivers 7% if every coupon can be redeployed at 7% — an assumption that rarely survives contact with reality. Zero-coupon bonds are the one clean escape, having no coupons to reinvest.

The risk the issuer fails to pay. Unlike rate risk, this is not symmetric — you gain a little yield for bearing it, but a default can cost you most of your principal.

Credit is measured by the spread over a comparable government bond, and by ratings from agencies. Two bonds with identical duration can carry entirely different risk if one is a sovereign and the other a low-rated corporate.

Liquidity risk is the cost of exiting: wide bid-ask spreads or no bid at all. It is often underestimated in corporate bond markets.

Inflation risk is the erosion of your fixed payments' purchasing power. A 7% nominal yield with 6% inflation is a 1% real return — the nominal figure flatters the outcome.

The present-value-weighted average time until you receive your cash flows, expressed in years. A zero-coupon bond's Macaulay duration equals its maturity exactly; a coupon bond's is always less, because some money arrives earlier.

Think of it as the balance point of the cash-flow timeline.

The approximate percentage price change for a 1% (100 bp) move in yield. This is the number traders actually use.

A modified duration of 6.5 means roughly a 6.5% price fall if yields rise 1%. "Roughly", because the relationship is curved — see convexity.

The change in a position's value, in currency terms, for a 1 basis-point move in yield. Where modified duration speaks in percentages, DV01 speaks in rupees.

This makes it the practical hedging unit: to neutralise a portfolio you match DV01s, not durations, because a large position in a short bond can carry the same rupee risk as a small position in a long one.

Duration assumes the price-yield relationship is a straight line. It isn't — it's curved. Convexity is the second-order correction that captures that curvature.

Positive convexity is a genuine benefit to the holder: prices rise more when yields fall than they fall when yields rise by the same amount. Duration alone systematically understates gains and overstates losses.

The plot of yield against maturity for bonds of equivalent credit quality. Its shape encodes market expectations about growth, inflation and policy.

Normal (upward) is most common; inverted curves — where short yields exceed long — have historically preceded slowdowns, though the signal is far from infallible.

The yield on a single payment at one future date, with no intervening cash flows — in other words, the yield of a zero-coupon bond of that maturity.

Spot rates are the theoretically correct discount rates: every cash flow should be discounted at the spot rate matching its own date, rather than all at one YTM. They're derived from coupon bond prices by bootstrapping.

The rate for borrowing between two future dates, implied by today's spot curve. If one-year and two-year spots are known, the market-implied one-year rate starting a year from now follows by no-arbitrage.

Forwards are implied expectations, not forecasts — they embed risk premia and are frequently wrong about realised rates.

Sensitivity to a change in one segment of the curve, holding the others fixed. Standard duration assumes the whole curve shifts in parallel — real curves steepen, flatten and twist.

Two portfolios with identical overall duration can respond very differently to a steepening. Key-rate durations expose that difference.

The issuer may redeem the bond early at a set price. That option benefits the issuer — they call when rates have fallen and they can refinance cheaper — so investors demand a higher yield in compensation.

Economically: callable = straight bond − call option you have sold. You are short an option, which is why the upside is truncated.

The investor may sell the bond back to the issuer at a set price. This option benefits you, so putable bonds carry lower yields than otherwise identical straight bonds.

Economically: putable = straight bond + put option you have bought. It caps your downside when rates rise.

The yield you'd earn if the bond were called at a specific date and price. Computed exactly like YTM, but on a cash-flow schedule truncated at the call date, with the call price replacing the final redemption.

Where several call dates exist, each yields a scenario; the lowest is normally the one that matters, since the issuer will act in their own interest.

The yield if you exercise the put at a given date. Same mechanics as YTC, but the option is yours.

Because exercise is at your discretion, YTP is not folded into "worst case" the way calls are — you'd only put when it helps you.

The lowest yield across maturity and all call scenarios — the market's standard conservative quote.

Note it combines YTM with calls but excludes puts, precisely because calls are outside your control while puts are within it.

Spread measures quantify yield pickup over a risk-free benchmark, with increasing rigour:

Nominal / G-spread — simple yield difference over a comparable government bond. I-spread — over the swap curve. Z-spread — the constant amount added to every spot rate that makes discounted cash flows equal the price, correctly handling curve shape.

Z-spread is the right foundation for OAS, which strips option value out of it.

Duration computed by actually repricing the bond under small upward and downward yield shifts, rather than from a closed-form formula.

This is the only valid duration for bonds with embedded options, because the cash flows themselves change when rates move — an assumption analytic duration cannot accommodate.

The same repricing approach applied to the second-order term. For callable bonds it frequently comes out negative, which analytic convexity would never produce.

When falling yields stop lifting the price — because the call becomes likely and the bond's value compresses toward the call price. The price-yield curve bends the wrong way.

This is the defining hazard of callable bonds and mortgage-backed securities: you keep full downside when rates rise, but your upside is capped when they fall.

A lattice of possible future short rates, calibrated so it reproduces today's observed curve and volatility. Bonds are valued by working backwards from maturity, node by node.

Trees are what make option-embedded valuation tractable: at each node you can test whether the issuer would call, and value accordingly.

The spread over the benchmark curve that remains after removing the value of embedded options — computed by adjusting the spread in an interest-rate tree until the model price matches the market price.

OAS is what lets you compare a callable bond with a straight one on equal terms. A wide Z-spread may be nothing but compensation for a short call position; OAS reveals whether real credit compensation remains.

Coupons reset periodically against a benchmark plus a fixed spread, so the coupon rises and falls with market rates.

Because the coupon re-fixes, price stays near par and duration is very short — roughly the time to the next reset, not to maturity. Rate risk is largely replaced by spread and credit risk.

Limits written into a floating-rate coupon: a cap ceilings it, a floor supports it, a collar does both.

Each is an option, so each must be valued as one. Holding a capped floater means you have sold a cap — you keep upside only to the ceiling.

A bond convertible into a fixed number of the issuer's shares at the holder's option — a debt instrument with an equity call attached.

Valuation is genuinely hybrid: bond-like when the share price is low (the "bond floor"), increasingly equity-like as it rises. Requires both credit and equity-volatility inputs.

Debt with engineered payoffs — linked to an index, basket, or formula, sometimes with principal protection.

They decompose into a plain bond plus one or more derivatives. Analysing them means valuing each component; the embedded derivatives, not the bond, usually drive both the risk and the fees.

Real instruments frequently break the neat pattern: step-up coupons, amortising principal, sinking funds, stub periods, irregular first coupons.

None of this changes the underlying method — every cash flow still gets discounted from its own exact date. It simply means the schedule must be built explicitly rather than generated from a coupon rate.