What is a Bond?
A bond is like a loan — but you are the one giving the loan, and the government, company, or organization is borrowing money from you.
- You lend them money today.
- They promise to pay you back later (on a maturity date).
- Meanwhile, they pay you interest regularly (this is called the coupon).
In return, you earn steady income and get your original money (principal) back after a fixed period.
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Key Features of a Bond:
| Feature | Meaning |
|---|---|
| Face Value | The original amount you lend (e.g., ₹1,000) |
| Coupon Rate | The interest rate paid to you (e.g., 5% yearly) |
| Maturity Date | When the bond ends, and you get your money back |
| Issuer | The borrower (government, corporation, etc.) |
| Price | What you pay today (could be above or below face value) |
| Yield | The real return you earn, based on price and coupon |
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Simple Example:
- You buy a bond with ₹1,000 face value.
- It pays 5% coupon → you get ₹50 every year.
- After 5 years, you get your ₹1,000 back.
✅ You earn income through coupons.
✅ You get back your principal at maturity.
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Why People Invest in Bonds:
- Steady income (safe, predictable)
- Lower risk than stocks (especially government bonds)
- Diversification in investment portfolios
- Capital preservation (protecting your money)
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Key Point:
A bond = you lend money → you earn interest → you get repaid later.
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Main Types of Bonds
| Type of Bond | Meaning |
|---|---|
| Government Bonds | Issued by national governments (e.g., U.S. Treasury Bonds, Indian G-Secs) — usually safest |
| Municipal Bonds | Issued by states, cities, or local governments — often tax-free in some countries |
| Corporate Bonds | Issued by companies to raise money — higher return, but higher risk than governments |
| Zero-Coupon Bonds | No regular interest payments; bought at a discount and paid full face value at maturity |
| Floating Rate Bonds | Interest rate changes over time based on a benchmark (like LIBOR, SOFR) |
| Convertible Bonds | Can be converted into company shares at a later stage |
| Callable Bonds | Issuer can repay (call back) the bond early before maturity |
| Puttable Bonds | Investor can ask for early repayment from the issuer |
| Perpetual Bonds | Bonds with no maturity date — pay coupons forever (or until called) |
| Inflation-Linked Bonds | Interest and principal adjust with inflation (e.g., TIPS in the U.S., CIBs in India) |
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Quick Summary:
- Government bonds = safer, lower return
- Corporate bonds = higher return, higher risk
- Zero-coupon bonds = no regular income, gain at maturity
- Convertible bonds = bond + stock option
- Floating rate bonds = protection against rising rates
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Extra Tip:
- Investment-grade bonds = safer companies (e.g., Apple, Microsoft)
- High-yield bonds (“junk bonds”) = riskier companies, higher returns
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Formula for Future Price of a Bond
The future price (or theoretical price) of a bond is mainly calculated based on the present value of its future cash flows — coupon payments and principal repayment — discounted at the yield to maturity (YTM).
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The Formula:
\text{Bond Price} = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}Where:
- C = Coupon payment per period
- F = Face value (principal)
- y = Yield per period (YTM divided appropriately)
- t = Time period (1, 2, 3…n)
- n = Total number of periods to maturity
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Meaning:
- Each coupon and the final principal are treated as separate cash flows.
- Each cash flow is discounted back to its present value.
- Adding them up gives you the bond’s price today or at any future point (adjusted for time left).
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Example:
Suppose:
- Face value = ₹1,000
- Annual coupon = 5% → ₹50/year
- YTM = 6%
- Maturity = 3 years
Then:
\text{Price} = \frac{50}{(1.06)^1} + \frac{50}{(1.06)^2} + \frac{1050}{(1.06)^3}Calculating each term:
- Year 1: ₹50 ÷ 1.06 = ₹47.17
- Year 2: ₹50 ÷ (1.06)^2 = ₹44.50
- Year 3: ₹1050 ÷ (1.06)^3 = ₹881.68
Adding up:
\text{Price} = 47.17 + 44.50 + 881.68 = ₹973.35
Thus, the bond’s price would be approximately ₹973.35 today.
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Important Notes:
- If calculating future price, you adjust n (remaining periods).
- If interest rates rise, future bond price falls.
- If interest rates fall, future bond price rises.
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What is Duration in a Bond?
Duration measures how sensitive a bond’s price is to changes in interest rates.
It tells you:
“If interest rates change by 1%, how much will the bond price move?”
It’s a way to measure the risk in a bond — especially interest rate risk.
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Key Points About Duration:
- Higher duration → Bond price is more sensitive to interest rate changes.
- Lower duration → Bond price is less sensitive.
- Duration also roughly tells you the average time you’ll wait to get your money back through cash flows.
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Types of Duration:
| Type | Meaning |
|---|---|
| Macaulay Duration | Weighted average time to receive all cash flows (in years) |
| Modified Duration | % change in price for 1% change in yield |
| Effective Duration | Duration considering bonds with embedded options (e.g., callable bonds) |
| Dollar Duration | How much ₹/$ value changes for a 1% interest rate change |
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Simple Formula for Modified Duration (approximate):
\text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \text{Yield per period}}
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Example to Understand:
Suppose:
- A bond has a Modified Duration of 5 years.
Then:
- If interest rates rise by 1%, the bond’s price will fall by approximately 5%.
- If interest rates fall by 1%, the bond’s price will rise by approximately 5%.
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Summary:
Duration = How much bond prices react to interest rate changes.
Higher duration = higher sensitivity = higher risk.
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What is Sensitivity of a Bond?
In bonds, sensitivity means:
How much the bond’s price will change when interest rates (yields) change.
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Main Concept:
- Interest rates ↑ → Bond prices ↓
- Interest rates ↓ → Bond prices ↑
✅ Bond prices and interest rates move in opposite directions.
✅ The amount by which the price moves is the sensitivity.
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How Do We Measure Sensitivity?
| Measure | What It Tells You |
|---|---|
| Duration | Approximate % price change for a 1% yield change |
| Modified Duration | More precise % price change for a 1% yield change |
| Convexity | How much duration itself changes when rates change (second-order adjustment) |
| Dollar Duration | ₹ or $ amount by which bond price changes for a 1% change in yield |
| Key Rate Duration | Sensitivity to yield changes at specific points on the yield curve |
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Simple Example:
Suppose:
- A bond has Modified Duration = 6.
Then:
- If interest rates increase by 1%, bond price will decrease by approximately 6%.
- If interest rates decrease by 1%, bond price will increase by approximately 6%.
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Important Points:
- Longer maturity bonds = Higher sensitivity (bigger price changes)
- Lower coupon bonds = Higher sensitivity
- Zero-coupon bonds = Highest sensitivity (duration = maturity)
- High convexity bonds = Price behaves better (more protection) when rates move sharply
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Simple Rule to Remember:
Higher duration = Higher sensitivity = More interest rate risk.
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Relationship Between Duration and Sensitivity
✅ Duration directly measures a bond’s sensitivity to interest rate changes.
- Higher Duration → Higher Sensitivity → Bigger price changes when interest rates move.
- Lower Duration → Lower Sensitivity → Smaller price changes when interest rates move.
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Simple Way to Remember:
| Duration | Sensitivity to Interest Rates | Risk |
|---|---|---|
| High | High (price moves a lot) | High |
| Low | Low (price moves less) | Low |
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Mathematically:
\%\text{ Change in Bond Price} \approx - (\text{Modified Duration}) \times (\%\text{ Change in Yield})- The minus sign (–) shows inverse relationship:
- When yields go up, bond prices go down.
- When yields go down, bond prices go up.
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Example:
Suppose:
- Bond A has Modified Duration = 3
- Bond B has Modified Duration = 7
Now, if market interest rates increase by 1%:
- Bond A’s price drops by ~3%
- Bond B’s price drops by ~7%
Thus, Bond B is more sensitive because it has a higher duration.
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Key Factors That Affect Duration (and Sensitivity):
| Factor | Effect on Duration |
|---|---|
| Longer maturity | Increases duration |
| Lower coupon | Increases duration |
| Higher coupon | Decreases duration |
| Higher yield | Decreases duration slightly |
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In Simple Words:
Duration tells you how emotionally sensitive your bond is to the drama of interest rate changes!
(Higher duration = More emotional reaction! 😅)
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What is Convexity in Bonds?
Convexity measures how the duration of a bond changes when interest rates change.
It captures the curvature — the bending — of the bond price vs. interest rate graph.
In simple words:
If Duration tells you the speed of bond price change,
Convexity tells you how the speed itself accelerates or slows down when interest rates change a lot.
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Why Convexity Matters:
- When interest rate changes are small, duration alone is good enough.
- When interest rate changes are big, convexity becomes very important to predict bond price more accurately.
- Bonds with higher convexity:
- Lose less money when rates rise
- Gain more money when rates fall
✅ Higher convexity = better bond behavior in volatile markets.
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Mathematical Idea (optional, for clarity):
- First derivative of bond price with respect to yield = Duration
- Second derivative of bond price with respect to yield = Convexity
So when we estimate bond price change:
\Delta P \approx - \text{Duration} \times \Delta r + \frac{1}{2} \times \text{Convexity} \times (\Delta r)^2- \Delta P = Change in bond price
- \Delta r = Change in yield
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Visual Intuition:
- Without convexity = Straight line (linear movement)
- With convexity = Smooth curve (more realistic)
That’s why real bond prices don’t move in straight lines when rates change — they curve!
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In Very Simple Words:
| Term | Meaning |
|---|---|
| Duration | “If interest rates change, how much does the price change?” |
| Convexity | “When interest rates change a lot, how much does the behavior of price change itself bend?” |
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Important Practical Point:
- Normal bonds (like government bonds) have positive convexity → Good.
- Callable bonds (issuer can call back bond early) can have negative convexity → Bad in falling rate environments.
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Relationship of Bond with Duration and Convexity
When you invest in a bond, two very important risk measures describe how its price behaves when interest rates change:
| Concept | Meaning |
|---|---|
| Duration | Measures how much the bond price will change for a small interest rate change. |
| Convexity | Measures how much the duration itself changes when interest rates change more significantly. |
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Simple Relation:
- Duration is the first layer of bond price sensitivity (linear effect).
- Convexity is the second layer (curvature effect).
Thus:
Bond Price Change ≈ Effect of Duration + Effect of Convexity
(Especially important when interest rates change a lot.)
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Important Observations:
| Aspect | Impact |
|---|---|
| High Duration | More sensitive to small interest rate changes (bigger price movements). |
| High Convexity | Price drops less when rates rise, and price rises more when rates fall — better protection. |
| Low Convexity | Price moves more “straight-line,” less protection in volatile markets. |
| Negative Convexity | (Rare) Price behavior worsens when rates fall (e.g., callable bonds). |
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Price Change Approximation Formula:
\Delta P \approx -\text{Duration} \times \Delta r + \frac{1}{2} \times \text{Convexity} \times (\Delta r)^2Where:
- \Delta P = change in bond price
- \Delta r = change in interest rate
- First term = Duration effect
- Second term = Convexity effect
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In Simple Words:
| Term | Simple Meaning |
|---|---|
| Duration | “How fast is the price moving?” |
| Convexity | “Is the movement getting faster or slower as we move further?” |
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Ultimate Key Point:
A bond with higher duration is more sensitive to rates, and a bond with higher convexity reacts more favorably to large rate changes.
Both duration and convexity together give you a full picture of bond price behavior.
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Specifically:
- The dotted line is the straight line showing how Duration alone predicts bond price movement when interest rates change.
- Duration assumes that the bond price changes in a perfectly straight line (like a tangent line touching a curve).
- However, real bond prices don’t behave perfectly linearly — they curve — that’s where Convexity comes in.
Notes–
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What is Interest Rate?
- Interest rate is the cost of borrowing money or the reward for saving money.
- It is usually set by banks or governments (like the central bank rate).
- Example: You borrow ₹1,00,000 from a bank at 10% interest rate, you must pay ₹10,000 per year as interest.
In bonds, the coupon rate (fixed interest payment) is like an interest rate based on face value.
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What is Yield (in bonds)?
- Yield is the return you actually earn from a bond investment.
- It depends on the bond’s market price and coupon payments.
- If bond prices fall, yields rise (and vice versa).
Example:
- You buy a ₹1,000 bond paying ₹50/year.
- If you buy it at ₹1,000 → yield = 5%.
- If you buy it at ₹900 → yield = 5.56% (because you’re earning ₹50 on a smaller investment).
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Relationship Between Interest Rate and Yield:
| Concept | Meaning |
|---|---|
| Interest Rate | The base level of rates set in the economy (by central banks, banks, etc.) |
| Yield | The return you earn, which is influenced by interest rates and market bond prices |
✅ When market interest rates rise, bond prices fall, and yields rise.
✅ When market interest rates fall, bond prices rise, and yields fall.
Thus, yields and interest rates move closely together, but yield depends on both the bond’s coupon and its current market price.
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Simple Final Thought:
Interest rate is the environment. Yield is your personal return.
They are connected, but not exactly the same thing!