What is the difference between Macaulay and Modified Duration?

Macaulay Duration is the weighted average time (in years) to receive all bond cash flows — it's a time measure. Modified Duration = Macaulay Duration / (1 + YTM/m) — it's a price sensitivity measure, telling you the % price change for a 1% yield change. For exam questions: if asked for price impact of a yield change, use Modified Duration. If asked for weighted average time to cash flows, use Macaulay Duration.

Why does bond price fall when yields rise?

Existing bond coupons are fixed. When new bonds are issued at higher yields, existing bonds paying lower coupons become less attractive — their prices must fall to make their effective yield (YTM) competitive with new issues. This is the fundamental price-yield inverse relationship. Think of it this way: if you paid ₹1,000 for a 5% coupon bond and new 1-year bonds are offering 7%, no one will pay ₹1,000 for your 5% bond — they'll only pay enough that their total return (coupon + price discount) equals 7%.

CFA Level 1 Fixed Income — Bond Pricing, Duration, Credit Analysis | PrepClever

Fixed Income is 11–14% of CFA Level 1 (~20 questions). The emphasis at Level 1 is bond mechanics — pricing, yield measures, duration, convexity, and an introduction to credit analysis. Fixed income deepens significantly at Level 2 (interest rate models, spread analysis, structured products) and Level 3 (portfolio construction).
CFA Level 1 · Fixed Income

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Key points

checkBond price = PV of coupon payments + PV of face value — discounted at YTM (Yield to Maturity)
checkYTM: the single discount rate that makes PV of bond cash flows equal to price — it's a promised yield, not guaranteed return
checkPrice-yield relationship is inverse: as yields rise, prices fall (and vice versa)
checkMacaulay Duration = weighted average time to receive bond's cash flows (in years)
checkModified Duration = Macaulay Duration / (1 + YTM/m) — % price change per 1% yield change
checkConvexity correction: actual price change = (−Modified Duration × ΔY) + (½ × Convexity × ΔY²)
checkCredit spread = yield of corporate bond − yield of comparable maturity government bond
checkInvestment grade: BBB−/Baa3 and above; High yield (junk): BB+/Ba1 and below

Bond Pricing

A bond's price is the present value of all its future cash flows:

Price = Σ [Coupon / (1+YTM)^t] + [Face Value / (1+YTM)^n]

For a semi-annual bond: use semi-annual coupon = Annual Coupon / 2, and semi-annual YTM = YTM / 2, and n = years × 2.

Example: 5% annual coupon, 3-year bond, face value ₹1,000, YTM = 6%:

Year 1: 50 / 1.06 = 47.17
Year 2: 50 / 1.06² = 44.50
Year 3: 50 / 1.06³ = 41.98 + 1000/1.06³ = 839.62
Price = 47.17 + 44.50 + 41.98 + 839.62 = ₹973.27 (discount bond — coupon < YTM)

Price-Yield Relationship

When YTM = Coupon rate: Price = Face Value (par bond)
When YTM > Coupon rate: Price < Face Value (discount bond)
When YTM < Coupon rate: Price > Face Value (premium bond)

Yield Measures

YTM (Yield to Maturity): Discount rate equating PV of cash flows to price. Assumes coupons reinvested at YTM and held to maturity.
Yield to Call (YTC): YTM calculated to the call date at call price — relevant for callable bonds in falling rate environments.
Current Yield: Annual Coupon / Current Price. Simple but ignores capital gain/loss and reinvestment income.
Effective Yield: For bonds with embedded options, uses option-adjusted spread (OAS) instead of YTM.

Duration

Macaulay Duration

The weighted average time (in years) to receive all cash flows, where weights = PV of each cash flow / Total Price.

Zero-coupon bond: Macaulay Duration = Maturity (all cash flow at end)

Coupon bond: Macaulay Duration < Maturity

Modified Duration

Modified Duration = Macaulay Duration / (1 + YTM/m)

where m = compounding periods per year.

Interpretation: If Modified Duration = 5, then a 1% rise in yield → approximately 5% price decline.

% Change in Price ≈ −Modified Duration × ΔYield

Example: Bond with Modified Duration = 7. Yield rises 0.50%: Price change ≈ −7 × 0.005 = −3.5%

Dollar Duration (DV01)

DV01 (Dollar Value of a Basis Point) = Modified Duration × Price × 0.0001

Used in portfolio hedging — tells you $ change in bond value per 1bp yield move.

Convexity

Duration is a linear approximation of price-yield relationship. Convexity captures the curvature.

More accurate price change = (−Modified Duration × ΔY) + (½ × Convexity × ΔY²)

Positive convexity is desirable: for a given yield move, price rises more than duration predicts and falls less. Higher convexity bonds command a premium.

Callable bonds have negative convexity at low yield levels (price is capped by call price even as yields fall).

Credit Analysis

Credit Rating Framework

S&P/Fitch	Moody's	Category
AAA, AA, A, BBB	Aaa, Aa, A, Baa	Investment Grade
BB, B	Ba, B	Speculative / High Yield
CCC, CC, C, D	Caa, Ca, C	Distressed / Default

Four Cs of Credit Analysis

Capacity: Ability to repay — EBITDA coverage, leverage ratios, free cash flow generation
Collateral: Asset quality — liquidation value in default scenario
Covenants: Debt covenants protecting bondholders (maintenance covenants, incurrence covenants)
Character: Management quality, track record, governance

Credit Spread Analysis

Credit Spread = Bond YTM − Risk-Free Rate (comparable government bond)

Spread reflects: Default risk premium + Liquidity premium

In recessions, spreads widen dramatically (flight to quality). In expansion, spreads narrow.

Practice questions on CFA Level 1 Fixed Income — Bond Valuation, Duration, and Credit Analysis

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Fixed Income is 11–14% of CFA Level 1 (~20 questions). The emphasis at Level 1 is bond mechanics — pricing, yield measures, duration, convexity, and an introduction to credit analysis. Fixed income deepens significantly at Level 2 (interest rate models, spread analysis, structured products) and Level 3 (portfolio construction).
CFA Level 1 · Fixed Income

lightbulb

Key points

checkBond price = PV of coupon payments + PV of face value — discounted at YTM (Yield to Maturity)
checkYTM: the single discount rate that makes PV of bond cash flows equal to price — it's a promised yield, not guaranteed return
checkPrice-yield relationship is inverse: as yields rise, prices fall (and vice versa)
checkMacaulay Duration = weighted average time to receive bond's cash flows (in years)
checkModified Duration = Macaulay Duration / (1 + YTM/m) — % price change per 1% yield change
checkConvexity correction: actual price change = (−Modified Duration × ΔY) + (½ × Convexity × ΔY²)
checkCredit spread = yield of corporate bond − yield of comparable maturity government bond
checkInvestment grade: BBB−/Baa3 and above; High yield (junk): BB+/Ba1 and below

Bond Pricing

A bond's price is the present value of all its future cash flows:

Price = Σ [Coupon / (1+YTM)^t] + [Face Value / (1+YTM)^n]

For a semi-annual bond: use semi-annual coupon = Annual Coupon / 2, and semi-annual YTM = YTM / 2, and n = years × 2.

Example: 5% annual coupon, 3-year bond, face value ₹1,000, YTM = 6%:

Year 1: 50 / 1.06 = 47.17
Year 2: 50 / 1.06² = 44.50
Year 3: 50 / 1.06³ = 41.98 + 1000/1.06³ = 839.62
Price = 47.17 + 44.50 + 41.98 + 839.62 = ₹973.27 (discount bond — coupon < YTM)

Price-Yield Relationship

When YTM = Coupon rate: Price = Face Value (par bond)
When YTM > Coupon rate: Price < Face Value (discount bond)
When YTM < Coupon rate: Price > Face Value (premium bond)

Yield Measures

YTM (Yield to Maturity): Discount rate equating PV of cash flows to price. Assumes coupons reinvested at YTM and held to maturity.
Yield to Call (YTC): YTM calculated to the call date at call price — relevant for callable bonds in falling rate environments.
Current Yield: Annual Coupon / Current Price. Simple but ignores capital gain/loss and reinvestment income.
Effective Yield: For bonds with embedded options, uses option-adjusted spread (OAS) instead of YTM.

Duration

Macaulay Duration

The weighted average time (in years) to receive all cash flows, where weights = PV of each cash flow / Total Price.

Zero-coupon bond: Macaulay Duration = Maturity (all cash flow at end)

Coupon bond: Macaulay Duration < Maturity

Modified Duration

Modified Duration = Macaulay Duration / (1 + YTM/m)

where m = compounding periods per year.

Interpretation: If Modified Duration = 5, then a 1% rise in yield → approximately 5% price decline.

% Change in Price ≈ −Modified Duration × ΔYield

Example: Bond with Modified Duration = 7. Yield rises 0.50%: Price change ≈ −7 × 0.005 = −3.5%

Dollar Duration (DV01)

DV01 (Dollar Value of a Basis Point) = Modified Duration × Price × 0.0001

Used in portfolio hedging — tells you $ change in bond value per 1bp yield move.

Convexity

Duration is a linear approximation of price-yield relationship. Convexity captures the curvature.

More accurate price change = (−Modified Duration × ΔY) + (½ × Convexity × ΔY²)

Positive convexity is desirable: for a given yield move, price rises more than duration predicts and falls less. Higher convexity bonds command a premium.

Callable bonds have negative convexity at low yield levels (price is capped by call price even as yields fall).

Credit Analysis

Credit Rating Framework

S&P/Fitch	Moody's	Category
AAA, AA, A, BBB	Aaa, Aa, A, Baa	Investment Grade
BB, B	Ba, B	Speculative / High Yield
CCC, CC, C, D	Caa, Ca, C	Distressed / Default

Four Cs of Credit Analysis

Capacity: Ability to repay — EBITDA coverage, leverage ratios, free cash flow generation
Collateral: Asset quality — liquidation value in default scenario
Covenants: Debt covenants protecting bondholders (maintenance covenants, incurrence covenants)
Character: Management quality, track record, governance

Credit Spread Analysis

Credit Spread = Bond YTM − Risk-Free Rate (comparable government bond)

Spread reflects: Default risk premium + Liquidity premium

In recessions, spreads widen dramatically (flight to quality). In expansion, spreads narrow.

Practice questions on CFA Level 1 Fixed Income — Bond Valuation, Duration, and Credit Analysis

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CFA Level 1 Fixed Income — Bond Valuation, Duration, and Credit Analysis

Bond Pricing

Price-Yield Relationship

Yield Measures

Duration

Macaulay Duration

Modified Duration

Dollar Duration (DV01)

Convexity

Credit Analysis

Credit Rating Framework

Four Cs of Credit Analysis

Credit Spread Analysis

Related Topics

CFA Level 1 Derivatives

CFA Level 1 Financial Statement Analysis

Keep Exploring

CFA Level 1

Financial Glossary

All NISM Certifications

FAQ

Put these notes into practice

CFA Level 1 Fixed Income — Bond Valuation, Duration, and Credit Analysis

Bond Pricing

Price-Yield Relationship

Yield Measures

Duration

Macaulay Duration

Modified Duration

Dollar Duration (DV01)

Convexity

Credit Analysis

Credit Rating Framework

Four Cs of Credit Analysis

Credit Spread Analysis

Related Topics

CFA Level 1 Derivatives

CFA Level 1 Financial Statement Analysis

Keep Exploring

CFA Level 1

Financial Glossary

All NISM Certifications

FAQ

Put these notes into practice