Modified Duration
Also written MD
Macaulay's duration divided by (1 + yield) — the percentage by which a bond's price moves for a one percentage point change in interest rates, and so the standard measure of interest rate risk.
In plain language
Every bond falls in price when interest rates rise. The question an analyst actually needs answered is by how much, and modified duration is the number that answers it.
Start with Macaulay's duration, which measures the weighted average time by which an investor gets the invested money back, using the present values of the bond's cash flows as the weights. Divide that by one plus the yield and the unit changes from years into sensitivity: a modified duration of 2.58 means a one percentage point rise in yields costs roughly 2.58% of the price.
How it works
Three features of a bond drive its duration, and the workbook states all three:
- Longer time to maturity → higher duration → higher interest rate risk.
- Lower coupon → higher duration. A small coupon means most of the money arrives at redemption, far away.
- Lower yield → higher duration. A low discount rate pushes weight onto the distant cash flows.
Duration is also not a fixed property of the bond. As a bond approaches maturity its duration shrinks and it becomes less rate-sensitive, which is why a fund manager who wants less risk shortens the portfolio rather than selling everything.
The formula
Σ [ CFt ÷ (1+y)^t × t ]
Macaulay Duration = ───────────────────────
CMP
Modified Duration = Macaulay Duration ÷ (1 + y)
Approximate price change = − Modified Duration × change in yield
CFt is the cash flow in year t, y the yield to maturity, and CMP the current market price.
A worked example
A 3-year bond, face value Rs 1,000, coupon 8% paid annually, yielding 8% — so it trades at par, Rs 1,000.
| Year | Cash flow | PV at 8% | PV × t |
|---|---|---|---|
| 1 | 80 | 74.07 | 74.07 |
| 2 | 80 | 68.59 | 137.17 |
| 3 | 1,080 | 857.34 | 2,572.02 |
| 1,000.00 | 2,783.26 |
Macaulay Duration = 2,783.26 ÷ 1,000 = 2.78 years
Modified Duration = 2.78 ÷ 1.08 = 2.58
Now yields rise 100 basis points, to 9%. The estimate says the price falls 2.58%, to Rs 974.23. Recompute the bond exactly at 9% and it is Rs 974.68 — the approximation is off by 45 paise, and the gap widens as the rate move gets larger.
Scale it up. A debt fund holding Rs 500 crore of this bond loses 2.58%, or Rs 12.9 crore, on that one percentage point. Had it held a 10-year bond with a modified duration of 6.7 instead, the same move would have cost Rs 33.5 crore — the same rupees, the same rate move, two and a half times the damage.
Why NISM asks about it
Chapter 3 (Terminology in Equity and Debt Markets, section 3.2.11) introduces duration as part of debt market terminology, and Chapter 12 lists interest rate risk among the systematic risks. Expect the formula relationship — modified equals Macaulay divided by (1 + y) — and the three directional statements: higher maturity, lower coupon and lower yield each raise duration and therefore interest rate risk.
Common exam traps
- Divide by (1 + y), not by (1 + y)^t. The exponent belongs in the discounting inside Macaulay's duration, not in the conversion.
- Duration is usually less than the term to maturity, because coupons return money early. Only a zero-coupon bond has a Macaulay duration equal to its maturity.
- It is an approximation, and a one-sided one. For a large rate move it overstates the loss on a rise and understates the gain on a fall.
- Duration is not static. It falls as the bond ages and it changes with the yield, so a stated duration belongs to a date.
- Higher duration is not "riskier" in every sense — it is specifically more sensitive to interest rates. It says nothing about whether the issuer will pay.
- Do not read modified duration as the bond's payback period. Macaulay's duration is measured in years; modified duration is a percentage sensitivity that merely inherits the same number.
Check yourself
1.A bond issued at face value ₹100 with a 10% coupon will, if market interest rates subsequently increase, likely quote:
- a)At a price above face value
- b)At the face value
- c)At a price below face value
- d)At a price the issuer announces
Show the answer
Answer: (c) At a price below face value
Because the value of a bond is the aggregate of the discounted value of the future cashflows from the bond, discounted at the market interest rates or required rate of returns of the investors, market price of bond and the interest rates are negatively related or inversely related. When the market interest rates increase the bond prices fall down, and when the interest rates fall down, the bond prices increase.
The bond's coupon is fixed at 10%. If new bonds now pay more, the only way this bond can offer a competitive return is for its price to fall.
How the price is set: the various players in the market collectively bring their intelligence to price the bond based on the ongoing market interest rates, inflation, and the default risk perceived in the bond. The market always compares the coupon rate of the bond and the ongoing market interest rates and then gives its verdict on the price.
Option D is wrong on a basic point — the issuer has no say in the secondary market price.
And a related rule about repayment: irrespective of the price at which a debt security is bought or sold in the secondary market, the issuer is liable to repay only the principal, represented by the face value, on maturity.
The measure of how far the price moves is Modified Duration: bonds with high modified duration experience greater increases in price when interest rates decline and greater losses in price when rates increase.
Where this is taught
- Series IV · Chapter 1: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Marketsintroduced here
- Series XXI-B · Chapter 4: Investing in Fixed Income Securitiesintroduced here
- Series V-A · Chapter 10: Risk, Return and Performance of Fundsintroduced here
- Series XV · Chapter 3: Terminology in Equity and Debt Marketsintroduced here
- Series X-A · Chapter 9: Investing in Fixed Income Securitiesintroduced here
- Series V-D · Chapter 18: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Marketsintroduced here
- Series XXI-A · Chapter 4: Investing in Fixed Income Securitiesintroduced here
- Series XV · Chapter 12: Fundamentals of Risk and Return
Related terms
- Interest rate riskThe risk that an investor in a debt instrument loses return because rates rise — existing instruments carrying the old, lower coupon fall in value until their yield matches the new market rate.
- Reinvestment riskThe risk that the coupons or other intermediate cash flows from an investment have to be put back to work at a lower rate than the original investment earned, pulling the total return below the promised yield.
- Yield to MaturityThe single discount rate at which a bond's future coupons and redemption amount add up to exactly its market price today — the return you actually earn if you hold it to maturity.
- Coupon rateThe rate of interest a bond pays, applied to its face value and never to its market price — which is why the coupon tells you the cash flow but not the return.
- Macaulay durationThe weighted average time, in years, to receive a bond's cash flows, each weighted by the present value of that cash flow — the bond's effective payback period.
- Price Value of a Basis PointThe rupee change in a bond's price for a one basis point change in its yield — the unit in which a fixed income desk actually measures and hedges interest rate risk.
- ConvexityThe curvature of the price-yield relationship — the correction duration misses, because duration is a straight line and the true relationship bends.
- Accrued interestCoupon earned from the last coupon date up to settlement, paid by the buyer to the seller on top of the negotiated price, because the issuer will pay the whole coupon to whoever holds the bond next.
- Term structure of interest ratesInterest rate plotted against term — one curve per credit quality, with the risk-free curve as the base and every other borrower quoted as a spread over it.
- Duration-based hedge ratioThe number of interest rate futures that drives a bond portfolio's duration to zero — portfolio modified duration times market value, divided by futures modified duration times futures price over par.
- UnderhedgingHolding fewer futures than the exposure needs — the specific failure of a duration-based hedge in a large yield move, because duration draws a straight line through a curved relationship.
- Parallel shiftA change in the term structure in which all rates move in the same direction by the same extent, leaving the spread between short and long rates unchanged — the only kind of move duration handles well.
- VarianceA measure of how widely a scheme's periodic returns fluctuate around its own average return — the square of standard deviation, and a risk measure relevant to both debt and equity schemes.
- Weighted average maturityThe average time to maturity of a debt portfolio, each security weighted by its share of the portfolio — a simple indicator of how sensitive a debt scheme's NAV is to interest rate changes.
- Macaulay duration bucketThe duration band an index provider uses to sort bonds — NIFTY bond indices run six such buckets, from Ultra-Short at 3 to 6 months up to Long Duration above 7 years.
- Positive convexityThe favourable asymmetry of an option-free bond: when yields fall its price rises more than duration predicts, and when yields rise it falls less — so buying convexity is an active fixed income strategy.
- TwistA steepening or flattening of the yield curve in which short-term and long-term rates move in opposite directions — as distinct from a convexity change, where they move the same way by different amounts or one stays put.
- Holding period returnThe total of coupons, income earned on reinvesting them and any capital gain, expressed as a percentage of the purchase price — a crude return for the period actually held, with no compounding in it.
- Deep Discount BondA zero-coupon bond of very long tenure, issued at a steep discount to its face value and redeemed at par. IDBI issued such bonds; the Kisan Vikas Patra is another example.
- Credit spreadThe extra yield a non-government borrower must pay over a government security of the same tenor — the market price of credit risk, quoted as an add-on over the risk-free rate.
- Short duration fundAn open-ended debt scheme in the short end of the curve: Chapter 2 defines the "Short Term Fund" by a Macaulay duration of 1 to 3 years; Chapter 12 describes maturities of 1 to 3 years.
- Spread riskThe risk that a floating rate security loses value because its spread over the benchmark moves adversely, even though the benchmark yield itself has not changed.
- Duration managementA debt fund manager's deliberate strategy of lengthening or shortening a portfolio's duration based on a view on interest rates, used only where the scheme's mandate allows it.
- Downgrade riskThe risk that a rating agency lowers an issuer's credit rating after an investor has bought its bonds, pushing the market price of those bonds down even if no payment is ever missed.