Concept of Duration

Duration is a measure of the sensitivity of a bond's price to a change in yield. It represents the weighted average time (in years) until the bond's cash‑flows are received, where the weights are the present values of those cash‑flows.

The concept was introduced by Frederick Macaulay in 1938 and is widely used by investment advisers to gauge interest‑rate risk. A higher duration means the bond price will move more for a given change in yields, which is crucial when advising risk‑averse clients.

In the NISM exam, duration appears in questions on bond pricing, portfolio immunisation, and the impact of yield changes on fixed‑income holdings. Remember: the term “duration” on its own usually refers to Macaulay duration unless the question specifies Modified or Effective duration.

• Duration is expressed in years, not months or days.
• It is a linear approximation; large yield shifts require convexity adjustments.

Macaulay duration (D_M) calculates the weighted average time until each cash‑flow is received, weighting each period by its present value. The formula sums the product of time (t) and the present value of the cash‑flow at that time, then divides by the total present value of all cash‑flows (i.e., the bond price).

This measure is useful for comparing bonds with different coupon structures because it reflects the timing of cash‑flows rather than just the maturity date. A zero‑coupon bond’s Macaulay duration equals its time to maturity, while a high‑coupon bond will have a lower duration.

For the exam, you may be asked to compute Macaulay duration for a simple annual‑coupon bond, or to interpret a given duration value to infer price volatility.

Modified duration (D_Mod) converts Macaulay duration into a direct measure of price sensitivity. It adjusts for the level of yield by dividing Macaulay duration by (1 + y/k), where k is the number of compounding periods per year. For most NISM questions, bonds are assumed to have annual compounding, so k = 1.

The formula is: D_Mod = D_M / (1 + y). This gives the approximate percentage change in price for a 1% (or 100 basis‑point) change in yield. For example, a bond with D_M = 5 years and YTM 8% will have D_Mod ≈ 5 / 1.08 ≈ 4.63. This means a 1% rise in yield reduces the bond price by roughly 4.63%.

Exam questions often present a Macaulay duration and ask you to compute the Modified duration, or vice‑versa. Remember to use the same yield that was used to compute the Macaulay duration.

Effective duration measures price sensitivity for bonds with embedded options (callable, putable, or mortgage‑backed securities). Because cash‑flows can change when the issuer exercises an option, the simple Macaulay formula is insufficient.

The standard approximation uses price changes for a small parallel shift in yield (Δy). The formula is: D_eff = (P_{-} - P_{+}) / (2 × P_0 × Δy), where P_{-} and P_{+} are bond prices after decreasing and increasing the yield by Δy respectively, and P_0 is the current price.

In the exam, you may be given two bond prices after a 100‑basis‑point shift and asked to compute effective duration. This helps you assess the interest‑rate risk of option‑adjusted securities.

Coupon rate: Higher coupons shift more cash‑flows to earlier periods, reducing duration. Conversely, low‑coupon or zero‑coupon bonds have longer durations because most of the value is received at maturity.

Maturity: All else equal, longer‑maturity bonds have higher duration. However, the effect is moderated by the coupon – a 10‑year bond with a 10% coupon may have a similar duration to a 5‑year bond with a 2% coupon.

Yield level: As yields rise, the present value of distant cash‑flows falls faster, shortening duration. This inverse relationship is captured in the Modified duration formula.

Embedded options: Callable bonds typically have lower effective duration because the issuer may call the bond when rates fall, truncating cash‑flows. Putable bonds show higher effective duration as the holder can force early repayment when rates rise.

• Remember: Duration ≈ (1 + y) / (y) for a zero‑coupon bond.
• Convexity complements duration; high convexity means the linear approximation of duration is less accurate for large yield moves.

Investment advisers use duration to construct "immunised" portfolios, where the portfolio's duration matches the investor's liability horizon. When the duration of assets equals the duration of liabilities, small parallel shifts in interest rates have minimal impact on the surplus or deficit.

For example, if a client needs funds in 5 years, an adviser might select a mix of bonds whose weighted average duration is also 5 years. This reduces the risk that a sudden rise in rates erodes the portfolio value before the cash need.

Exam questions may present a client with a target horizon and ask you to compute the required mix of two bonds to achieve a specific portfolio duration. The key steps are: (1) calculate each bond's duration, (2) set up the weighted‑average equation, and (3) solve for the weightings.

Duration assumes a linear relationship between price and yield, which holds only for small yield changes (typically ≤ 50 basis points). For larger moves, the price‑yield curve is curved; convexity measures this curvature.

For bonds with embedded options, effective duration provides a better estimate, but even it is an approximation that depends on the chosen Δy. The actual price change can differ if the option is exercised at a different point.

In exam scenarios, if a question mentions a 200‑basis‑point shift, you should be cautious about using duration alone. The correct approach is to compute price at the new yields or to apply convexity if the data is provided.