NISM Series IV: Term Structure of Interest Rates, Yield Curves, Bond Pricing & Duration

Welcome to Part 2 of our NISM Series IV Interest Rate Derivatives study notes. In this post, we cover the term structure of interest rates, types of yield curves, bond pricing concepts including YTM, duration, modified duration, convexity, and the debt market in India. These are high-weightage topics in the NISM IV certification exam.

Term Structure of Interest Rates

The term structure of interest rates shows the relationship between interest rates (or bond yields) and different maturity periods. The interest rates of different terms are determined by the demand and supply of money for different tenors.

Types of Yield Curves — Very Important for NISM IV Exam

The yield curve can take different shapes. There are four main types you must know:

1. Normal Yield Curve

This is an upward sloping yield curve. It indicates higher yield for higher maturity. Long-term yields are higher than short-term yields because the risk premium is higher for longer maturities. This is the most common shape.

2. Inverted Yield Curve

Here, the short-term yields are higher than the long-term yields. This happens when policy rates are kept high to bring down excess demand and reduce financial bubbles. Severe asset-liability mismatch can also produce an inverted yield curve.

3. Flat Yield Curve

Yields remain constant regardless of time to maturity. There is no difference between short-term and long-term yield, indicating no extra premium for higher maturities.

4. Humped Yield Curve

Short-term and long-term yields are both lower than medium-term yields, creating a hump in the middle.

Term Structure of Rates: Shifts

The yield curve can shift in different ways:

Shift Type	Description
Steepening	Difference between long-rate (LR) and short-rate (SR) rises or widens. Curve shifts in anti-clockwise direction.
Flattening	Difference between LR and SR falls or narrows. Curve shifts in clockwise direction.
Parallel	All rates move in the same direction by the same extent.

Simple Interest vs Compound Interest

Simple Interest (SI) is basically an interest rate without any reinvestment option.

Formula: SI = Principal × Interest Rate p.a. × Time in years

Compound Interest (CI) — Under compound interest, the money received at various points in time is reinvested to earn a higher effective rate of return.

When interest is accrued for more than one period, it is important to distinguish between simple and compound interest.

Day Count Fraction (Day Count Basis)

Day count fraction specifies the conversion of a payment period into a year fraction. The two main conventions are:

Actual/Actual Day Counting — Takes into account the actual number of days between the last coupon date and the next coupon date. Widely used in India for bond securities.
30/360 (European) Day Counting — Considers all months as having 30 days and a year as having 360 days.

Accrued Interest, Clean Price & Dirty Price

This is a market practice specific to the bond market. For secondary market trades of coupon bonds, there are two prices:

Clean Price — The price at which the bond is negotiated.
Dirty Price (Invoice Price) — The price at which the bond is settled.

Dirty Price = Clean Price + Accrued Interest

Dirty price is always higher than clean price by the amount of accrued interest.

Yield Measures — Important for NISM Derivatives Exam 1. Current Yield

The simplest measure of yield on a bond:

Current Yield = Coupon / Clean Price × 100

2. Yield to Maturity (YTM)

YTM is the discount rate that equates the discounted future cash flows and principal to be received with the present value or current price of the bond. It is essentially the Internal Rate of Return (IRR) or the expected rate of return on the bond.

To make the different zero rates / spot rates easier for interpretation, we average all the rates into a single number — this is called the YTM.

3. Discount Yield (Money Market)

Yield measures for money market instruments are annualized but not compounded.

Discount Yield = (Face Value – Price) × 360 / (Face Value × Days to Maturity)

4. Spot Rate (Zero Rate)

The spot rate is the true return on investment. It considers premium/discount in bond price, capital gain/loss at redemption, and reinvestment of interim income.

Relation Between Coupon Rate, Yield, Price and Par Value

Bond Selling at	Coupon Rate (CR)	Current Yield (CY)	YTM
Par	CR = CY = YTM
Discount	CR < CY < YTM
Premium	CR > CY > YTM

Also:

If CR = R → Bond valued at par
If CR < R → Bond valued at discount
If CR > R → Bond valued at premium

Price-Yield Relationship

The relationship between bond price and yield is inverse in nature. Key points:

The inverse relation between a bond's price and rate of return is given by the negative slope of the price-yield curve. The movement is non-linear.
A bond with a larger maturity will have higher sensitivity to interest rate changes.
A bond with a lower coupon rate will have greater price sensitivity.

Risk Measures: Price Risk vs Reinvestment Risk

These two risks always work in opposite ways:

If market interest rates rise → Bond price falls but reinvestment income rises.
If market interest rates fall → Bond price rises but reinvestment income falls.

The change in bond price is instant after a change in interest rate, but the effect on reinvestment income is slow over time.

Duration of a Bond — Key Concept for NISM IV

Duration of a bond is the time-weighted average of the present value of a bond's future known cash flows. It is also called the weighted average maturity or payback period of the bond.

Macaulay Duration

Macaulay duration is the weighted average of the time to get the future cash flows from a bond. It is measured in units of years. In simple terms, it tells you the weighted average time you need to hold a bond so that the total present value of future cash flows equals the current market price of the bond.

Formula:

Mac Duration = [Sum of (PV(CFt) × t)] / Market Price of Bond

Important Duration Relationships

Coupon is inversely related to Duration — Higher coupon = Lower duration (early cash flows reduce payback period).
YTM is inversely related to Duration — Higher yield = Lower duration.
Duration increases with maturity.
For a zero coupon bond, duration equals its maturity.
For a simple coupon-paying bond with no embedded features, duration is less than its maturity.
The duration of a portfolio is equal to the weighted average of the duration of the bonds in the portfolio.

Modified Duration

Modified Duration is an adjusted measure of Macaulay duration that helps estimate a bond's price sensitivity to changes in interest rates. It illustrates the effect of a 100 basis points (1%) change in interest rates on the price of a bond.

Formula:

Modified Duration = Macaulay Duration / [1 + (YTM / Number of Coupon Periods per Year)]

Important: Modified duration shows the volatility of the dirty price.

Price Value of a Basis Point (PVBP)

PVBP is the change in price (in currency terms) of a bond if the yield changes by 1 basis point (0.01%).

Convexity

Duration is derived from the first derivative of the bond price equation — it is a first approximation of the price/yield relationship. However, modified duration is accurate only for very small and parallel shifts in the yield curve.

The actual price change is more convex (curved) than the straight line suggested by modified duration. Convexity measures how the bond's duration (and by implication, its price) will change depending on how much interest rates change.

Change in Price = Duration Effect + Convexity Effect

Convexity is a measure of the sensitivity of a bond's price to changes in yield that is not captured by duration, due to the non-linear relationship between price and yield.

Summary Table: Duration Quick Reference

Factor	Effect on Duration
Higher Coupon Rate	Lower Duration
Higher YTM	Lower Duration
Longer Maturity	Higher Duration
Zero Coupon Bond	Duration = Maturity

Quick Recap for NISM Series IV Exam

4 types of yield curves: Normal, Inverted, Flat, Humped.
3 types of yield curve shifts: Steepening, Flattening, Parallel.
Dirty Price = Clean Price + Accrued Interest.
Price and yield have an inverse relationship.
Modified Duration = Macaulay Duration / (1 + YTM/n).
Change in Price = Duration Effect + Convexity Effect.

Next Blog Post: We will cover the Role and Importance of the Debt Market, Primary & Secondary Debt Market in India, and Money Market instruments — all essential for your NISM IV exam.

For mock tests and question banks for the NISM Series IV Interest Rate Derivatives exam, visit passnism.in.