Clean Price, Dirty Price and Accrued-Interest Calculations - Article

Introduction

When a coupon-bearing bond is bought or sold between two coupon-payment dates, the quoted market price is normally not the entire amount paid by the buyer.

The seller has held the security since the previous coupon date and has therefore earned part of the next coupon.

Consequently, the buyer normally compensates the seller for this accrued interest at settlement.

This creates three important concepts:

1. Clean Price

2. Accrued Interest

3. Dirty Price

The relationship is:

Clean Price + Accrued Interest = Dirty Price

The current IIBF CAIIB BFM syllabus includes Securities Market Products under Module C – Treasury Products.

RBI's Government Securities Market Primer specifically explains clean price, accrued interest, consideration amount and the day-count convention used in the Indian Government-securities market.

Learning Objectives

After studying this topic, you should be able to:

• distinguish clean price from dirty price;

• understand why accrued interest is paid to the seller;

• calculate accrued coupon interest;

• determine the total settlement consideration;

• apply the 30/360 convention for Indian Government securities;

• distinguish face value from quoted price;

• understand what happens on or near coupon dates; and

• avoid common CAIIB numerical errors involving price quotation and settlement amount.

What Is Clean Price?

The clean price is the quoted price of a coupon-bearing bond excluding accrued interest.

RBI describes the market-quoted price of a Government security as clean price when the accrued-interest component has not been added.

For example, if a bond is quoted at:

₹101.25

this generally means ₹101.25 for every ₹100 face value, before adding accrued interest.

Clean price is useful because it allows market participants to observe changes in the bond's underlying market value without the mechanical build-up of coupon interest between coupon dates.

What Is Accrued Interest?

Coupon interest is paid only on specified coupon dates.

However, interest economically accumulates between those dates.

Suppose a bond pays coupons every six months.

If the seller holds the bond for part of the six-month coupon period and then sells it, the seller has earned interest for the period during which the security was held.

The buyer will receive the entire next coupon from the issuer if the buyer holds the security on the relevant payment date.

Therefore, the buyer compensates the seller for the seller's share of the upcoming coupon through accrued interest.

RBI describes accrued interest as interest calculated for the broken period from the previous coupon date up to the prescribed point before settlement.

What Is Dirty Price?

The dirty price is the total price including accrued interest.

Dirty Price = Clean Price + Accrued Interest per ₹100 Face Value

This gives the full price including the interest accumulated since the last coupon date.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Dirty price is also commonly called the:

• full price; or

• invoice price,

depending on market terminology.

Conceptually:

Clean Price = Market quotation

Dirty Price = Actual economic settlement price including accrued coupon

Why Dirty Price Is Higher Than Clean Price

Between normal coupon dates, accrued interest is positive.

Therefore:

Dirty Price > Clean Price

Immediately after the relevant coupon accrual resets, accrued interest falls substantially, bringing dirty price back close to clean price.

As the next coupon date approaches, accrued interest normally builds again.

This produces a useful pattern:

Coupon Date → Accrued Interest resets → Accrued Interest builds → Next Coupon Date

The bond's clean price may not change much during this process, but dirty price can rise simply because more coupon interest has accrued.

Clean Price vs Dirty Price

A numerical question may use the words price, quoted price, consideration, or settlement value differently.

Always identify what the amount represents before calculating.

Accrued-Interest Formula

For a conventional coupon-bearing Government security under the applicable 30/360 convention:

Accrued Interest = Face Value × Annual Coupon Rate × Broken-Period Days ÷ 360

This determines coupon interest accumulated for the relevant broken period.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

When working with percentages, convert the coupon rate correctly into decimal form before multiplication.

Indian Government-Security Day-Count Convention

Day-count convention tells us how the time period is measured for interest calculations.

RBI states that the Indian bond market uses the:

30/360 convention

for Government-security accrued-interest calculations.

Under 30/360:

• each month is treated according to a 30-day convention; and

• the year is treated as 360 days.

This should be distinguished from the Indian money-market convention:

Actual/365

which is used for instruments such as Treasury Bills and repo-interest calculations.

Broken Period

The period between the previous coupon date and the relevant settlement date is called the broken period for accrued-interest purposes.

RBI explains that accrued interest belongs to the seller for the period the seller held the security before settlement.

Therefore, a numerical question may provide:

• previous coupon date;

• settlement date; and

• coupon rate.

You must first determine the applicable broken-period days before calculating accrued interest.

Important Date Point

Do not automatically count every calendar date up to and including settlement.

Follow the market convention and the wording of the question carefully.

If the question directly provides the number of accrued days, use those days rather than independently creating a different number.

Accrued Interest per ₹100 Face Value

Government-security prices are commonly quoted per ₹100 face value.

For price-comparison purposes, accrued interest may therefore be calculated per ₹100 face value.

Accrued Interest per ₹100 = 100 × Coupon Rate × Broken-Period Days ÷ 360

This gives the accrued coupon amount corresponding to ₹100 face value.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Once this is known:

Dirty Price per ₹100 = Clean Price + Accrued Interest per ₹100

This converts the quoted clean price into the full price per ₹100 face value.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Price Amount for the Entire Transaction

If the security quotation is given per ₹100 face value, first convert that quotation into the price amount for the total face value traded.

Clean Price Amount = Face Value Traded × Clean Price ÷ 100

This converts the per-₹100 market quotation into the clean value of the entire transaction.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

This step is important when the question gives, for example:

• a large face-value amount; and

• a quoted security price per ₹100.

Do not multiply the quoted price directly by the entire face value without allowing for the per ₹100 quotation basis.

Total Settlement Consideration

The buyer normally pays:

• clean price amount; plus

• accrued interest.

Therefore:

Settlement Consideration = Clean Price Amount + Accrued Interest

This gives the full amount payable for the bond transaction before any separate charges or adjustments specified in the question.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

RBI's Government Securities Primer describes the consideration amount in this manner: the buyer pays the agreed quoted price plus the accrued-interest amount.

Why the Seller Receives Accrued Interest

Consider a bond that pays coupon every six months.

The seller holds it for several months after the previous coupon payment and then sells it.

If there were no accrued-interest adjustment:

• the seller would receive no coupon for the holding period; while

• the buyer might receive the entire next coupon despite holding the security only for the remaining part of the coupon period.

Accrued interest solves this problem.

Economically:

Seller receives interest for seller's holding period

and

Buyer ultimately retains interest relating to buyer's holding period

subject to the market's coupon and settlement conventions.

Coupon Payment and Dirty Price

Accrued interest gradually builds as the coupon period progresses.

Immediately around a coupon payment, the accrued-interest component resets according to the applicable market convention.

Therefore, the dirty price can show a noticeable adjustment around coupon dates even when the clean market price does not move by the same amount.

This is why bond charts and quotations often focus on clean price: it removes the predictable mechanical accumulation of coupon interest.

Face Value Is Not Market Price

This is one of the most important numerical distinctions.

Suppose a security has:

• face value of ₹100; and

• quoted clean price of ₹102.

The investor does not pay only the ₹100 face value.

Nor does ₹102 automatically represent the entire settlement amount.

Instead:

• ₹100 represents contractual principal;

• ₹102 represents clean market value per ₹100 face value; and

• accrued interest must still be considered to obtain dirty/full price.

Therefore, keep these concepts separate:

Face Value ≠ Clean Price ≠ Dirty Price

Clean Price and Bond Yield

Bond yield calculations and settlement calculations are related but not identical.

A clean price is used as a standard market quotation because it makes comparisons between securities easier.

Accrued interest does not represent a change in the bond's underlying required market yield. It is largely the accumulation of earned coupon between payment dates.

Therefore:

clean-price movement is more informative about changes in:

• market yields;

• interest-rate expectations;

• credit spread, where applicable; and

• security demand.

Dirty price additionally contains the accrued-interest component.

Accrued Interest Is Not Capital Gain

Another exam trap is treating accrued interest as a price appreciation.

Suppose dirty price increases between coupon dates while clean price remains unchanged.

The increase may simply arise because additional coupon interest has accrued.

This does not necessarily mean the bond has generated a market-price capital gain.

A capital gain or loss relates to movement in the underlying market price, whereas accrued interest represents earned coupon.

Zero-Coupon Securities

A zero-coupon security does not make periodic coupon payments.

Therefore, there is no coupon accrued interest of the same type as in a conventional coupon-bearing bond.

For example, a Treasury Bill:

• is issued/traded on a discount basis; and

• has no periodic coupon accrual to add to clean price in this manner.

Do not automatically apply coupon accrued-interest formulas to zero-coupon instruments.

Practical Calculation Sequence

For CAIIB numerical questions, follow this order.

Step 1: Identify Face Value

Determine the total nominal value of securities traded.

Step 2: Identify Clean Price

Check whether the price is quoted per ₹100 face value.

Step 3: Determine Annual Coupon

Use face value and coupon rate.

Step 4: Identify Previous Coupon Date and Settlement Date

This determines the broken period.

Step 5: Apply the Correct Day-Count Convention

For Indian Government-security accrued interest, use the RBI-stated 30/360 convention unless the question provides a specific different basis.

Step 6: Calculate Accrued Interest

Calculate the seller's earned coupon for the broken period.

Step 7: Calculate Clean Price Amount

Convert the quoted price per ₹100 into the value applicable to the total face value traded.

Step 8: Add Accrued Interest

This gives the dirty/full settlement consideration.

Numerical Decision Path

Before starting the arithmetic, identify exactly what the question asks.

Asked for Accrued Interest?

You need:

• face value;

• coupon rate;

• broken-period days.

Asked for Dirty Price?

You need:

• clean price; and

• accrued interest per ₹100.

Asked for Total Consideration?

You need:

• total clean-price amount; plus

• total accrued interest.

Dirty Price Given and Clean Price Required?

Reverse the relationship:

Clean Price = Dirty Price − Accrued Interest

This removes the accrued coupon component from the full price.

Refer to the Formula Section for a detailed explanation, variables, and solved examples.

Reasonableness Checks

A quick logical check can prevent calculation errors.

Between Normal Coupon Dates

Expect:

Dirty Price > Clean Price

because accrued interest is normally positive.

Immediately After Coupon Reset

Accrued interest should be low or reset according to the applicable convention.

As Next Coupon Date Approaches

Accrued interest should generally become larger.

Larger Face Value

Total accrued interest should increase proportionately.

Higher Coupon Rate

For the same face value and broken period:

Accrued Interest ↑

Longer Broken Period

For the same bond:

Accrued Interest ↑

If your answer violates these relationships, review the inputs.

CAIIB Exam Focus

Clean price excludes accrued interest.

Dirty price includes accrued interest.

Dirty Price = Clean Price + Accrued Interest.

Accrued interest belongs economically to the seller for the relevant holding period before settlement.

Indian Government-security accrued interest uses the 30/360 convention stated by RBI.

Money-market Actual/365 should not be automatically used for G-Sec coupon accrued interest.

Government-security quotations are commonly expressed per ₹100 face value.

Therefore, convert the quoted price correctly when total face value is given.

Face value, clean price and dirty price are different concepts.

Dirty-price movement does not automatically represent capital gain.

Part of the movement may simply be accrued coupon interest.

Zero-coupon instruments do not have conventional coupon accrued interest.

Common Exam Traps

Trap 1: Clean Price Already Includes Accrued Interest

Incorrect.

Clean price excludes accrued interest.

Trap 2: Dirty Price Excludes Accrued Interest

Incorrect.

Dirty price is the full price including accrued interest.

Trap 3: Clean Price and Face Value Are the Same

Incorrect.

Face value is the contractual principal. Clean price is the market quotation.

Trap 4: A Quote of ₹102 Means ₹102 for the Entire Transaction

Incorrect if the quotation is per ₹100 face value.

The quote must be applied to the total face value traded.

Trap 5: Use Actual/365 for Government-Bond Accrued Interest

Incorrect under the RBI-stated Indian G-Sec convention.

Use 30/360 for accrued coupon interest.

Trap 6: Use 30/360 for Every Treasury Calculation

Incorrect.

Money-market calculations such as T-Bill and repo interest use different conventions.

Trap 7: Accrued Interest Is Calculated on Market Price

Incorrect.

Coupon accrual is fundamentally based on the security's face value and coupon terms.

Trap 8: Accrued Interest Is a Capital Gain

Incorrect.

It represents coupon earned during the broken period.

Trap 9: Dirty Price Can Never Change Unless Market Yield Changes

Incorrect.

Dirty price changes as accrued interest builds even if clean price remains unchanged.

Trap 10: Seller Receives No Coupon Benefit Unless the Bond Is Held Until Coupon Date

Incorrect.

Accrued-interest settlement compensates the seller for the coupon earned during the relevant holding period.

Trap 11: A Zero-Coupon T-Bill Requires Coupon Accrued-Interest Calculation

Incorrect.

A T-Bill does not pay a periodic coupon.

Trap 12: Always Add Accrued Interest Even When the Question Already Gives Dirty Price

Incorrect.

If dirty/full price is already given, accrued interest has already been incorporated unless the question states otherwise.

Trap 13: Use Original Issue Price to Calculate Settlement

Incorrect.

A secondary-market bond transaction uses the applicable current market quotation and settlement terms.

Trap 14: Settlement Consideration Is Always Equal to Face Value

Incorrect.

The settlement amount reflects market price and accrued interest, not merely the amount repayable at maturity.

Memory Technique

Remember:

C + A = D

• C – Clean Price

• A – Accrued Interest

• D – Dirty Price

For Government securities:

30/360 = Bond coupon accrual

For money-market instruments:

Actual/365 = Money

And remember the transaction:

Quote Clean → Add Earned Interest → Pay Dirty