### H. Key Terms

**Face value: ** Value is the value of a bond as stated on it. Interest is calculated on face value. Also called Par Value or Nominal Value.

**Accrued interest and purchase price:**

In practice investors are likely to purchase bonds between coupon payments so the next coupon is less that 6 months or 1 year away. Under these circumstances, how do we calculate the fair price of the bond?

To calculate the fair price of a bond bought between coupon payments we must go through the following three steps:

1. Calculate the number of days until the next coupon payment.

2. Determine the present value of the cash flows over a fractional period.

3. Calculate by how much the buyer must compensate the seller for the coupon interest he would have earned during the fractional period that the bond was held.

To calculate the fair price of a bond bought between coupon payments we must go through the following three steps:

1. Calculate the number of days until the next coupon payment.

2. Determine the present value of the cash flows over a fractional period.

3. Calculate by how much the buyer must compensate the seller for the coupon interest he would have earned during the fractional period that the bond was held.

**Number of days until the next coupon payment**

The number of days until the next coupon payment is not as straightforward a question as it would appear.

The answer will depend on the market conventions for the type of bond in question.

Here we will look at two conventions: “actual/actual” and “30/360”. The former method is used for US Treasury securities, the latter for Eurobonds.

Consider a US government bond whose last coupon payment was March 1st. US Treasuries pay a semi-annual coupon so the next coupon payment would be six months later on September 1st.

The bond is purchased with a settlement date of May 10. The actual number of days between May 10 and September 1st is calculated as follows:

The answer will depend on the market conventions for the type of bond in question.

Here we will look at two conventions: “actual/actual” and “30/360”. The former method is used for US Treasury securities, the latter for Eurobonds.

Consider a US government bond whose last coupon payment was March 1st. US Treasuries pay a semi-annual coupon so the next coupon payment would be six months later on September 1st.

The bond is purchased with a settlement date of May 10. The actual number of days between May 10 and September 1st is calculated as follows:

However, if the security was a Eurobond - because each month is assumed to have 30 days - the number of days until the next coupon payment is 111.

**Accrued interest and clean price**

The buyer must compensate the seller for the fraction of the next coupon payment the seller is due but will not receive.

This amount is called accrued interest.

While different markets may differ slightly in their exact methods of calculating accrued interest, all major bond markets follow the practice of quoting clean prices (as opposed to full or dirty prices, which include accrued interest).

It is important that you check the precise method used for calculating accrued interest in the particular market being used, but two typical systems are represented by US government bonds and Eurobonds. The US Treasury market uses an "actual/actual, semi-annual" method, i.e. the actual number of days divided by the actual number of days in the interest period, all on a six-monthly basis. For Eurobonds, the system is "30/360, annual", meaning 12 months of 30 days, divided by 360 (not 365)

This amount is called accrued interest.

While different markets may differ slightly in their exact methods of calculating accrued interest, all major bond markets follow the practice of quoting clean prices (as opposed to full or dirty prices, which include accrued interest).

It is important that you check the precise method used for calculating accrued interest in the particular market being used, but two typical systems are represented by US government bonds and Eurobonds. The US Treasury market uses an "actual/actual, semi-annual" method, i.e. the actual number of days divided by the actual number of days in the interest period, all on a six-monthly basis. For Eurobonds, the system is "30/360, annual", meaning 12 months of 30 days, divided by 360 (not 365)

**Duration:**

**Macaulay duration:**

Macaulay duration is a weighted average of a bond's life, taking into account:

• The size of each cash flow

• Its timing.

Duration is a measure of market risk: the higher the bond's duration, the greater is its risk.

• The size of each cash flow

• Its timing.

Duration is a measure of market risk: the higher the bond's duration, the greater is its risk.

**Example:**

Macaulay Duration calculation.

Security: 5% US Treasury note maturing 21 January 2005

Type: Semi-annual, actual/actual

Settlement date: 3 June 2003

Price: 95.48

Accrued: 1.84

Yield: 8.00%

Security: 5% US Treasury note maturing 21 January 2005

Type: Semi-annual, actual/actual

Settlement date: 3 June 2003

Price: 95.48

Accrued: 1.84

Yield: 8.00%

In the table below we proceed in three stages:

1. Calculate the present value of each cash flow (column 2)

2. Multiply each PV by its corresponding time period (column 3)

3. Divide the sum of column 3 by the sum of column 2 (the bond's dirty price).

1. Calculate the present value of each cash flow (column 2)

2. Multiply each PV by its corresponding time period (column 3)

3. Divide the sum of column 3 by the sum of column 2 (the bond's dirty price).

Macaulay duration = 303.309/97.32

= 3.12 coupon periods

= 1.56 years

= 3.12 coupon periods

= 1.56 years

**Modified Duration**

Macaulay duration allows us to rank fixed income securities in terms of their market risk, but it doesnot tell us how risky the securities are in terms of profit or loss.

However, Macaulay duration can be easily modified to quantify risk in those terms.

However, Macaulay duration can be easily modified to quantify risk in those terms.

**Example**

Modified duration calculation

Security: 5% US Treasury note maturing 21 January 2005

Type: Semi-annual, actual/actual

Settlement date: 3 June 2003

Price: 95.48

Accrued: 1.84

Yield: 8.00%

Macaulay duration: 1.56 years

Amount held: USD 1 million.

What is the potential loss on this investment if yields rose to 9%?

We calculated the Macaulay duration of this bond in section Macaulay Duration!

Modified duration = 1.56

(1 + 0.08/2)

= 1.50%

A rise in yield from 8% to 9% would result in a capital loss on this investment of approximately 1½%.

Risk in cash terms = 1.50 x (95.48 + 1.84) x 1,000,000

100 100

= USD 14,598

Security: 5% US Treasury note maturing 21 January 2005

Type: Semi-annual, actual/actual

Settlement date: 3 June 2003

Price: 95.48

Accrued: 1.84

Yield: 8.00%

Macaulay duration: 1.56 years

Amount held: USD 1 million.

What is the potential loss on this investment if yields rose to 9%?

We calculated the Macaulay duration of this bond in section Macaulay Duration!

Modified duration = 1.56

(1 + 0.08/2)

= 1.50%

A rise in yield from 8% to 9% would result in a capital loss on this investment of approximately 1½%.

Risk in cash terms = 1.50 x (95.48 + 1.84) x 1,000,000

100 100

= USD 14,598

**Bid/Ask Price:**

Traders quote the bond prices as bid and ask price in the market. As an investor you purchase the bond at the ask price and if you wish to sell the bond you sell it at the bid price.