# Price

The price is determined by discounting the future cash flows to the present value using the appropriate discount rate.

The price of a bond is calculated as the present value of all the future cash flows (coupon payments and principal)

In this case the present value of the future coupon payments plus the present value of the principal at maturity.

You can think of the price as the intrinsic value because it is equivalent to the payoff that you receive.

When you calculate the price of a bond you are calculating the **maximum price you would want to pay**.

### How to calculate the Price ?

The price of a bond is best thought of as 'our valuation' of the bond (not the markets valuation).

If your calculated price is higher than the markets then you should buy the bond

If your calculated price is lower than the markets then you should sell the bond.

Yield is quoted as annual interest rate

You must alter depending on compounding before plugging into PV/Bond price equation

Must alter period numbers the same way

### Different Approaches to Pricing a Bond

There are several different formulas you can use to calculate the price of a bond.

Some are obviously more accurate than others.

1) Using Yield to Maturity

2) Relative Price - Using Yield to Maturity of a benchmark. This approach prices the bond relative to a benchmark, usually government bonds

3) Arbitrage Free - Using Spot Yields. Here each cash flow is priced separately and is discounted at the rate corresponding to zero coupon government bonds.

4) Arbitrage Free - Using Spot Yields - Interest Rate Swaps market

### Different Benchmarks

On the Run Treasuries (bills, notes and bonds)

Some bonds are priced relative to specific treasury bonds - for example the on-the-run 10 year treasury might be used to price a 10 year corporate bond)

Bonds that have a maturity date that does not coincide exactly with any treasuries are priced relative to a benchmark pricing curve

Constructed using several different benchmark interest rates or other securities with maturities from 3 months to 30 years where are gaps the curve is interpolated

Other popular benchmark curves

spot rate treasury curve

swap curve (LIBOR interest rate swap curve)

eurodollar curve

agency curve

### Bond Prices

The price of a bond is always expressed as a percentage of the principal value

Bond prices are always quoted as a percentage of their par value.

Bond prices move inversely to interest rate changes.

Long dated bonds are highly sensitive to interest rate changes (interest rates up, bond price down, interest rates down, bond prices up), whereas short term bonds are only marginally sensitive.

If interest rates rise the bond price falls since they are receiving less than they could get elsewhere.

For investors that are going to hold their bonds until maturity, the changes in interest rates are less important.

There is an inverse relationship between bond prices and required returns (or discount rates).

### Pricing a Bond between payment Periods

Of course you will rarely be buying and selling bonds on their coupon payment dates so we need to be able to price a bond on any day including those days between payment periods.

If the bond price is higher than its par value then it will sell at a premium because its interest rate is higher than the interest rate elsewhere.

If the bond price is lower than its par value then it will sell at a discount because its interest rate is lower than the interest rate elsewhere.

Bond Pricing

1) Based on discounted cash flows - we are taking the future stream of cashflows (coupons and redemption) and calculating the yield that would equate this to todays bond price

2) You could use a desired yield to calculate a price.

Example - suppose we have a bond issued in 2000 which matures in 25 years (2025)

The bond pays an interest of 10% a year

What is the yield ? The answer is 10% on the assumption that you paid the full price for the bond !!!!

The price of a bond is expressed as a percentage of the par value

If the price is 90 then the price for £100 par value bond is £90

Example - Suppose you buy a 25 year UK government bond at issue for £5,000.

2 years later you need the money and decide to sell.

The interest rate has obviously changed since you bought it and the yield on the bond is now 12.5%

As you try and get full price for your 10% bond no one is interested.

Why would they pay you £100 for £10 when they could buy a new issue for £100 and get £12.50

They might pay you £80 since 12% of £80 is £10

The face rate of interest is 10% but by buying it at a discount, the yield has increased to 12.5%

Lets say that for some reason you didn't sell it and 2 years later the interest rate is 8%

Therefore if you try and sell it for £100 you wont have any problem selling it as it is a much better investment than anything else as the yield is 10%

Buyers would actually be prepared to buy this bond for £125 since 8% of £125 is 10%

Lets suppose you sell this bond on 20th April and that the price is the saem (£90)

When you sell your bond you will get the accrued income to date. As you held the bond for 3 months

The calculation is simply multiplying The yearly interest payments by The number of days The bond is held as a proportion of The days in The year.

The accrued interest is almost always quoted separately and the price without the accrued interest is the clean price.

The price that is quoted is always the clean price

When you buy the bond you will have to pay the accrued interest.

There must be a cut off, at which time everyone on the register gets the interest. After this the bond is marked as __XD (ex dividend)__

For example if the payment is on the 30 June, the bond may go XD on the 10th May, so anyone buying the bond on the 7th will not accrue any interest until

- interest accrued from 30th Jan - 30th Jun

- anyone selling on 10th May will get all the interest

- anyone buying on 10th May will not start accruing any interest until 30th June

As a consequence when bonds are sold in the XD period - the seller (who gets all the interest) will pay accrued to the buyer.

### Important

At the time of issuance the bonds coupon rate is set at approximately the prevailing yield in the market. The price is approximately equal to par.

When the required yield is equal to the coupon rate, the price of the bond is equal to par.

When the price is less than par, the coupon rate is less than the required yield. (ie the required yield it greater than the coupon rate)

When the price is greater than par, the coupon rate is more than the required yield. (ie the required yield is less than the coupon rate).

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