**Day Counts**

There are different kinds of day counts being used for financial calculations

These can be

- Actual/Actual (used for US treasury bonds)
- 30/360 (used for corporate and municipal bonds in US)
- Actual/360 (used for money market instruments in US)

The 360 can sometimes be 365 based on the conventions used.

**US Treasury Bonds**

**The quoted price:** the price quoted for the bond. This is not the same as paid by the purchaser.

**The Cash Price: **This is the price the purchaser paid.

Cash Price = Quoted price + interest accrued since last coupon date

**Example:**

It is March 24, 2017, and the bond under consideration is a 13% coupon bond maturing on July 5, 2022 with a quoted price of $94.6. Find the cash price of the bond?

Coupon payments are done every 6 months.

Last coupon payment date = 5^{th} Jan, 2017

Next coupon payment date = 5^{th} July, 2017

Days since last coupon = 78

Number of days between last coupon payment date and next coupon payment date = 181

Interest accrued till March 24^{th} = (55/181) × 6.5 = $2.80

The cash price = 94.6 + 2.8 = $97.4

**Treasury Bond Futures**

- Prices are quoted as fractional value of 32.

Ex: 94 – 05 =

- Delivery can be done through any government bond
- One contract is for $100,000

**Conversion Factor**Since delivery is through a different bond, there is a conversion factor for this futures priceThe quoted price is = (most recent settlement price × conversion factor)The cash price is = (most recent settlement price × conversion factor) + accrued interestAs rules for conversion factor,

- the interest rate for all maturities are assumed to be 6% (might be different in specific cases)
- The bond maturity and the times to the coupon payment dates are rounded down to the nearest 3 months for the purpose of calculation
- If after rounding, the bond lasts for an exact number of 6 month periods, the first coupon is assumed to be paid in 6 months.
- If after rounding the bond does not lasts for an exact number of 6 month periods, the first coupon is assumed to be paid after 3 months and accrued interest is subtracted.

**Example:**12% coupon bond with 10 years and 1 months to maturity.

As per the rules stated above, the bond is assumed to have 10 years to maturity

Coupon payments are made every 6 months; in all there are 20 payments

Discount rate is 6%

Value of the bond (for delivery) is =

= $144.63

Conversion factor = 144.63/100 = 1.4463

**Example:**10% coupon bond with 20 years and 5 months to maturity.

As per the rules stated above, the bond is assumed to have 20 years 3 months to maturity

Coupon payments are made every 6 months; in all there are 40 payments

Discount rate is 6%

Value of the bond (for delivery) is

= $151.23

Interest rate for 3-month period is (1.03)

^{0.5}-1 = 0.014889 or 1.4889%Discounting back to present = 151.23/1.014889 = 149.01

Accrued interest = 2.5

Hence value of bond = 146.51

Conversion factor = 1.4651

**Cheapest to deliver bond**The party with the short position can chose a bond that is cheapest to deliver.

This party receives

(most recent settlement price × conversion factor) + accrued interest

The cost of purchasing is

Cash price = Quoted price + accrued interest

Cheapest to deliver is one for which

[Quoted Price ─ (most recent settlement price × conversion factor)] is least

**Example 6.1 from ‘Options, Futures and other Derivatives’ by John C. Hull**The party with the short position has decided to deliver and is trying to choose between the three bonds mentioned below. The most recent settlement price is 93-08

Bond Quoted Bond Price Conversion Factor 1 99.50 1.0382 2 143.50 1.5188 3 119.75 1.2615 Most recent settlement price is 93.25

The cost of delivering the bonds are as shown below

Bond 1: 99.50 ─ (93.25 × 1.0382) = 2.69

Bond 2: 143.50 ─ (93.25 × 1.5188) = 1.87

Bond 3: 119.75 ─ (93.25 × 1.2615) = 2.12

Hence, the cheapest to deliver is Bond 2.

Factors determining the cheapest to deliver bond:

- Bond yield > 6%; the conversion factor tends to favour the delivery of low-coupon long maturity bonds.
- Bond yield < 6%; the conversion factor tends to favour the delivery of high-coupon short maturity bonds.
- Yield curve upward sloping; long maturity bonds are favoured.
- Yield curve downward sloping; short maturity bonds are favoured.

**Determining the Futures Price**The futures price is calculated in a manner similar to the forward price

**F**_{0}= (S_{0}– I)*e*^{rT}Where

F

_{0}= futures priceS

_{0}= spot priceI = Present value of the coupons that will be paid during the life of the futures contract

T = time until the futures contract matures

r = risk free interest rate applicable for the period T

**Example 6.2 from ‘Options, Futures and other Derivatives’ by John C. Hull**Cheapest to deliver bond:

Coupon = 12%

Conversion factor = 1.4

Delivery time = 270 days

Coupons are payable semi-annually on the bond and last coupon payment was made 60 days ago, the next coupon date is 122 days later. The current quoted price is $120. The term structure is flat and the rate of interest is 10% p.a. (continuously compounded)

Find the quoted futures price?

Time chart for the bond can be used to show the calculations involved

- Interest accrued since last coupon date needs to be added to the current quoted price.=120 + 6× (60/182) = 121.978

- Coupon payment in 122 days needs to be subtracted and future value of this is calculated

Coupon value = 6

Coupon value today = 6e

^{ (–0.1 × 122/365)}= 5.803The present value = 121.978 – 5.803 = 116.175

Taking this value forward to 270 days = 116.175 e

^{(0.1 × 270/365)}= 125.095- Next coupon payment is in 183 days so interest accrued till 148 days is to be subtracted

Interest accrued for 148 days = 6 × (148/183) = 4.852

Futures price of the bond = 125.095 – 4.852 = 120.242

As the conversion factor means that the 1.4 standard bonds are considered equivalent to each 12% bond, the quoted futures price should be

120.42/1.4 = 85.887