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
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 = 5th Jan, 2017
Next coupon payment date = 5th 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 24th = (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 FactorSince 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 interest
As 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.
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 =
Conversion factor = 144.63/100 = 1.4463
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
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
F0 = (S0 – I) e rT
F0 = futures price
S0 = spot price
I = 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) = 5.803
The 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