Lecture 8

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Lecture 8 Yield to maturity: Single discount rate that equates the bond price to the present value of all future cash flows of the bond. Bond yield: Bond's internal rate of return given the current price Example: Price YTM YTM Price YTM Comparative Statics As ytm increases, P 0 decreases As C increases, P 0 increases When the price is calculated an instant after a coupon payment: o If ytm < c, then P 0 > Par o If ytm > c, then P 0 < Par o If ytm = c, then P 0 = Par o Note: These relationships are not necessarily true if the price is calculated at t ime between coupon dates. Bond pricing in Australia The earlier bond pricing formula is a simplification of reality in three main aspects: 1. It assumed annual coupon a. But most bonds pay half-yearly coupon 2. It assumed that the first coupons will be paid exactly one year from the pricing date a. So it can be used only twice a year for most bonds 3. It assumed that bonds are paid of ('settled') on the same day that the transaction is done (the 'trade date') a. But for most Australian government bonds the settlement date is a few days after the trade date
We require a formula that will take account of these three features (and some others too). The formula we will develop is the one used by the RBA and the Australian Financial Markets Associations. Viewed from the payment date of the first coupon: 1. This is an annuity-due of n cash flows of \$C, plus Par 2. Or equivalently, it is an immediate CF of \$C plus an ordinary annuity of n - 1 cash flows of \$C, plus Par We will use the second of these descriptions Using the second description, given a yield of ytm per half-year, the value as at time 1 is: We now need to discount V 1 back to time 0 The length of this period is f/h of a half-year Therefore: P 0 = 1 ( 1 + ytm ) f h ×V 1 Putting both parts together, we get the RBA pricing formula for coupon bonds: P 0 = 1 ( 1 + ytm ) f h { c + c ytm [ 1 1 ( 1 + ytm ) n 1 ] + Par ( 1 + ytm ) n 1 Where: Seven further technicalities: 1. In trading and reporting, the yield will be quoted per year (not per half-year): a. The quoted yield (pa) will be 2 x ytm 2. The coupon rate (c) is also stated on an annual basis. Therefore: C = 1 2 ×c ×Par 3. For bonds with more than 6 months to run to maturity, the settlement date is 3 business days after the trade date. Non-business days are Saturday, Sundays an days that are public holidays in both Sydney and Melbourne 4. The price (per \$100 par value) is taken to 3 decimal places. 5. There is an "ex-interest" period beginning a week (7 days) before every coupon date. a. If a bond is bought during that week (that is, if the settlement date falls in that week), the buyer is not entitled to receive the next coupon b. So, delete the first C from the pricing equation 6. If the first coupon payment date falls on a non-business day (eg on A Saturday or Sunday), then: a. F is calculated as the number of days from the settlement date to the first business day after the next coupon date but b. H is calculated as the number of days in the half-year ending on the next coupon date. 7. When a bond goes ex-interest for the second-last time (ie 6 months plus 7 days before maturity), the CF consist of a single payment at maturity, which is equal to the par value plus one half-yearly coupon.
a. To maintain consistency with the procedures for Treasury notes and other short-term securities, simple interest is used b. Also for these bonds, the settlement date is: i. For trades completed before 12.00 noon, the same day; ii. For trades completed after 12:00 noon, the next business day Example of the 7 th Technicality Example of the Main Formula Suppose that Friday 7 May 2021 the market yield on the 3.25% April 2025 CW gvt bond is 0.45% pa. The maturity date is 21 April 2025 If the par value was \$10,000,000 what would the bond price be? In general, yields are more important than prices The price of a bond with \$1000 par value will be higher than the price of a bond with a \$100 par value, but this doesn't mean that the \$100 par value bond is a better investment In the previous example, if we had been given the price \$111.116, we could have backed out the yield as 0.45% pa and compared this to yields of other bonds
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