Topic 2 Price Yield Conventions and Repo Markets
Nature of cash flows in Debt Markets • Coupons – paid periodically (annual, semi-annual, quarterly, etc.) • Balloon payment – paid at maturity. • Other cash flows – payments when the debt is called, converted into equity, or when the issuer defaults and settles through a bankruptcy procedure. • We will take a look at cash flows when there is no credit risk, and assume that fixed coupons and a terminal balloon payments are promised by the issuer. Treasury debt securities fall in this category.
Bond Basics • Two basic yield measures for a bond are its coupon rate and its current yield.
Straight Bond Prices and Yield to Maturity • The price of a bond is found by adding together the present value of the bond’s coupon payments and the present value of the bond’s face value. • The Yield to maturity (YTM) of a bond is the discount rate that equates the today’s bond price with the present value of the future cash flows of the bond. Is also the IRR of a bond’s cash flow….
Price as present value of future cash flows With annual compounding, the price P of a bond that pays annual dollar coupons of C for N years, per $100 of face value, is PV of balloon payment PV of an annuity
Price as present value of future cash flows With semi-annual compounding, the price P of a bond that pays semi annual dollar coupons of C/2 for N periods, per $100 of face value, is PV of balloon payment PV of an annuity
Price as present value of future cash flows In reality, bonds are bought and sold on dates that are not necessarily their coupon dates. This leads to partial first coupons – the buyer will have to compensate the seller for the partial coupon earned. We need to modify the formulas in previous slides to reflect this practice. Let z = number of days from settlement date to next coupon date. x = number of days between the last coupon date and the next coupon date. N = number of remaining coupons.
Market conventions and quotes Treasury debt with coupons are called T-notes (with 2 to 10 yrs at-issue maturity) and T-bonds (> 10yrs at-issue maturity). Treasury debt with no coupons with an at-issue maturity of 1 year or less are known as Treasury bills. A plus sign (+) means 1/64th. ++ means 1/128th Prices are quoted in 1/32nd
Market conventions and quotes Treasury bills are quoted on a discount yield basis. The procedure for obtaining the invoice price from discount quotes is illustrated in the next example, in which quotations for settlement on June 26, 2008, for three- and six-month U.S. Treasury bills are shown in Table 2.4 .
Market conventions and quotes The invoice price, P , using the discount yield is then calculated as follows: where the T-bill has n days remaining from the settlement date to maturity and has a discount yield d per $100 face amount. For the three-month T-bill, the price is computed as follows:
T-bills with a maturity of less than 182 days Discount yield of a T-bill: • where the T-bill has n days remaining from the settlement date to maturity date and has a discount yield d per $100 face amount. • Gain is divided by 100 rather than the price, which is • the initial investment. • b. Annualization is done by using 360 days per year.
BEY - T-bills with a maturity of less than 182 days BEY of a T-bill: where the T-bill has n days remaining from the settlement date to maturity and has a discount yield d per $100 face amount. T-bills are quoted in discount yields. BEY is a better measure of returns.
BEY - T-bills with a maturity of more than 182 days EXCEL automatically takes the maturity into account in computing BEY of T-bills.
Coupon paying Treasury Securities – Notes and Bonds Accrued interest: Number of days of coupon that has accrued but not paid. Clean price: Price excluding accrued interest. Dirty price: Clean price plus accrued interest.
Coupon paying Treasury Securities – Notes and Bonds Last Coupon Date (LCD or PCD) Settlement Date (SD) Next Coupon Date (NCD)
Price-Yield relationship is generally convex Price Yield
Price-Yield relationship is for callable debt can exhibit negative convexity Par Price Call option is unlikely To be exercised as yields are high. The bond behaves like a non-callable security. Bond can be called at par: this “compresses” the price to par as yields fall. Yield
Examples of Debt Securities with Negative Convexity • Many debt securities issued by Fannie and Freddie are callable (Why?). They exhibit negative convexity. • Mortgage-Backed Securities exhibit negative convexity. (Why?) • Spreads between MBS and “otherwise identical” non-callable debt securities will widen as yields decrease – explain why this is the case.
Duration • Bondholders know that the price of their bonds change when interest rates change. But, • How big is this change? Point of tangency…. • How is this change in price estimated? 1st derivative of the pricing function wrt to yield (modified) • Macaulay Duration, or Duration, is the name of concept that helps bondholders measure the sensitivity of a bond price to changes in bond yields. That is: • Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.
Example: Using Duration • Example: Suppose a bond has a Macaulay Duration of 11 years, and a current yield to maturity of 8%. • If the yield to maturity increases to 8.50%, what is the resulting percentage change in the price of the bond?
Calculating Macaulay’s Duration • Macaulay’s duration values are stated in years, and are often described as a bond’s effective maturity. • For a zero-coupon bond, duration = maturity. • For a coupon bond, duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.
Duration Properties • All else the same, the longer a bond’s maturity, the longer is its duration. • All else the same, a bond’s duration increases at a decreasing rate as maturity lengthens. • All else the same, the higher a bond’s coupon, the shorter is its duration. • All else the same, a higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.
Topic 2 - Conclusions/Main insights • Debt securities are often quoted in terms of yields. • There is a direct relationship between prices and yields. • Market conventions differ from one segment (Treasury, for example) to another (Corporate, for example) within the same country. • EXCEL has useful functions to take out the tedious calculations – you should familiarize yourself with these functions. • Quoted prices exclude accrued interest, which must be added to the quoted price to get the actual price that must be paid to complete the transaction.