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Analytic Example: Given: 0.5-year spot rate, Z1 = 4%, and 1-year spot rate, Z2 = 4.3% (we can get these rates from T-Bills which are zero-coupon); and the par rate on a 1.5-year semi-annual coupon bond, R3 = 4.5%. We then use these rates to calculate the 1.5 year spot rate. We solve the 1.5 year spot rate, Z3, by the formula below:
The Z-spread of a bond is the number of basis points (bp, or 0.01%) that one needs to add to the Treasury yield curve (or technically to Treasury forward rates) so that the Net present value of the bond cash flows (using the adjusted yield curve) equals the market price of the bond (including accrued interest). The spread is calculated iteratively.
The standard broker valuation formula (incorporated in the Price function in Excel or any financial calculator, such as the HP10bII) confirms this; the main term calculates the actual (dirty price), which is the total cash exchanged, less a second term which represents the amount of accrued interest.
Here’s an example of how much a Series EE U.S. Savings bond purchased in October 1994 would be worth today. EE bonds are guaranteed to double in value after 20 years. Denomination
Duration is a linear measure of how the price of a bond changes in response to interest rate changes. It is approximately equal to the percentage change in price for a given change in yield, and may be thought of as the elasticity of the bond's price with respect to discount rates. For example, for small interest rate changes, the duration is ...
For example, for bond options [3] the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff.
If a trade is agreed, the price is calculated. In this example, SOFR +0.25% would be roughly equivalent to a price of 99.75. This can be calculated as par, minus the difference between the coupon and the price that was agreed (0.05%), multiplied by the maturity (5 year).
There is a time dimension to the analysis of bond values. A 10-year bond at purchase becomes a 9-year bond a year later, and the year after it becomes an 8-year bond, etc. Each year the bond moves incrementally closer to maturity, resulting in lower volatility and shorter duration and demanding a lower interest rate when the yield curve is rising.