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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:
Even though the yield-to-maturity for the remaining life of the bond is just 7%, and the yield-to-maturity bargained for when the bond was purchased was only 10%, the annualized return earned over the first 10 years is 16.25%. This can be found by evaluating (1+i) from the equation (1+i) 10 = (25.84/5.73), giving 0.1625.
Expression (3) which uses the bond's yield to maturity to calculate discount factors. The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield , not varying by term to payment. [10]
The yield to maturity (YTM) is the discount rate which returns the market price of a bond without embedded optionality; it is identical to (required return) in the above equation. YTM is thus the internal rate of return of an investment in the bond made at the observed price.
The forward rate is the future yield on a bond. It is calculated using the yield curve . For example, the yield on a three-month Treasury bill six months from now is a forward rate .
The British pound yield curve on February 9, 2005. This curve is unusual (inverted) in that long-term rates are lower than short-term ones. Yield curves are usually upward sloping asymptotically: the longer the maturity, the higher the yield, with diminishing marginal increases (that is, as one moves to the right, the curve flattens out).
A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (< % and terms =10–30 years), the monthly note rate is small compared to 1. r << 1 {\displaystyle r<<1} so that the ln ( 1 + r ) ≈ r {\displaystyle \ln(1+r)\approx r} which yields the simplification:
Par yield is based on the assumption that the security in question has a price equal to par value. [5] When the price is assumed to be par value ($100 in the equation below) and the coupon stream and maturity date are already known, the equation below can be solved for par yield.