Yield Measures, Spot Rates, and Forward Rates参考.ppt

Deriving a 6-Month Forward Rate To compute a 6-month forward rate, it is necessary to utilize a yield curve and the corresponding spot rate curve. The following 2 investments should have the same value: 1-year Treasury bill and 2 six-month Treasury bills (one purchased now and the other in six months) An investor should be indifferent since they should produce the same investment income over the same investment horizon. Deriving a 6-Month Forward Rate Although an investor does not know the interest rate of the second 6-month T-bill, it is possible to compute it because the “forward” rate must such that it equalizes the dollar return between the two alternatives. Exhibit 11 shows the timeline for the two investment alternatives: The value of first six-month T-bill is: X(1 + z1) The value of the total investment following the second six-month T-bill is: X(1 + z1)(1 + f) Where z1 is one-half the bond-equivalent yield of the 6-month spot rate and f is one-half the forward rate on a 6-month Treasury bill available 6 months from now. X is the amount of the investment. Deriving a 6-Month Forward Rate Relationship Between Spot Rates and Short-Term Forward Rates The value of alternative investment (a 1-year T-bill) is computed as: X(1 + z2)2 Because the two alternatives should generate identical returns: X(1 + z1)(1 + f) = X(1 + z2)2 Solving for f = [(1 + z2)2 / (1 + z1)] -1 Multiplying f by 2 to get the forward rate on a bond-equivalent yield basis. Forward rates can be computed on various combinations of short- and longer-term interest rates. Exhibit 12 provides the six-month forward rates for the entire yield curve. Exhibit 13 is a graph of the forward rate curve. 3. Solve for the 1-year spot rate. $3.0/(1.025)1 + $103.0/(1+z2/2)2 = $100 where z2 is the annualized 1-year spot rate. Solve for z2/2 as: $103.0/(1+z2/2)2 = $100 - $3/1.025 = $100 - $2.927 = $97.073 or: $103.0/$97.073 = (1+z2/2)2 So: sq. root of ($103.0/$97.073) -1 = z2/