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where RMi is the individual benchmark returns. Note that the fund and risk-free rates are the annual average and that the standard deviations are the annualized standard deviations. Exhibit 12.5 illustrates the calculation of M-squared for our sample fund, which equals 25.51%. We also calculate M-squared for the benchmark, which equals the benchmark return of 23.81%. Using M-squared we can say that the fund had a higher return than the benchmark, even on a risk-adjusted basis. We can compare the fund M-squared return to the benchmark return. If M-squared is higher than the benchmark, the fund has a positive risk-adjusted return. We also can compare and rank funds using the M-squared return as each of the fund returns has been adjusted to have the same volatility as the benchmark return. The relative ranking will be the same as a ranking of funds by the Sharpe ratio, but the results should be easier to interpret.

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data 1H nuclear magnetic resonance (NMR), infrared (IR), and mass spectrometry (MS) were identical to the data reported for the natural product ( )-calanolide A. The four-step synthesis of racemic calanolide A was accomplished with 16% overall yield. This four-step procedure to synthesize calanolide A (1) is not only shorter, but also the reagents are cheaper and commercially available. 8.4.1.2 Total Synthesis of Racemic Inophyllum ( )-Inophyllum B (3), which is isolated from the C. inophyllum in 1972, has a similar skeleton of ( )-calanolide A with a phenyl group at the C4 position instead of the n-propyl group. The synthetic procedure of ( )-inophyllum B was accomplished similar to that of ( )-calanolide A as shown in Scheme 8-4.46 The reaction of phloroglucinol with 4-phenyl-acetoacetate in the presence of hydrogen chloride afforded 5,7-dihydroxy4-phenyl coumarin using the Pechman reaction. Then acylation of the coumarin using tigloyl chloride in the presence of AlCl3 in PhNO2 and CS2 gave 8-position acylated coumarin (11) in 70% yield. This result was different from the case of

[RDM-LOM]

model for the evolution of the term structure of rates, but is not practical since all zero-coupon bonds may not trade and the initial transaction costs will be high. The second method to synthetically replicate a swap is to use a portfolio of forward contracts written on the spot rate of futures dates. This method is also independent of any term structure of rates. The third method is to use a dynamic portfolio consisting of a single zero-coupon bond (for a one-factor model) and the money market account. This method requires a speci cation of the term structure of rates and can be synthetically replicated using single-factor models such as those discussed in this chapter. At each node, the amount of shares held in the money market account and those in the zero-coupon bond are determined as well as the net cash ows and swap value. At the rst node, the swap has a zero value and zero cash ow. We value synthetic swaps in section 12.12 using the Heath-Jarrow-Morton (HJM) single-factor model. There are many other types of swaps, including currency swaps, xed-for oating dual currency swaps, commodity swaps, equity swaps, forward start (deferred) swaps, capped swaps, callable and putable swaps, constant maturity swaps, and index amortizing swaps. Index amortizing swaps are valued in section 14.7 using the HJM model.

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