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Denote the price of a Bermudan swaption by Ss,x,e that is characterized by three dates: the lockout date (the rst exercise date), Ts; the last exercise date, Tx; and the nal swap maturity date, Te. We assume that Ts < Tx < Te, and that all three dates coincide with the exercise schedule dates; that is, s, x, and e are all integers in {0, . . . , K + 1 }. Moreover, to be consistent with market-traded swaptions, we assume Tx = Te 1. Early exercise of the Bermudan swaption is restricted to dates in the discrete set {Ts, Ts+1, . . . , Tx}. If exercise takes places at, say, * = Ti, then the swaption holder receives at time Ti, Ss,x,e(Ti ) = Si,e(Ti ) where Si,e(Ti ) which is the value of the remaining European swaption at Ti , with an underlying notional amount of $1, given by e 1 S s ,e (Ts ) = E Q P(Ts , Tk +1 ) k max(Fk (Ts ) X, 0) k=s e 1 = E Q max (1 P(Ts , Te ) X k P(Ts , Tk +1 ), k=s

For example, the function fibonacci() in the module fib.py is called fib. fibonacci(). There can be only one module with a given name that can be loaded on the Python interpreter. Therefore, fib. fibonacci() cannot conflict with another name. If there is another module named fibnew.py containing the function fibonacci(), the function will be called fibnew. fibonacci(). Therefore, there is no chance of conflict between the names of attributes.

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Along with composite returns, the standards recommend that managers present appropriate composite risk statistics. The most commonly used measure is volatility, or the annualized standard deviation of the composite returns. Analytics such as the average P/E of an equity composite or the duration of a xed income composite also help disclose the historical risk associated with the composite. When quoting or using the composite standard deviation it is important to differentiate between the two types of standard deviation commonly reported in an AIMR-PPS presentation, Exhibit 20.11 illustrates the calculation of the two types: 1. The standard deviation used as a proxy for the investment risk inherent in the composite is a measure of the variability in the total composite returns over time. It is calculated by rst taking the weighted average composite return for each period. Then we take the standard deviation of the periodic returns. We calculate the standard deviation, as well as the other risk statistics covered in Part II, to provide an indication as to the absolute and relative risk of the composite strategy. 2. The standard deviation used as a measure of dispersion. This is a measure of the variability in the cross section of fund returns within a composite over a particular time period. The period can be either a single period or a cumulative return over multiple periods. This standard deviation is calculated by deriving the standard deviation of the individual constituent fund returns around the composite average return for the period.

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4. Click Add. 5. Repeat Steps 3 and 4 to create additional calculated items for Qtr-2 (=Apr+May+Jun), Qtr-3 (=Jul+Aug+Sep), and Qtr-4 (=Oct+Nov+Dec). 6. Click OK to close the dialog box.

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Figure 6-7

The Math object provides a standard library of mathematical constants and functions. The constants are simply the properties of the Math object. Similar to the Global object, you cannot create instances of the Math object because it is not an object type but is a predefined object. Table 6.6 lists the properties of the Math object. Table 6.6: Math Object Properties Method E LN2 LN10 LOG2E LOG10E PI SQRT2 Usage Euler s constant Natural logarithm of 2 Natural logarithm of 10 Base 2 logarithm of e Base 10 logarithm of e Constant value of Square root of 2

N incoming terrestrial channels are compressed into M outgoing satellite channels through DSI operation, where the ratio N/M is the DSI gain

B RF = 2 ( f + f m ) i.e. the "Carson's rule bandwidth", where f is the multichannel peak frequency deviation.

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r2 [10 mm ] 2.91 3.08 3.37 3.61 0.651 0.642 0.629 0.612

Reagents and Conditions: (a) TiCl4, Zn, DME, 0 C, 81%; (b) MsCl, Py then K2CO3, MeOH, 65%.

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