# Arbitrary Cash-Flow Model

 

Pricing Arbitrary Cash-Flow

 

An Arbitrary Cash-Flow (ACF) security interface values future known cash-flows. These cash-flows must be in a single (potentially foreign) currency. The present value of these cash-flows is determined by prevailing market interest and foreign exchange rates.

 

Suppose ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png) denote known future cash-flows, which occur at the known times ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) If ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) denotes the risk-free discount factor to time *T* for the cash-flow currency, then GET applies the following valuation formula to obtain the present value (PV) of the above cash-flows

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Here, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) is the spot market foreign exchange (FX) rate from the cash-flow currency into the domestic currency.

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If ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png) denote Imagine interest rate maturity dates for the simply compounded rates ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png) then our benchmark assumed piece-wise constant continuously compounded instantaneous forward rates *f(t)* given by. In addition, the following is satisfied for all ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png)

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Moreover, for ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png) we have

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We note that, in the above, we equivalently apply log-linear interpolation of discount factors.

In the case, when we considered an Imagine bond par yield curve input, we performed the following calculations:

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Suppose ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image024.png) denoted annually compounded par yields, with respect to maturities ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png) for bonds ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png). Thus, bond ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) has an annual coupon equal to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png)

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We performed a bootstrap to obtain the discount factors ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png) In particular,

we applied the following recursive formulas

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The equations above are derived from the following observation; given a notional of 100

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the price of the par yield bond ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) is 100. Thus,

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Finally, we assume as above that continuously compounded instantaneous forward rates  *f(t)* satisfy

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that is, log-linear interpolation of discount factors.

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The implementation requires the following input parameters:

·         Future cash-flow amounts,

·         Future cash-flow times,

·         Cash-flow currency,

·         Domestic currency,

·         Spot FX rate from the cash-flow to the domestic currency,

·         Imagine interest rate curve for the cash-flow currency.

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For a rate curve, the user must specify the following:

·         a series of interest rates with start and end dates,

·         interest rate compounding,

·         day count convention.

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For a bond yield curve (ref <https://finpricing.com/lib/IrCurveIntroduction.html>), the user must specify the following:

·         a series of bond yields with maturity dates,

·         bond yield compounding,

·         day count convention,

·         coupon frequency,

·         coupon amounts (as percentage of notional amount).

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For a discount curve, the user must specify a series of discount factors and discount dates.

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### We test the model with respect to the following representative interest rate curve inputs:

·         Flat continuously compounded interest rate curve,

·         Simply compounded spot yield curve,

·         Discount factor curve,

·         Annual coupon par yield curve.

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