## Inflation Adjusted Cash Flow

Given a cash flow amount

**at time**

*CF(t)***, the**

*t***is a cash flow**

*Inflation Adjusted Cash Flow***that occurs at**

*CF'(t)***and given by the formula:**

*t*

*CF'(t) = CF(t)Q(t)***is the index quotient ascribed to time**

*Q(t) = I(t-lag)/I(t₀)***that is defined as the raw inflation index**

*t***that applies at time**

*I(t-lag)***divided by the raw inflation index**

*t-lag***that applies at some fixed initial time**

*I(t₀)*

*t₀*This type of cash flows is used in the Deriscope Types Inflation Swap and Inflation Bond

Effectively,

**represents the raw inflation index at time**

*Q(t)***in units of the raw inflation index at time**

*t-lag*

*t₀*Note that the raw inflation index

**is ascribed to time**

*I(t)***but becomes known (published) with a certain delay**

*t***at a later time**

*delay*

*t+delay*In order to know the index value

**at time**

*Q(t)***, it must hold that**

*t*

*lag >= delay*In most cases, the amount

**is computed but never realized as a cash flow event, whereas the amount**

*CF(t)***is the actually occurred cash flow.**

*CF'(t)*The amount

**is referred as**

*CF'(t)***because it is the number recorded in the corresponding transaction.**

*nominal*The

**is referred as**

*CF(t)***because it expresses the**

*real***amount**

*nominal***in**

*CF(t)Q(t)***currency units**

*real***that are imaginary units defined as follows:**

*RU*At each time

**, the**

*t***currency unit**

*real***is assumed to be equivalent to**

*RU***nominal (i.e. regular) currency units**

*Q(t)*

*NU*So, by definition it holds:

**for every time**

*RU = Q(t)NU*

*t*Therefore the

**amount**

*nominal***paid in nominal currency units**

*CF(t)Q(t)***can be written as**

*NU*

*CF(t)RU*In words, all this translates to the following two equivalent statements:

At time

*t***units of the nominal currency**

*CF(t)Q(t)***are paid.**

*NU*or equivalently

At time

*t***units of the real currency**

*CF(t)***are paid.**

*RU*Just like one talks about different economies associated with different currencies (eg the USD or EUR economy), one may also talk about the

**and**

*real***economies associated with the**

*nominal***and**

*real***currencies.**

*nominal*Similarly, just like one may convert any price or rate from one currency to another, one may also convert a

**price or rate to its equivalent**

*nominal***one and vice versa.**

*real*The corresponding

**exchange rate is the ratio of the two currency units, i.e.**

*real*

*RU/NU = Q(t)NU/NU = Q(t)*where we used the relation

*RU = Q(t)NU*