# IDENTITY BETWEEN EV AND "ALL-IN" CASH FLOWS (CAPITAL AND INTEREST)

The reason why EV and the present value of Nils remain equal is that the EV discounts all-in flows, interest and capital. The relationship between the two calculations stems from the relation between all-in cash flows and interest flows.

## Separating All-in Flows into Capital and Interest Flows

By definition, the EV discounts the all-in cash flows, capital plus interest. We can split the two types of cash flows by sub-period. Consider t - 1 to t:

all-in (capital + interest) flows (t- 1, t) — capital flows (t- 1, t) + interest flows (t—,i)

Net capital flows over periods are amortizations of assets and liabilities and equal to the variations of the liquidity gaps. Net interest flows over the same periods are the NIL The EV is the present value of all-in cash flows:

EV — PV (future interest margins) + PV (A liquidity gaps)

Present values are calculated at market rates corresponding to each maturity of each cash flow generated by assets and liabilities, whether interest or capital. From the previous example of a simplified balance sheet with equity, we see that the liquidity gap is equal to equity at date 3. Therefore, the EV should be the present value of such liquidity gap -100 (with the sign convention asset - liability) as of 3 plus that of Nils for all three periods. It is easy to see that the identity works for any horizon and any liquidity gap profile. A final example illustrates the general above property of economic value and NIL

## Economie Value and NiL: General Example

Taking the same balance sheet as before but allowing assets and liabilities to amortize progressively, we show that the identity applies to all horizons. We stick to the same values of assets and liabilities, the same spreads, and the same original risk-free rate, but we remove the assumption of full amortization at horizon 3. Instead assets and liabilities amortize progressively and there is a liquidity gap at the cut-off horizon 3.

The rates and spreads are identical to the previous example (Table 26.8). For generality, we now assume that there is an increase of the risk-free rate from 5% to 6% after the first period. Assets are fixed rate as well as debts. The equity is 100. The asset interest rate is fixed for all three periods, and is 8%, 3% above the initial risk-free rate of 5%. The debt interest rate is fixed at 6% for all non-amortized balances. Subsequently, the risk-free rate increases to 6%. All discount factors use the new risk-free rate of 6%.

Consider the projected balance sheet with amortizing assets and liabilities. Note that the capital flows measures the variations of the liquidity gaps and are marginal liquidity gaps. With sign conventions, the amortization of assets is an inflow and the amortization of liabilities is an outflow. The capital flows are the net values of such inflows and outflows. They are identical to the variations of liquidity gaps (Table 26.9). The interest revenues and costs of period t are calculated from the outstanding balances of assets and liabilities as of t - 1 using the fixed rates applicable to assets and debt (Table 26.10).

TABLE 26.8 Interest rate data TABLE 26.9 Balance sheet amortization up to a fixed horizon TABLE 26.10 Interest revenues and costs TABLE 26.1 I Discount factors

 Dotes 0 1 2 3 Discount factors at 6% 0.943 0.89 0.84

TABLE 26.12 Present values of all-in flows, capital and interest flows From capital flows and N11 flows, we derive the present value of each. The economic value is the sum of the present values of capital flows and of Nils. The discount factors use the final risk-free rate of 6% (Table 26.11). Finally, the present value of all-in flows provides the economic value and is identical to the present value of all Nils and capital flows up to the cut-off horizon (Table 26.12).

The present value of Nils is the economic value minus the present value of capital flows over any horizon:

PV (N11) economic value - PV (capital flows)

PV (Nils) = 64.4 - (-5) = 69.4

These identities hold for all horizons. When extending the horizon to the longest horizon, we see that the present value of all Nils over sub-periods is simply the economic value minus the final liquidity gap, which becomes then equal to equity in absolute value.

Economic value is a target variable of ALM because it summarizes in a single value the present value of the entire stream of net interest income up to any cut-off horizon. This corrects a major drawback of gaps, which is that they are periodical. Gaps are fine for hedging the N11 of particular periods, if there are no options in the balance sheet, but the economic value it is better to measure is the entire stream of Nils up to a cut-off horizon. Managing the risk of EV should be conducted simultaneously with gap management because it targets a longer horizon. The next chapter explains how EV risk can be hedged or controlled.