# EV and Projected Interest Income at Risk-free Rate

For calculating the economic value, we use the risk-free rate, 5%. Expanding the calculations, the formulas are:

PV(asset) = 80/(1 + 5%) + 80/(1 + 5%)2 + 1080/(1 + 5%)3 = 1053.5 PV(debt) = 60/(1 + 5%) + 60/(1 + 5%)2 + 1060/(1 + 5%)3 = 1027.5

TABLE 26.3 The asset and liability cash flows

 Dotes 0 1 2 3 Assets 1000 80 80 1080 Liabilities 1000 60 60 1060 Revenues _ 80 80 80 Costs — 60 60 60 Nil — 20 20 20

The economic value is the difference:

EV = PV(asset) - PV(debt) = 54.5

Because we used the risk-free rate, the economic value of the debt is above the par value since it costs more than the risk-free rate. The asset value is also above the par value for the same reason. Next, we calculate the present value of all future interest incomes at the prevailing market rate 5%:

20/(1 + 5%) + 10/(1 + 5%)2 +10/(1 + 5%)3 = 54.5

The two calculations provide exactly the same value, showing that EV from cash flows is exactly equal to the present value of interest incomes at the same rate:

EV — present value of N11 54.5

We can now change the discount rate, taking an extreme example with a sudden large jump of the risk-free interest rate up to 15% - we have a negative EV. The cash flows of assets do not change. The debt, when renewed at the new rate costs much more, 15% + 1% — 16%. Table 26.4 shows the cash flows and the N11, and the present values of both cash flows and N11 at the risk-free rate of 15% are shown in Table 26.5. We now find that both present values are equal to -95.7 and are negative. The present values of assets and of debt are now below par because we discount at a rate higher than the initial rate.

We can understand why the economic value is negative, and equal to -95.7. This is because the EV discounts highly negative Nils, which result from mismatch risk combined with a high increase of short-term rates.

TABLE 26.4 Using 15% as cost of debt for the bank TABLE 26.5 Present values of cash flows and of Nil at the risk-free rate

 Dotes 0 1 2 3 PV (assets) 840.2 69.6 60.5 710.1 PV (liabilities) 935.9 52.2 121.0 762.7 EV (balance sheet) -95.7 PV(NII) -95.7 17.4 -60.5 -52.6

Note that changes of EV result from the different variations of asset values and liability values when we change the discount rate. The magnitude of their changes is different because they do not have the same sensitivity to a change of interest rate. Should we need to control the risk of the economic values, we would need to adjust their sensitivities to interest rates. This is the purpose of the next chapter. The choice of discount rates equal to risk-free rates becomes clear at this stage. This choice allows valuing any spread between customers' rates or market rates, that would show up in banks' NIL