# The Cost of the Mirror Debt

The cost of the "mirror" debt is its yield. For a fixed-rate loan, the mirror debt can be decomposed in several zero-coupon debts, with volumes and maturities matching the time profile of the amortizing debt. The yield can be expressed as a function of zero-coupon rates. In the above example, there are two layers of debt: 60 for 2 years, and 40 for 1 year. The relevant rates are the spot rates for these maturities. However, there should be a unique transfer price for a given loan. It is the average cost of funds of the two debts. Its exact definition is that of a yield, or the discount rate making the present value of the future flows generated by the two debts equal to the amount borrowed.

The future outflows are the capital repayments and interests. The interest flows cumulate those of the one-year and the two-year market rates. If the debts are zero-coupon, the interest payments are at maturity. The flows are 40(1 + r ) in one year and 60(1 + r2)2 in two years. The yield *y* of this composite funding is a discount rate *y* such that:

100 = 40(1 + rj)/(l *+y)* + 60(1 + r2)7(l + *yf*

The discount rate is somewhere between the two market rates. An approximate solution uses a linear approximation of the exact formula:

100 = 40(1 + *rx-y)* + 60(1 + 2r2 - *2y)*

*y = (40r1* + 60 x 2 x r2)/(40 + 2 x 60)

The rate *y* is the weighted average of the spot rates for one and two years, using weights combining the size of each debt and its maturity. With rates *r* and r2 equal to 8% and 9%, *r* - 8.75%. The rate is closer to 9%, because the two-year debt is the one whose amount and maturity are the highest.

In practice, transfer prices tables provide immediately the yield of the mirroring debt, which is a composite of market rates, given the time profile of loans and the current market rates.

# The Benefits from the Mirror Debt

Using the cost of funds of a debt that fully replicates the assets offers numerous economic benefits. Perfect matching implies that:

• the margin of the asset is immune to interest rate movements

• there is no need of conventions for assigning existing resources to usages of funds

• there is no transfer of income generated by collecting resources to the income of lending activities

• the calculation of a transfer prices is mechanical and easy.

Using the yield of this notional funding mimicking the loan as the transfer price complies with all consistency requirements.

The replicating debt is defined for a loan which has a maturity schedule. However, contractual maturities are not effective maturities, due to prepayments of mortgage loan for example. In such cases, we have to plug in the option cost in the pricing, adding the option-adjusted spread in the all-in cost of funding. Most other credit lines are floating rates, such as committed lines of credit. The financing replicates the time profile of the drawn fraction using floating rates and reset dates matching the reference index used for the line of credit. However, the issue of lines without maturity is even more important with deposits.

The ALM unit does not have to use a funding policy that actually immunizes the interest margin of the bank. The debt replicating the asset characteristics is a "notional debt." The ALM policy generally deviates from perfect matching of the flows, simply because it wishes to keep an open position, for example, due to the existence of demand deposits which do not require raising financial debt in the market.