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The Case of Floaters

The value of a floater providing interest revenues calculated at a rate equal to the discounting rate does not depend on prevailing rates. This is obvious in the case of a single period, since the asset provides 1 + r at the end of the period and this flows discounted at r has a present value of (1 + r)/(l + r) — I. The property extends to any multi-period floater, since such a bond can be replicated by a series of one-period bonds. At each date, a one-period bond is repaid and the proceeds reinvested in another one.[1] The payoff of a multiple period floating rate note can be replicated by entering into a series of one period floating rate notes. At the end of each period, we borrow again for another period with the proceeds of the preceding period floating rate note. Since, at the beginning of each period, the value of the floating rate debt is par, all single period notes have values equal to par value. Moving recursively back in time from one period to the previous one, the value of the floating rate note today is also par value.

Continuous Time

When using continuous compounding instead of discrete compounding, the basic formulas for future and present values are very simple, with a continuous discount rate r and a horizon T:

The future value of a today flow is obtained as the exponential of r(T - t), or exp[r(!T- t)]. Using discounted cash flow formulas for valuing the present value of a stream of cash flows simply substitute exp(-rt) to the discount factor 1/(1 + R)'. The formulas in continuous time for several cash flows occurring at date k, k being between t and T are:

Note that the rate r is allowed to vary across dates.

  • [1] This is an application of the replication principle, which allows replicating assets by simpler ones, as illustrated in the chapters on derivatives.
 
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