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Time Value of Money and Discount Factors

Present value
PV=FV(1+r)tor continuouslyPV=FV e−rt\text{PV} = \frac{\text{FV}}{(1+r)^t} \qquad \text{or continuously} \qquad \text{PV} = \text{FV}\,e^{-rt}

Discrete compounding on the left, continuous on the right. The same statement: money later is worth less than money now.

Money today is worth more than money later, because today's money can be invested. Discounting makes cash flows at different dates comparable, which is the precondition for valuing anything.

Discount factors

D(t)=1(1+r)tD(t) = \frac{1}{(1+r)^t}

The value today of $1 received at time tt. Working with discount factors rather than rates is often cleaner: the value of any cash flow stream becomes a dot product.

V=∑tCt D(t)V = \sum_t C_t\,D(t)

This is bond pricing, and with D(t)D(t) read off a curve rather than from a single rate, it is how a desk actually values cash flows.

Compounding conventions

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