Discount Factors

Discount Factors

A discount factor $d(t)$ is the present value of $1 received at time$t$. It is the building block for pricing all fixed-income instruments.

Discrete Compounding

$d(t) = \frac{1}{(1+r)^t}$

Continuous Compounding

$d(t) = e^{-rt}$

Relationship to Spot Rates

The discount factor and spot rate are equivalent representations of the same information:

• Given $d(t)$, the spot rate is: $s = d(t)^{-1/t} - 1$
• Given $s$, the discount factor is: $d(t) = 1/(1+s)^t$

Applications

Any fixed-income price can be written as:

$P = \sum_t CF_t \cdot d(t)$

Discount factors provide a model-free way to price instruments once the yield curve is known.

Example

At 5% annual rate:

• $d(1) = 1/1.05 \approx 0.9524$
• $d(5) = 1/1.05^5 \approx 0.7835$
• $d_{cont}(1) = e^{-0.05} \approx 0.9512$