**Using discount factors to find the final discount factor in a series**

If I already have several discount factors I can use them to calculate the following years’ discount factor; What I need to know is the number of years to maturity, the coupon and the current price.

With those values I can form an equation;

PV = CF or CPN * sum of given discount factors + (FV + final CPN) * x

x = discount factor in the final year

102.72 = 5 * (0.9709 + 0.9335 + 0.8916) + 105 * x

x = 0.8451

**Using the discount factor to calculate the spot rate in year 4**

Info; required knowledge: 1/(1 + s_{t})^{t} = the discount factor

df_{4} = 1 / (1 + S_{4})^{4} =>

0.8451 = 1 / (1 + x )^{4} => | * (1 + x )^{4 } | then / [division] by 0.8451

(1 + x )^{4 } = 1 / 0.8451 => | ^1/4 [take the fourth root]

(1 + x ) = (1 / 0.8451)^{1/4} | calculate right side

1 + x = 1.042973 | subtract 1 , rewrite 0.042973 as %

x = 4.2973% = 4.30%

The discount factor can be used to calculate the spot rate of the same year.

So the above solution answers the question: How do I get the spot rate using the discount factor.