# Get discount factors with limited knowledge?

I am facing the problem of just having this information:

6% coupon bond with 2.5 years to maturity, traded at a 100% clean price 4% coupon bond with 1.5 years to maturity, traded at a 98% clean price A 6M deposit with 6M rate of 5% (MMY)

And I want to get the discount factors for each maturity.

My problem is, I do not know where to start to get to each discount factor. Is there a straight method for this kind of problem?

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## 1 Answer

The general idea is to bootstrap the discount factors in the correct order, based on the data you have given. I'm going to make some assumptions that your bonds are paying annual coupons. The longest maturity is 2.5 years, meaning you need discount factors for 6M, 1.5Y and 2.5Y.

The 6M deposit has a rate of 5%, this tells you that you should use the 5% rate to create the 6M discount factor (you wrote 6M rate = 5%, if the 5% was expressed in yearly terms the rate applicable would have been 2.5% = 5%/2). So know you have the discount factor for 6M, $DF_{6M} = e^{-0.05*0.5}$. To retrieve the remaining discount factors you need to formulate equations, equations relating the given bond prices to an analytical expression involving the sought discount factors. For example, assuming the notional is equal to 1, $0.98 = 0.04 DF_{6M} + (1+0.04) DF_{1.5Y}$. But you already know $DF_{6M}$, so from this equation you can solve for $DF_{1.5Y}$. Now write a similar equation for the longest maturity bond and solve for $DF_{2.5Y}$.

Hope it helps!

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