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# Tuckman - Short rates - Expected Retun & Risk Premium

#### Arnaudc

##### Member
Hi,

This morning, I came across a situation which I cannot logically solve (although I think this is really basic)

The goal is to find the price and expected return over one year of a ZC bond with 3 years to maturity.
The IR tree starts at 10% then move up to either 14% or 6% and then 18% / 10% / 2% (I do not know how to make it visually clear..)
We assume a risk premium of 20 bps / year.
1) Pricing of the 3Y ZC bond.
In order to price the bond, we start by discounting the 3 terminal values (1 , 1 , 1) with the associated rates: 18.4% , 10.4% and 2.4% which gives 0.844595, 0.905797 and 0.976563. We discounted these values including 20 bps x 2 as this is in 2 years from now (not sure).
Then these 3 values are further discounted using 14.20% and 6.20%/ weighted by 0.5 (we assumed probability up/down of 50%) to get 0.766371 and 0.886234. These 2 values are then discounted at 10%, weighted by 0.50 and we get the price of the 3Y ZC today = 0.751184
This part was fine...

But now I do not get how did he value the 3Y ZC in one year assuming the IR path is valid.
The terminal values (1, 1 , 1) are discounted simply at 18%, 10% and 2%. Why? He mention it is because at this point in time, there is no more interest rate risk (Ok but this was the case for pricing the 3Y ZC when we discounted the terminal value right?) So why is it a different treatment?
This is really this point bugging me... Why when pricing the 3Y ZC today, we discount the terminal value at IR + 40 bps (2 x 20bps/year) and to price a 2Y ZC in on year (the 3Y ZC will be a 2Y in one year) we do not assume any risk premium for terminal value discounting?....

Sorry if it is not clear... That(s page 218/219 of the GARP book on Market Risk Measurement...

Thanks!

Kind regards