Present Value When Interest Rates are Stochastic

[Oxymoron]

It's probably a basic question, but I'm confusing myself with this:

Suppose we're planning to lend for a 5 year period, $1 at 6% continuously compounded. The current (continuous) term structure, say, is also flat at 6% for all terms. So the present value now is 1*exp(-.06*5)*exp(.06*5) = 1. Straight forward.

Now suppose the borrower repays the money in 2 years - and the three year continuously compounded rate for re-lending is say, 10%, and let's say the term structure is also flat at that rate.

Is the present value:

1*exp(-.06*5)*exp(.06*2)*exp(.10*3) - i.e, reinvested for a period of 3 more years after the 2nd year

or

1*exp(-.06*2)*exp(.06*2) - where we ignore reinvestment?

 

[Mark Mitchell]

In your example you say you're going to lend $1 now. So the present value of this is $1, irrespective of what the term structure of interest rates is, as you have $1 today.

Any discounting and accumulating you do should be consistent with this fact.

In the first of your options, you accumulate for 2 years at 6% and 3 years at 10%, and then discount for 5 years at 6%, which is inconsistent.

 

[Oxymoron]

Thanks Mark. Sorry missed this somehow.