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?
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.
