When calculating the Price of a risk-free bond using the spot rates on the spot curve (after deriving them using the forward rates), is there no need to add the exponents for years when discounting? I think the picture explains it a little better than I.
No, they shouldn’t.
Would you mind explaining a little as to why not? It’s just that the other discounting problems all included the exponents, is there something different about this discount?
Look closely at the denominator for year 2, for example. You are already compounding time value by multiplying the year 1 spot rate times the “year 1 to year 2” forward rate, in the denominator. The 2 together will equal the year 2 spot rate squared already. No need to “double compound” it by squaring it another time on top of that, too.
Cheers - good luck - you got this
Right, I can see that. Thanks.
You can discount a cash flow two years from today in two ways: divide by:
- \left(1 + s_2\right)^2 = \left(1 + s_2\right)\left(1 + s_2\right), or
- \left(1 + 1y1y\right)\left(1 + s_1\right)
In each case, you’re dividing by \left(1 + something\right)\left(1 + something\right): two (annual) rates for two years of discounting.
Perfect explanation, thanks Magician