Given the following conversion of the spot interest rate R(t, T):

Please prove that whether you use periodic compounding or continuous compounding, the relationship between the discount factor B(t, T) and the spot interest rate can be simplified as:

Please derive another conversion of the spot interest rate R(t, T), so that whether it is using periodic compounding or continuous compounding, the discount factor B(t, T) and the spot interest rate are:

Please derive another conversion of the spot interest rate R(t, T ), so that whether it is using periodic compounding or continuous compounding, the discount factor B(t, T) and the spot interest rate are:

PIR. interest R't, T):= [1+R(t, T)) 1, *{IF the regular compound > If continuous compounding ER(I,T) 1, R'(t, T) 1 B(t, T) T= [1+ R"(t, t)]?- R"(t, T), R"(t, T), Bt, T) = 1 1+ (T t).R"(t, T) R'' (t, T), R'' (t, T), B(t, T) = e-(T-1).R" (1,7). PIR. interest R't, T):= [1+R(t, T)) 1, *{IF the regular compound > If continuous compounding ER(I,T) 1, R'(t, T) 1 B(t, T) T= [1+ R"(t, t)]?- R"(t, T), R"(t, T), Bt, T) = 1 1+ (T t).R"(t, T) R'' (t, T), R'' (t, T), B(t, T) = e-(T-1).R" (1,7)