Find the present value of $$800$ due in the future under each of these conditions:

a$.15\%$ nominal rate, semiannual compounding, discounted back $6$ years. Do not round intermediate calculations. Round your answer to the nearest cent.

b$.15\%$ nominal rate, quarterly compounding, discounted back $6$ years. Do not round intermediate calculations. Round your answer to the nearest cent. $

c$.15\%$ nominal rate, monthly compounding, discounted back $1$ year. Do not round intermediate calculations. Round your answer to the nearest cent. $

d$.$ Why do the differences in the PVs occur?