Assumptions in converting between nominal/effective interest/discount

Assume we have the present value $A_0$ and future value $A_1$. Using compounded interest we can write the equation $$ big(1+frac{i^{(m)}}{m}big)^mA_0=(1+i)A_0=A_1 $$ but to discount, switch the exponent $m$ to $-m$ so that $$ big(1+frac{i^{(m)}}{m}big)^{-m}A_1=(1-d)A_1=A_0. $$ Notice that I still use $i^{(m)}$ and for this construction to be valid it must hold that $$ d=1-big(1+frac{i^{(m)}}{m}big)^{-m} $$ to which we will try to match a defined compounded discount. Define the compounded discount by $$ big(1-frac{d^{(m)}}{m}big)^mA_1=(1-d)A_1=A_0. $$ From your question, you have interest to couple nominal interest $i^{(m)}$ with nominal discount $d^{(m)}$ which is, we have to solve $$ big(1+frac{i^{(m)}}{m}big)^{-m}=big(1-frac{d^{(m)}}{m}big)^m $$ which gives us $$ (1-frac{d^{(m)}}{m})(1+frac{i^{(m)}}{m})=1 $$ and we can easily see that $d^{(m)}neq i^{(m)}$ since if $d^{(m)}=i^{(m)}$ then $(1-frac{i^{(m)}}{m})(1+frac{i^{(m)}}{m})=1-(frac{i^{(m)}}{m})^2<1$.

How to find with discount rate if interest and discount have different compundings? Let interest be compunded $m_i$ times and discount be compunded $m_d$ times. Then $$ (1-frac{d^{(m_d)}}{m_d})^{m_d}(1+frac{i^{(m_i)}}{m_i})^{m_i}=1 $$ has the solution $$ d^{(m_d)}=m_d(1-frac{1}{(1+frac{i^{(m_i)}}{m_i})^{frac{m_i}{m_d}}}). $$ We choose total time to be 2 years since your choice was $m_d=0.5$. In 2 years interest is compounded 16 times an we discount only one time(!) during 2 years. To validate this, assume you have 100 dollars that you keep for 2 years, that is 16 periods in which each period gives 2% interest, so the future value becomes 137.28 dollars. We actually have $m_i=16$ and $m_d=1$ so $$ d^{(1)}=1-frac{1}{1.02^{16}}approx 0.27155. $$ Then $137.28cdot(1-0.27155)=100$.