Consider a $10-$year zero$-$coupon bond with a face value of $$10,000$ that isn't subject to income taxes. Suppose that bank deposits offer an interest rate of $5\mathrm{.}25\%$ per year, but interest earned from bank deposits is taxed at a rate of $20\%.$ Taxes on interest income are due at the time of withdrawal.

a$.$ Should the zero$-$coupon bond's yield$-$to$-$maturity be greater than, equal to$,$ or less than $5\mathrm{.}25\%?$ Why?

b$.$ How much would you need to deposit into a bank account today to enable a post$-$tax withdrawal of $$10,000$ in $10$ years? Assume that you will make no additional deposits and no withdrawals in the intervening years. Hint: after taxes are deducted, you have $100\%$ of your initial deposit plus $80\%$ of the interest you earned.

c$.$ Assuming that the untaxed zero$-$coupon bond is comparable to a bank deposit in terms of risk and liquidity, what will its current price be$?$

d$.$ What would the bond's current yield to maturity be if it was purchased at that price?