Today is Dec. $31,2020.$ Mark is planning to retire in $40$ years from today and wants to save some money to support his living for at least $20$ years, i$.$e$.,2061$ through $2080,$ after his retirement. Mark estimates that he will need $$60,000$ each year from Jan. $1,2061,$ to Dec. $31,2080.$ Assume that the interest rate is $10\%$ and will be constant over time. Answer questions a$)$ and b$)$ below.

Determine how much he needs to save by the end of $2060.[$free credit$($Part A only$)]$

Answer $($show the steps$/$calculation toward your results$)$:

Need $$60$k at the end of each year from $2061-2080,$ i$.$e$.,$ an annuity with C $=60$k; T $=20$; r $=10\%.$ Find the PV$,$ as of t$=2060,$ of the annuity:

$$PV$\_(($t$=2060))=60$k$/0\mathrm{.}10[1-(1/1\mathrm{.}10)^20]=$$$510,813\mathrm{.}82$

Fin. calculator: Key N I$/$Y PV PMT FV

Enter $2010(\%)[$solve for$]60000-$

Excel function: fn $=$ pv$($rate$,$ nper, pmt$,$ fv $)$

Enter $=$ pv$(10\%,20,60000,0)$

Suppose that Mark will start saving money one year from today by making $40$ annual installment deposits. How much per year, at the end of each year from $20212060,$ does he need to save?

$(*$Hint: To work in the present value as of today $($t$=2020),$ discount the funds needed by t$=2060$ back to the PV as of t$=2020$ and then solve for the annual payment for next $40$ years.$)$

Answer $($show the steps$/$calculation toward your results$)$: