Perry Industries has a defined benefit pension plan that specifies annual, year$-$end retirement benefits equal to $1\mathrm{.}3\%$ x Service years x Final year's salary. Carol was hired by Perry at the beginning of $2017.$ Clark is expected to retire at the end of $2056$ after $40$ years of service. Her retirement is expected to span $15$ years. At the end of $2026,10$ years after being hired, her salary is $$60,000.$ The company's actuary projects Clark's salary to be $$210,000$ at retirement. The actuary's discount rate is $6\%.$

Percentage of final year's salary $1\mathrm{.}30\%$

Total years of service $40$ years

Service years through December $31,202610$ years

Clark's salary at December $31,2026$ $$60,000$

Retirement is expected to span $15$ years

Clark's projected salary at retirement $$210,000$

Actuary's discount rate $6\%$

Required:

$1.$ Estimate the amount of Carol$$s annual retirement payments for the $15$ retirement years earned as of the end of $2026.$

$$27,300$

$2.$ Suppose Perry$$s pension plan permits a lump$-$sum payment at retirement in lieu of annuity payments. Determine the lump$-$sum equivalent as the present value as of the earned retirement annuity at the expected date of retirement $($the end of $2056).$

$3.$ What is the company$$s projected benefit obligation at the end of $2026$ with respect to Carol?