Sarah Khan is a $30-$year$-$old marketing manager working for a large tech company. She is thinking seriously about planning for her retirement. Sarah wants to retire at age $65$ and expects to live until age $90.$ She wants to maintain a lifestyle that will require an annual income of $$80,000$ per year $($in today$$s dollars$)$ once she retires. Sarah plans to invest in a retirement account that will earn an annual return of $7\%.$ Inflation is expected to average $2\%$ per year over the long term. Sarah currently has $$25,000$ saved in her retirement account. $$ Part $1$: Retirement Goal $$ How much will Sarah need? $1\mathrm{.}1.$ At retirement $($age $65),$ Sarah wants to receive the equivalent of $$80,000$ per year in today$$s dollars, but the payments will be inflation$-$adjusted. What will be her first year$$s withdrawal $($nominal dollars$)$ at age $66?1\mathrm{.}2.$ Assuming Sarah makes annual withdrawals for $25$ years $($from age $65$ to $90),$ and the portfolio continues earning $7\%$ annually, what is the present value at age $65$ of the income stream she will need $($i$.$e$.$ how much money does she need saved at retirement$)?($Hint: This is a growing annuity $$ annual income growing at $2\%,$ discount rate $7\%.)2\mathrm{.}1.$ How much time does Sarah have to save for retirement? $($Hint: $65-30)2\mathrm{.}2.$ If Sarah invests a fixed amount at the end of each year starting now until she turns $65(35$ years of saving$),$ how much does she need to contribute annually to reach her target savings at age $65?$ Dont forget the initial $25,000$ saving $2\mathrm{.}3.$ Suppose Sarah expects to receive an inheritance of $$50,000$ at age $45($in $15$ years$)$ and will add it to her retirement account. How will this affect her required annual contributions $($What is the recalculated contribution?$)?$ art $3$: Perpetuity $$ A lifetime income option At retirement, Sarah$$s company offers her the option to purchase a life annuity: a perpetual income stream $($indexed to inflation$)$ paying $$80,000$ per year in today$$s dollars. $3\mathrm{.}1.$ If the real discount rate is $5\%($after adjusting for inflation$),$ how much would the insurance company charge Sarah today for this inflation$-$adjusted perpetuity? $($Hint: Use perpetuity present value formula.$)$