# You are trying to decide how much to save for retirement. Assume you plan to save $6,500 per year... ## Question: You are trying to decide how much to save for retirement. Assume you plan to save$6,500 per year with the first investment made one year from now. You think you can earn 7.0 per year on your investments and you plan to return in 29 years, immediately after making your last $6,500 investment. a. How much will you have in your retirement account on the day you retire? b. If, instead of investing$6,500 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need be?

c. If you hope to live for 24 years in retirement, how much can you withdraw every year in retirement? (Starting one year after retirement) so that you will just exhaust your savings with the withdrawal (assume our savings will continue to earn 7.0% in retirement)?

d. If instead, you decide to withdraw $114,000 per year in retirement (again with the first withdrawal one year after retiring), how many years will it take until you exhaust your savings? (Use trial-and-error, a financial calculator: solve for ''N'', or Excel function NPER) e. Assuming the most you can afford to save$1,300 per year, but you want to retire with $1,000,000 in your investment account, how high of a return do you need to earn on your investment? (Use trial-and-error, a financial calculator: solve for the interest rate, or Excel function RATE) ## Valuing Annuities: An annuity is a series of level cash flows over a set period of time. With a given interest rate, you can calculate the annuity's present value or future value. This is particularly valuable when figuring out how to invest for a future expense, or the sustainable level of withdrawals in retirement. ## Answer and Explanation: #### a. How much will you have in your retirement account on the day you retire? Use the FV of Annuity formula: {eq}FV \ of \ Annuity = C * \cfrac{(1+r)^t - 1}{r} \\ FV = 6,500 * \cfrac{(1.07)^{29} - 1}{0.07} = 567,752.44 {/eq} #### b. If, instead of investing$6,500 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how much would that lump sum need be?

{eq}PV = \cfrac{567,752.44}{1.07^{29}} = 79,804.88 {/eq}

#### c. If you hope to live for 24 years in retirement, how much can you withdraw every year in retirement? (Starting one year after retirement) so that you will just exhaust your savings with the withdrawal (assume our savings will continue to earn 7.0% in retirement)?

Use the Annuity PV Formula:

{eq}Withdrawal = PV * \cfrac{r}{1-1/(1+r)^t} \\ Withdrawal = 567,752.44 * \cfrac{0.07}{1-1/(1.07)^{24}} = 49,501.78 {/eq}

#### d. If instead, you decide to withdraw $114,000 per year in retirement (again with the first withdrawal one year after retiring), how many years will it take until you exhaust your savings? (Use trial-and-error, a financial calculator: solve for ''N'', or Excel function NPER) {eq}114,000 = 567,752.44 * \cfrac{0.07}{1-1/(1.07)^{n}} \\ 1-1/(1.07)^n = 0.07 * \cfrac{567,752.44}{114,000} \\ \cfrac{1}{1.07^n} = 1 - 0.07 * \cfrac{567,752.44}{114,000} \\ 1.07^n = \cfrac{1}{1 - 0.07 * 567,752.44/114,000} \\ n = log_{1.07}(\cfrac{1}{1 - 0.07 * 567,752.44/114,000}) = 6.34 \ years {/eq} #### e. Assuming the most you can afford to save$1,300 per year, but you want to retire with \$1,000,000 in your investment account, how high of a return do you need to earn on your investment? (Use trial-and-error, a financial calculator: solve for the interest rate, or Excel function RATE)

{eq}FV \ of \ Annuity = C * \cfrac{(1+r)^t - 1}{r} \\ 1,000,000 = 1,300 * \cfrac{(1+r)^{29} - 1}{r} {/eq}

r = 18.72%