Question
In: Finance
Aaron McCarthy Aaron McCarthy is 28 years old and is making his first attempt at determining...
Aaron McCarthy
Aaron McCarthy is 28 years old and is making his first attempt at determining how much he needs to save each year in order to fund his retirement. He works for a Fortune 500 corporation as a project manager. His employer offers a 401(k) plan featuring a 1:1 match for every dollar contributed to a maximum of 6% of his salary. Aaron is eligible to participate but currently is not enrolled in the plan. He is in the 25% marginal tax bracket.
Based on an analysis of his last 3 annual cash flow statements, he has determined that his average gross income has been about $58,000 and his annual living expenditures have been about $46,700. He expects both his income and expenses to match an assumed inflation rate of 3.5% going forward indefinitely.
In order to be conservative, Aaron is working under the assumption that he will be able to maintain his current standard of living on 85% of his current spending after he retires (that is, he expects his annual expenses to fall by 15% in retirement without reducing or lowering his current standard of living).
According to the annual benefit statement he received before his last birthday, Aaron can expect to receive $15,000 (expressed in today’s dollars) per year in Social Security retirement benefits at his normal retirement age (67). Aaron plans to retire at age 67, and expects to live until age 90. His mother’s parents lived until their early 80’s, as did both his father’s parents.
Again, to be conservative, Aaron is assuming a 3.5% inflation rate between now and retirement age and that any future investments he makes can be compounded at an annual rate of 7% (before taxes).
Aaron has a combined total of $15,000 currently invested in three IRAs that he owns. Assume the same 7% annual growth rate for these funds.
Over the last 3 years, Aaron’s savings rate has averaged 3% of his gross wages.
- How much will Aaron need to save each year
in order to meet his retirement goal?
- Assuming he invests these funds in a non-qualified account?
- Assuming he enrolls in and funds his 401(k) to the maximum amount possible?
- What savings rate does this represent under each of the scenarios above?
Solutions
Expert Solution
We need to calculate how much will Aaron need to save each year in order to meet his retirement goal. For this, we first need to calculate his retirement goal, i.e., the total savings that he should have at the time he retires.
| Particulars | Value | Remarks |
| Current Age (a1) | 28 | Given |
| Retirement Age (a2) | 67 | Given |
| Expected to live until Age (a3) | 90 | Given |
| 401(k) plan with 1:1 match (r) | 6.00% | Given |
| Tax rate (t) | 25.00% | Given |
| Gross annual income (I) | 58,000.00 | Per year till retirement |
| Expenses (e1) | 46,700.00 | Per year till retirement |
| Inflation (i) | 3.50% | Given |
| Expenses post retirement (e2) | 39,695.00 | In current dollar terms, 85% of Expenses, post retirement |
| Social Security Retirement Benefits (b) | 15,000.00 | To be Received per year post retirement |
| Current Investments | 15,000.00 | Given |
| Last 3 years savings' rate (sr) | 3.00% | Given |
| Nominal growth rate for investments (g) | 7.00% | Given |
| Calculations | ||
| Particulars | Value | Remarks |
| Earning Years (A1) | 39.00 | a2 - a1 |
| Life (in years) as a retired person (A2) | 23.00 | a3 - a2 |
| Real growth rate for investments (gr) | 3.38% | [(1 + g) / (1 + i)] - 1 |
| Current savings per year | 1,740.00 | Gross annual income*Saving's rate |
| Total saving's goal at retirement | 403,622.78 | In current dollar terms, {[(e2-b)*(1+gr)]*[(1+gr)^A2 - 1] / gr*(1+gr)^A2} |
| Detailed estimation for explanantion is shown as follows |
| Estimating Total Saving's Goal at the time of retirement | |||||
| Age | Savings at year start | Social Security Benefit | Expenses | Remaining Savings | Amount post real growth |
| S | b | e2 | S + b - e2 | (S-e2)*(1+gr) | |
| 68 | 403,622.78 | 15,000.00 | 39,695.00 | 378,927.78 | 391,741.77 |
| 69 | 391,741.77 | 15,000.00 | 39,695.00 | 367,046.77 | 379,458.98 |
| 70 | 379,458.98 | 15,000.00 | 39,695.00 | 354,763.98 | 366,760.83 |
| 71 | 366,760.83 | 15,000.00 | 39,695.00 | 342,065.83 | 353,633.27 |
| 72 | 353,633.27 | 15,000.00 | 39,695.00 | 328,938.27 | 340,061.78 |
| 73 | 340,061.78 | 15,000.00 | 39,695.00 | 315,366.78 | 326,031.36 |
| 74 | 326,031.36 | 15,000.00 | 39,695.00 | 301,336.36 | 311,526.48 |
| 75 | 311,526.48 | 15,000.00 | 39,695.00 | 286,831.48 | 296,531.10 |
| 76 | 296,531.10 | 15,000.00 | 39,695.00 | 271,836.10 | 281,028.62 |
| 77 | 281,028.62 | 15,000.00 | 39,695.00 | 256,333.62 | 265,001.91 |
| 78 | 265,001.91 | 15,000.00 | 39,695.00 | 240,306.91 | 248,433.23 |
| 79 | 248,433.23 | 15,000.00 | 39,695.00 | 223,738.23 | 231,304.26 |
| 80 | 231,304.26 | 15,000.00 | 39,695.00 | 206,609.26 | 213,596.04 |
| 81 | 213,596.04 | 15,000.00 | 39,695.00 | 188,901.04 | 195,289.00 |
| 82 | 195,289.00 | 15,000.00 | 39,695.00 | 170,594.00 | 176,362.88 |
| 83 | 176,362.88 | 15,000.00 | 39,695.00 | 151,667.88 | 156,796.74 |
| 84 | 156,796.74 | 15,000.00 | 39,695.00 | 132,101.74 | 136,568.95 |
| 85 | 136,568.95 | 15,000.00 | 39,695.00 | 111,873.95 | 115,657.13 |
| 86 | 115,657.13 | 15,000.00 | 39,695.00 | 90,962.13 | 94,038.14 |
| 87 | 94,038.14 | 15,000.00 | 39,695.00 | 69,343.14 | 71,688.08 |
| 88 | 71,688.08 | 15,000.00 | 39,695.00 | 46,993.08 | 48,582.22 |
| 89 | 48,582.22 | 15,000.00 | 39,695.00 | 23,887.22 | 24,695.00 |
| 90 | 24,695.00 | 15,000.00 | 39,695.00 | (0.00) | (0.00) |
Now we understand that Aaron's retirement goal is $403,622.78 in current dollar terms, and can calculate his required saving's rate to reach this goal.
(a) In a non-qualified account, his investments will grow at the usual 7% nominal growth rate.
| Estimating Saving's Rate | ||
| Particulars | Value | Remarks |
| Goal (G) | 403,622.78 | Total saving's goal at retirement |
| Current Investment (C) | 15,000.00 | |
| Current Investment value at retirement (Cr) | 54,876.32 | In current dollar terms, C*(1+gr)^A1 |
| Amount to be saved | 348,746.47 | In current dollar terms, G - Cr |
| Any amount saved is assumed to grow at the real growth rate for investment, since savings will be invested | ||
| Per year savings (x) | 4,291.12 | In current dollar terms, detailed estimation shown as follows |
| Per year saving's rate | 7.40% | x / I |
| Let the amount to be saved per year be x | ||
| We know the this will happen for the next 39 years | ||
| Thus, amount saved in year 1 will be invested for 39 years, amount saved in year 2 will be invested for 38 years, and so on. | ||
| Thus, total amount to be saved = x*[(1+gr)^39 + (1+gr)^38 + …… + (1+gr)] | ||
| Now we solve for x, since the term in bracket is a geometric progression, with first term a = (1+gr), r = (1+gr) and n=39 | ||
| Sum of geometric progression | 81.27 | (1+gr)*[(1+gr)^39 - 1] / (1+gr) - 1] |
| Per year savings (x) | 4,291.12 | Amount to be saved / Sum of geometric progression |
| Age | Income | Savings | Total Savings | Amount post real growth |
| I | s | s + P (previous year) | P | |
| 29 | 58,000.00 | 4,291.12 | 4,291.12 | 4,436.23 |
| 30 | 58,000.00 | 4,291.12 | 8,727.34 | 9,022.47 |
| 31 | 58,000.00 | 4,291.12 | 13,313.59 | 13,763.81 |
| 32 | 58,000.00 | 4,291.12 | 18,054.92 | 18,665.48 |
| 33 | 58,000.00 | 4,291.12 | 22,956.59 | 23,732.90 |
| 34 | 58,000.00 | 4,291.12 | 28,024.02 | 28,971.69 |
| 35 | 58,000.00 | 4,291.12 | 33,262.81 | 34,387.64 |
| 36 | 58,000.00 | 4,291.12 | 38,678.75 | 39,986.73 |
| 37 | 58,000.00 | 4,291.12 | 44,277.85 | 45,775.16 |
| 38 | 58,000.00 | 4,291.12 | 50,066.28 | 51,759.34 |
| 39 | 58,000.00 | 4,291.12 | 56,050.46 | 57,945.89 |
| 40 | 58,000.00 | 4,291.12 | 62,237.00 | 64,341.64 |
| 41 | 58,000.00 | 4,291.12 | 68,632.75 | 70,953.67 |
| 42 | 58,000.00 | 4,291.12 | 75,244.78 | 77,789.29 |
| 43 | 58,000.00 | 4,291.12 | 82,080.41 | 84,856.08 |
| 44 | 58,000.00 | 4,291.12 | 89,147.19 | 92,161.83 |
| 45 | 58,000.00 | 4,291.12 | 96,452.95 | 99,714.64 |
| 46 | 58,000.00 | 4,291.12 | 104,005.76 | 107,522.86 |
| 47 | 58,000.00 | 4,291.12 | 111,813.98 | 115,595.13 |
| 48 | 58,000.00 | 4,291.12 | 119,886.24 | 123,940.37 |
| 49 | 58,000.00 | 4,291.12 | 128,231.49 | 132,567.82 |
| 50 | 58,000.00 | 4,291.12 | 136,858.93 | 141,487.01 |
| 51 | 58,000.00 | 4,291.12 | 145,778.13 | 150,707.82 |
| 52 | 58,000.00 | 4,291.12 | 154,998.94 | 160,240.45 |
| 53 | 58,000.00 | 4,291.12 | 164,531.57 | 170,095.44 |
| 54 | 58,000.00 | 4,291.12 | 174,386.55 | 180,283.68 |
| 55 | 58,000.00 | 4,291.12 | 184,574.80 | 190,816.46 |
| 56 | 58,000.00 | 4,291.12 | 195,107.58 | 201,705.42 |
| 57 | 58,000.00 | 4,291.12 | 205,996.53 | 212,962.60 |
| 58 | 58,000.00 | 4,291.12 | 217,253.72 | 224,600.46 |
| 59 | 58,000.00 | 4,291.12 | 228,891.58 | 236,631.87 |
| 60 | 58,000.00 | 4,291.12 | 240,922.99 | 249,070.14 |
| 61 | 58,000.00 | 4,291.12 | 253,361.26 | 261,929.03 |
| 62 | 58,000.00 | 4,291.12 | 266,220.15 | 275,222.76 |
| 63 | 58,000.00 | 4,291.12 | 279,513.88 | 288,966.04 |
| 64 | 58,000.00 | 4,291.12 | 293,257.16 | 303,174.06 |
| 65 | 58,000.00 | 4,291.12 | 307,465.18 | 317,862.55 |
| 66 | 58,000.00 | 4,291.12 | 322,153.67 | 333,047.76 |
| 67 | 58,000.00 | 4,291.12 | 337,338.87 | 348,746.47 |
Thus, Aaron needs to save 7.40% of his gross wages per year to reach his goal in scenario (a).
(b) He enrolls in and funds his 401(k) to the maximum amount possible.
| Estimating Saving's Rate | ||
| Particulars | Value | Remarks |
| Amount to be saved | 348,746.47 | |
| Aaron can invest a maximum of 6% in 401(k), and an equal amount will be contributed by the employer | ||
| However, Aaron's savings has averaged 3% in the past 3 years | ||
| Thus, we assume that he can invest a maximum of 3% in the 401(k) | ||
| This will result in an additional 3% being contributed by the employer, making it a total of 6%. Since taxes will be levied at retirement, we ignore the annual effects. | ||
| Thus, Aaron needs to save an additional (7.40 - 6) = 1.40% |
Thus, Aaron needs to save 1.40% of his gross wages per year to reach his goal in scenario (b).
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