401k Math Question

Question

Tina contributes 9.5% of her monthly salary towards her 401(k) and her employer matches up to 5.4% of her contribution. If the interest rate of her 401(k) is 8.1% compounded monthly and her monthly salary is $2,461, determine the amount in her account after 25 years. Round to the nearest cent.

 

 

Answer 1

Every month the total contribution is:
(9.5%)($2,461)(1+5.4%) = (233.79)(1+5.4%) = $246.41

Monthly multiplier is: (1+0.081/12) = 1.00675

month contribution balance
0 $246.41 $246.41 = b0
1 $246.41 b0(1.00675)+$246.41 = b1
2 $246.41 b1(1.00675)+$246.41 = b2
…
n $246.41 b{n-1}*(1.00675)+$246.41 = bn
…
300 $246.41 b299*(1.00675)+$246.41 = b300

It’s easiest to do this on an electronic spreadsheet like Excel or Libre Calc.

Here’s an .xlx file:
http://www.wyzant.com/resources/files/264038/supporting_spreadsheet

The answer is $240,041.00

Answer 2

Quick and dirty solution, rounding the final value to the nearest cent.

assumption: investment is made at the end of the month

-------------------------------------------------------------------------------

 

Her monthly contribution = $2,461 * 9.5% = 233.795

Company match = $2,461 * 5.4% = 132.894

Total investment per month = 366.689

 

interest rate per month = 8.1% / 12 = .675% / month

Number of periods = 25 * 12

 

using Excel FV function (with "type"=1), you get $88,798.898 which would round to $88,798.90

 

---------------------------------------------------------------------------------------

But any real financial institution will round the employee and employer contribution to the nearest cent

 

Monthly contribution = 233.80+132.89 = 366.69

 

....and then it would round the increase every month,

so that at the end of the first period it would be 366.69

at the end of the second period it would be 735.86 (instead of a mathematically correct 735.8552) and I'm not going to figure out the other 298 intermediate values.

 

sorry for all the stuff below the dashed line, but I wanted you to know about the complexities in the real world.

 

 

Note: FV = A (1 + i/12)^nper - 1

                   -----------------------

                          i/12

 

where nper in this case is 25 years * 12 payments/year and i/12 is the monthly interest rate

Answer 3

The key to this problem is to simplify it. What are they asking you to do? You can tell it is a "future value" problem and, specifically a "future value of an annuity".  The twist is they have complicated it by making you calculate the annuity instead of giving it to you.

 

Calculate the annuity: (.095+.054)*2461=_______ (note they assume that this person never gets a raise)

 

Since the compounding period is a month you have to find out the compounding rate (i): .081/12=______

 

The period (n) is 25 years, but you have to express this in months: 25*12=_______.

 

how are you equipped to solve the problem?  Do you have a financial calculator or are using future value tables?