The simple answer is yes. A good portion of your savings should be going towards wealth building—which in some ways can be considered retirement savings. This money is what will allow you to enjoy your retirement stress free. It is essential that you begin saving for retirement as soon as you are able. You should save enough so that your budget seems slightly uncomfortable.
It is important to have a reason for saving your money. This gives you earmarks and attainable goals in order for you to work towards. You may be saving up for a down payment on a home, saving to fully fund your emergency fund, saving for your child’s education or for a dream vacation. Don’t forget that a good goal is to pay for most of your major purchases with cash, and the only way to do this is to save up for them in advance.
You can work on saving for these goals all at once or really focus on one or two at a time. If you are saving towards multiple goals you may want to keep a tally sheet of how much you have going towards each goal or you may choose to have a separate account for specific goals. It is generally a good idea to have a separate account for you emergency fund.

