Retirement: How much to save, when to retire, and how much you'll need

Question
How does retirement work? How much do I need to retire? How much will I have when I retire? How much do I need to put away to retire? Note: I don't cover Social Security benefits in this post. They are significant, though probably not sufficient.

These questions pop into my head relatively frequently, and I've been getting better at being realistic about them (I'm going to retire at 35 and live off of 200,000 every year, bwa-ha-ha-ha!).

First some definitions:

• Principal will refer to the amount you have in your retirement account on the day you say "screw you guys, I'm going home" for good.
• When we talk about interest rate, we're referring to a yearly rate of return. If it's 10% on $10, then after a year you'll have $11.

We'll follow a standard retirement plan:

1. Let's say you start with some retirement savings.
2. Then each year you add to it. You do this up until you retire.
3. This savings will be invested, so each year, your savings will make some money.
4. At the point you retire, you will have your retirement principal saved up.
5. Each year, you will take out the amount you need to live on for that year. That will cause your retirement account to shrink.
6. The principal is invested, so each year it will make interest which will cause it to grow.
7. We want to set it up to last as long as we do. It shouldn't run out before our life does, but it doesn't need to last much longer, now does it (feel free to set r_n to some amount instead of Zero if you do want to have some amount left over for progeny or charity or what have you)?

When I first started thinking about retirement, my plans weren't nearly this sophisticated. My first idea was to have enough money to live off of the interest. While that would be nice, it requires a lot of money and a high, stable interest rate. My next idea came from something I heard somewhere (maybe my Mom, in which case it comes from Suze Orman), and it was that you just need enough to last out your retirement (estimate to the end of your 99th year). Suppose I want 80,000 a year for 30 years (which means retiring on my 70th birthday). Then I would need to have 80,000 * 30 = 2.4 million. Yikes! That's when I came up with the idea laid out above. Make use of interest during the accumulation phase and during the retirement phase, and you only need enough to last until you're dead. This makes for a much more attainable retirement goal.

Math Time

Let's start by defining some variables:

• P = Retirement Principal. This is the amount that you start retirement with. For the sake of this math, it will be a constant number. When talking about retirement with people, it is common to talk about your principal changing referring to your retirement account balance changing (just FYI).
• Y = Yearly retirement salary. This is the amount you want to live on each year. Here it will be constant, but you might have more knowledge of how your needs will change over time. If that is the case, write a simulation program.
• i1 = Interest Rate during the accrual phase.
• n1 = Number of years until retirement, or number of years you have to build up the principal.
• i2 = Interest Rate during the retirement phase. This is often lower due to a decreased desire for risk. You can always just work a little more, but if you lose money on your investments during retirement, it can be a bigger problem.
• n2 = 99 - retirement age. Number of years you will be retired (i.e. until you're playing backgammon with worms).
• S = retirement savings you currently have saved up.
• A = Annual amount put into retirement savings.
• s_0, s_1, s_k, s_n1 = How much we have in savings after 0, 1, k, or n1 years.
• r_0, r_1, r_k, r_n2 = How much we have in our retirement account after 0, 1, k, or n2 years.

Here's an outline of what we're about to figure out:

1. First we'll figure out the relationship between Principal, Yearly retirement salary, number of years of retirement, and interest rate during retirement. The relationship between P, Y, n2, and i2, which will be r_n2. Once we know that, we can answer some questions like "if I retire at 65, how much Principal will I need to live off of 80,000 a year (at different interest rates)?" We will be able to answer this question by noting that r_n2, or the amount that we should have in our savings account by 99 years old, is zero. That will let us solve for P.
2. Then we'll figure out the relationship between Starting Savings, Annual deposits, interest, and number of years to save. The relationship between S, A, i1, and n1, which will be s_n1. This will allow us to answer questions such as "how much will I have after 20 years of saving 10,000 a year?"
3. Finally, we'll put it all together by noting that the amount we have after our final year of savings is our principal, which is also our starting amount in our retirement account. In other words s_n1 = P = r_0. This will relate all of our values and allow us to create a sort of retirement calculator.

A few more notes before we begin: In so many financial sequences, we deal with a summation of powers of interest rates. You can find this in the prerequisites page as well as the loan amortization page (which is probably more relevant). If you want to follow the math (and if you've read this far, I'm assuming you do), then make sure you understand that. I'll just use it here, as opposed to proving it again.

Time to get down to it! First, we'll try to understand r_k:

1. Obviously r_0 = P. We start with our retirement principal ... or do we? Perhaps we should be more realistic and say that r_0 = P - Y. We need to live on something for that first year. We'll have r_0 = P - Y. Let's think this through a little more as we go, so that when we say r_n2, when know what n2 should be.
2. r_1 is the interest on r_0 minus our Yearly retirement salary. r_1 = r_0 (1+i2) - Y = (P - Y)*(1+i2) - Y = P*(1+i2) - Y*(1+i2) - Y. Those of you who have read some of the other posts should see where this is going. Consider what r_1 represents. r_1 is the amount of money we have after one year, and after taking out what we need to live through our second year. So if we were retiring on our 98th birthday, and r_1 = 0, then we have lived through two full years. In this case n2 would be 1.
3. r_2 should be the interest on r_1 minus Y again. r_2 = (P*(1+i2) - Y*(1+i2) - Y)*(1+i2) - Y = P*(1+i2)^2 - Y*(1+i2)^2 - Y*(1+i2) - Y = P*(1+i2)^2 - Y*(1 + (1+i2) + (1+i2)^2). 
4. Already we can see that r_k = P*(1+i2)^k - Y*(1 + (1+i2) + (1+i2)^2 + ... + (1+i2)^k). 
5. The summation of powers of interest rates becomes ((1+i2)^(k+1) - 1)/(1+i2 - 1) = ((1+i2)^(k+1) - 1)/i2.
6. Putting it all together and looking at year n2, we get r_n2 = P*(1+i2)^n2 - Y*(((1+i2)^(n2+1) - 1)/i2)
7. We also know that r_n2 should be 0 since we don't want to have to save up more than we need.
8. r_n2 = 0 = P*(1+i2)^n2 - Y*(((1+i2)^(n2+1) - 1)/i2), so P*(1+i2)^n2 = Y*(((1+i2)^(n2+1) - 1)/i2)
9. Thus P = Y * ((1+i2)^(n2+1) - 1) / (i2*(1+i2)^n2). We could simply this a little more, but who cares?

Now let's try to understand s_k:

1. It's a similar story here. s_0 = S. We start with whatever we have already saved up. 
2. s_1 = s_0*(1+i1) + A. The interest on what we have plus our annual deposit. s_1 = S*(1+i1) + A
3. s_2 = s_1*(1+i1) + A = (S*(1+i1) + A)*(1+i1) + A = S*(1+i1)^2 + A*(1+i1) + A
4. s_3 goes the same way as r went, so to keep things to the point, s_3 = S*(1+i1)^3 + A*(1 + (1+i1) + (1+i1)^2)
5. s_k = S*(1+i1)^k + A*(1 + (1+i1) + ... + (1+i1)^(k-1))
6. Using our telescoping series trick we get s_n1 = S*(1+i1)^n1 + A*((1+i1)^n1 - 1)/i1

Let's put it all together now:

1. We know that s_n1 = P, since after n1 years of saving up, we should have our principal amount.
2. Thus P = Y * ((1+i2)^(n2+1) - 1) / (i2*(1+i2)^n2) = S*(1+i1)^n1 + A*((1+i1)^n1 - 1)/i1

Final Answers

There are many questions that can now be answered.

• If I want to retire at 70 and make 80,000 a year supposing I can get a 3%, 5%, or 7% interest rate on my retirement account, how much principal do I need? Note: you can put any one of these equations in google and it will give you the answer, so you can play with the numbers all you like.
• 3%: P = 80000*((1.03)^(99-70+1) - 1)/(.03*(1.03)^(99-70)) = 1,615,076.37. Still steep, but not as steep as 2.4 million!
• 5%: P = 80000*((1.05)^(99-70+1) - 1)/(.05*(1.05)^(99-70)) = 1,291,285.89. Getting better, but still pretty heavy.
• 7%: P = 80000*((1.07)^(99-70+1) - 1)/(.07*(1.07)^(99-70)) = 1,062,213.93. Just over a million is not too high to shoot for a lifetime of savings. Let's look at the savings side to see if it is attainable.
• If I start saving at 30 with plans to retire by 70 and I am starting with 5000 in savings, how much do I need to save each year supposing I can get a 3%, 5%, or 7% interest rate on my savings account? Let's assume I will get the same interest rate on the retirement account that I get on the savings account. First let's do some rearranging to solve for A. A*((1+i1)^n1 - 1)/i1 = P - S*(1+i1)^n1, so A = (P - S*(1+i1)^n1) / (((1+i1)^n1 - 1)/i1)
• 3%: A = (1615076.37 - 5000*(1.03)^(70-30)) / (((1.03)^(70-30) - 1)/.03) = 21,203.44. That's pretty steep!
• 5%: A = (1291285.89 - 5000*(1.05)^(70-30)) / (((1.05)^(70-30) - 1)/.05) = 10,398.08. That's starting to seem more attainable for some folks, though probably not folks in their 20's.
• 7%: A = (1062213.93 - 5000*(1.07)^(70-30)) / (((1.07)^(70-30) - 1)/.07) = 4,945.73. If you can save up 5000 a year and you have 40 years until you retire, and you can find an average annual rate of 7%, then you can live on $80,000 a year for 30 years.
• Here are some other scenarios:
• Y=40000, retire at 70, rates of 3%, 5%, and 7%
• 3%: P = 40000*((1.03)^(99-70+1) - 1)/(.03*(1.03)^(99-70)) = 807,538.18.
• 5%: P = 40000*((1.05)^(99-70+1) - 1)/(.05*(1.05)^(99-70)) = 645,642.94.
• 7%: P = 40000*((1.07)^(99-70+1) - 1)/(.07*(1.07)^(99-70)) = 531,106.96. Note that these are all half (they come out exact, but I rounded) of Y=80000. This pattern holds true, so Y=60000 would give you half again as much as 40000.
• Continuing the above scenario for starting at 30 with 5000 in savings, what is A?
• 3%: A = (807538.18 - 5000*(1.03)^(70-30)) / (((1.03)^(70-30) - 1)/.03) = 10,493.56. A little less than half (this isn't just rounding error).
• 5%: A = (645642.94 - 5000*(1.05)^(70-30)) / (((1.05)^(70-30) - 1)/.05) = 5,053.35. 
• 7%: A = (531106.96 - 5000*(1.07)^(70-30)) / (((1.07)^(70-30) - 1)/.07) = 2,285.34. 

These numbers are a hard truth to face. Luckily, what I have shown you is not the full picture. There are significant advantages to reality. For one, if you are investing monthly, then you will make significantly more on interest. Consider the first month of the year. Instead of that money sitting around for a year, it will be earning interest. It might not seem like much, but it adds up. It is significant. Also, if instead of taking out what you need each year, you do so each month, or bimonthly, or even weekly, then the money you don't take out will continue to earn interest as well. These changes will significantly improve your outlook on retirement. I will add a new post as a continuation of this one detailing those changes once I figure out the math. So don't despair as much, especially if you're still young. I will admit that these numbers caused me to start my own business and join the Austin Real Estate Investment Club. Stay tuned for the truth, and feel free to use google calculator and the equations above to test your own ideas about retirement.

Update
So the math is relatively similar:

• P = Retirement Principal.
• M = Monthly salary.
• i1 = Interest Rate during the accrual phase. We will be working with (1+i1/12)
• n1 = (Retirement age - current age) * 12. Number of months until retirement.
• i2 = Interest Rate during the retirement phase.
• n2 = (99 - retirement age) * 12. Number of months you will be retired.
• S = retirement savings you currently have saved up.
• D = Monthly Deposit into retirement savings.
• s_0, s_1, s_k, s_n1 = How much we have in savings after 0, 1, k, or n1 months.
• r_0, r_1, r_k, r_n2 = How much we have in our retirement account after 0, 1, k, or n2 months.

We can work through the same sequence math to solve for P:

1. r_0 = P - M
2. r_1 = r_0*(1+i2/12) - M = P*(1+i2/12) - M*(1 + (1+i2/12))
3. r_2 = P*(1+i2/12)^2 - M*(1 + (1+i2/12) + (1+i2/12)^2)
4. r_n2 = P*(1+i2/12)^n2 - M*((1+i2/12)^(n2+1) - 1)/(i2/12) = 0
5. P = M*((1+i2/12)^(n2+1) - 1) / ((i2/12)*(1+i2/12)^n2)

And also to solve for D:

1. s_0 = S
2. s_1 = s_0*(1+i1/12) + D = S*(1+i1/12) + D
3. s_2 = S*(1+i1/12)^2 + D(1 + (1+i1/12))
4. s_n1 = S*(1+i1/12)^n1 + D((1+i1/12)^n1 - 1)/(i1/12) = P
5. D = (P - S*(1+i1/12)^n1) / (((1+i1/12)^n1 - 1)/(i1/12))

Final Answers (cont.)

Again let's have M = 80000/12 = 6666.67, retiring at 70, and getting 3%, 5%, and 7%:

• 3%: P = 6666.67*((1+.03/12)^((99-70)*12+1) - 1) / ((.03/12)*(1+.03/12)^((99-70)*12)) = 1,554,916.38
• 5%: P = 6666.67*((1+.05/12)^((99-70)*12+1) - 1) / ((.05/12)*(1+.05/12)^((99-70)*12)) = 1,230,222.50
• 7%: P = 6666.67*((1+.07/12)^((99-70)*12+1) - 1) / ((.07/12)*(1+.07/12)^((99-70)*12)) = 998,538.69

Savings to achieve that get's better too:

• 3%: D = (1554916.38 - 5000*(1+.03/12)^((70-30)*12))) / (((1+.03/12)^((70-30)*12) - 1)/(.03/12)) = 1,661.17 per month or 19,934.04 per year.
• 5%: D = (1230222.50 - 5000*(1+.05/12)^((70-30)*12))) / (((1+.05/12)^((70-30)*12) - 1)/(.05/12)) = 782.05 per month or 9 384.60 per year. Compared to 10,398.08, that's a pretty huge difference.
• 7%: D = (998538.69 - 5000*(1+.07/12)^((70-30)*12))) / (((1+.07/12)^((70-30)*12) - 1)/(.07/12)) = 349.35 per month or 4,192.20 per year