My Prius

I own a 2006 Prius. In my actual usage of the car, it gets just about 45 miles per gallon. We are going to estimate how much money I save (if any).

But before we do that, note that actual usage can be different than the published EPA mileage numbers. For example, for the 2006 Prius, the EPA mileage for city driving is 60 miles per gallon, for highway driving is 51 miles per gallon, and for combined driving is 55 miles per gallon. In the computations below, we will use the actual miles per gallon that I get, which is 45 miles per gallon.

So how much money do I save on gas? Let’s compare my car with

1. Car A that (actually) gets 30 miles to the gallon and

2. Car B that (actually) gets 15 miles to the gallon.

Assume first that all three cars go 10,000 miles and gasoline costs $3.00 a gallon.

(Since I wrote the first draft of this post, the price of gasoline has started to come down. So I am going to include the results for $2.00 gallon gasoline in parentheses after the results for $3.00 per gallon gasoline.)

1. At 45 miles per gallon, my Prius uses 10,000 / 45 = 222.22 gallons. Since gas costs $3 per gallon, my cost is 222.22 * 3 = $666.66 (at $2.00 per gallon my cost is $444.44)

2. At 30 miles per gallon, Car A uses 10,000 / 30 = 333.33 gallons at a cost of $1,000 (a cost of $666.66)

3. At 15 miles per gallon Car B uses 666.67 gallons at a cost of $2,000 (a cost of $1,333.33)

So when I drive 10,000 miles,

1. Compared to Car A, I save $333.33 (I save $222.22)

2. Compared to Car B, I save $1,333.33 (I save $888.89)

Stated differently

1. Compared to Car A, I save 3 1/3 cents per mile (I save 2.22 cents per mile)

2. Compared to Car B, I save 13 1/3 cents per mile (I save 8.88 cents per mile)

Suppose I keep my Prius for 150,000 miles, which is the mileage my last Toyota had when I donated it to a charitable organization. My total savings over the life of my car would be

1. Compared with Car A, I would save $0.03333 * 150,000 = $5,000 (I would save $3,333)

2. Compared with Car B, I would save $0.13333 * 150,000 = $20.000 (I would save $13,333)

But did I really save anything when considering the extra cost of my car. After all, I certainly paid something extra for the hybrid mechanism. Taking that into account, did I save any money? It’s hard to determine how much I paid for the hybrid mechanism because Toyota doesn’t make a non-hybrid Prius. But they do make both a hybrid and a non-hybrid Camry. The base price for the 2007 hybrid Camry is $25,900, and the base price for the 2007 non-hybrid Camry is $18,445. Thus the base price for the non-hybrid Camry is 71.2% of the base price of the hybrid Camry. Using that same ratio and the base price of the (hybrid) Prius, which is $22,175, the base price of a non-hybrid Prius would be $15,788. Thus the extra cost I paid for the hybrid mechanism would be $6,387.

Hence, over the 150,000 mile estimated life of my Prius,

1. Compared with Car A (with mileage of 30 miles to the gallon), I would lose $1,387 (I would lose $3,054)

2. Compared with Car B (with mileage of 15 miles per gallon), I would gain $13,613 (I would gain $6,946)

However, I got a $3,145 income tax rebate for buying a 2006 Prius. Factoring that in

1. Compared with Car A, I would gain $1,758 (I would gain $91)

2. Compared with Car B, I would gain $16,758 (I would gain $10,071)

Suppose we consider a different car, Car C. How many miles per gallon would Car C have to get so that I would exactly break even after going 150,000 miles considering the cost of the hybrid mechanism (and ignoring my tax rebate)? A little math shows the answer to be about 27.5 miles per gallon (18.3 miles per gallon).

Now let’s consider a different question: what would gasoline have to cost so that the drivers of Car A and Car B would spend the same amount of money on gasoline to go 10,000 miles as I do in my Prius, $666.66 ($444.44). (Of course, the same price of gasoline would apply no matter how many miles we used in the computation.)

1. Since Car A used 333.33 gallons, gasoline would have to cost 666.66 / 333.33 = $2 per gallon ($1.33 per gallon)

2. Since Car B used 666.67 gallons, gasoline would have to cost 666.66 / 666.67 = $1 per gallon ($0.67 per gallon)

Unfortunately, I can remember when gasoline could be bought for all of these prices (and even cheaper).

Roth IRA

The Roth IRA, named after Senator William Roth of Delaware, came into existence in 1997. The conventional IRA came into existence in 1974. The Roth IRA is widely considered to have significant advantages over the conventional IRA. We compare the two types of IRAs in a number of ways and then summarize the advantages of the Roth IRA. The results might be somewhat surprising.

Differences in How the Roth and Conventional IRA are Taxed

Roth IRAs differs from conventional IRAs is in the way they are taxed:

1. In a conventional IRA, you do not pay any income tax on the money you put into the IRA. When you take money out of the IRA, you pay income tax on the money at your usual rate.

2. In a Roth IRA, you pay income tax on the money you put into the IRA. When you take money out of the IRA, you do not pay income tax on it.

Let’s see what effect this difference has on the amount of money you can take out of the IRA in later years. Consider a simple example. Suppose you decide that you can allocate $4,000 from your salary for either a conventional or a Roth IRA. Suppose that over the years the money you put into the IRA grows by a factor of 10. Suppose you are in the 25% tax bracket over the entire interval of the IRA

1. In a conventional IRA, the $4,000 grows to $40,000. When you take the $40,000 out of the IRA, you pay $10,000 tax on it, so you are left with $30,000

2. In a Roth IRA, before you put the $4,000 into the IRA, you must pay $1,000 tax on it, so you actually put only $3,000 into the IRA. Then it grows by a factor of 10 to $30,000 and you can take the $30,000 out of the IRA tax-free.

So you end up with the same amount of money in either case.

However, you might be in a different tax bracket when you retire than when you put the money in. If the retirement tax bracket is higher, the Roth IRA is better. If it is lower, the conventional IRA is better.

Difference in the Amount of Money You Can “Invest” in Roth and Conventional IRAs

You are allowed to “invest” more money in a Roth IRA than in a regular IRA, so its final value is higher.

That statement might not be clear from the way the regulations are written. The regulations say that (in 2006 and 2007) you can invest up to $4,000 a year into either a Roth or a conventional IRA. But the $4,000 you invest in a Roth IRA is after taxes. So if you are in the 25% tax bracket, you are really allocating $5,333 to the IRA. $1,333 goes to the government for taxes, and $4,000 goes into the IRA.

You are essentially being allowed to “invest” 33.33 % more in a Roth IRA than in a conventional IRA. So other things being equal, you will end up with 33.33% more in the Roth IRA than in a conventional IRA. In our example, where the amount in the IRA increases by a factor of 10:

1. The conventional IRA would grow to $40, 000. You would pay $10,000 in taxes and be left with $30,000

2. The Roth IRA would also grow to $40,000, which you could take out tax-free.

If you had been allowed to invest the entire $5,333 in the conventional IRA, you would also have ended up with $40,000.

Suppose you had $5,333 to invest, but since you were allowed to put only $4,000 in the conventional IRA, you invested the extra $1,333 in a taxable investment. You would first have to pay 25% income tax on the $1,333, leaving you with $1,000. You invest this $1000 in a taxable investment that also grows by a factor of 10, giving you $10,000. You have to pay 15% capital gains tax on the $9,000 profit, leaving you with $8,750. Adding that to the $30,000 you take out of the conventional IRA leaves with with a total of $38,750, compared with the $40,000 you would have with the Roth IRA

Thus, it is a significant advantage of the Roth IRA that you can “invest” more money in it.

The advantage depends on your tax bracket. Suppose you are in the x tax bracket, where x is expressed as a decimal (in the above example, x = .25). Suppose nc is the nominal Roth IRA contribution you are allowed (in the above example nc = $4,000). The actual contribution you are making, ac, ($5,333 in the example) is related to nc by the equation

ac – (x * ac) = nc

which can be rewritten as

ac = nc / (1 – x)

Hence the actual contribution is 1 / (1 – x) times the nominal contribution, and thus the amount you withdraw from the Roth IRA is 1 / (1 – x) times the amount you would withdraw from a conventional IRA with the same nominal contribution. Thus if x = .25 as in the example, then the amount you withdraw from the Roth IRA is 1 / .75 = 1.33 times the amount you would withdraw from a conventional IRA. Call this number the multiple. If x = .1 (the minimum non-zero tax rate), then the multiple is 1/ .9 = 1.11. If x = .35 (the maximum tax rate), the multiple is 1 / .65 = 1.54.

Note, however, that there are income limitations for eligibility to make contributions to a Roth IRA: $95,000 for single taxpayers and $150,000 for married taxpayers filing jointly. Thus not all people in the higher tax brackets are eligible to contribute to a Roth IRA.

Sidenote: You might think that the government is worse off with the Roth IRA than with the conventional IRA because, in our first example, they collect only $1,333 in taxes from the Roth IRA (when the money is invested), while they collect $10,000 in taxes from the conventional IRA (at a later time when the money is withdrawn). But if the government had invested the $1,333 they collected from the Roth IRA in some investment that returned a factor of 10, they would have $13,330 (of course tax free) at the time they collected the $10,000 from the conventional IRA.

Converting a Conventional IRA into a Roth IRA

Under certain conditions a conventional IRA can be converted into a Roth IRA. You have to pay income tax on the current value of the conventional IRA, but then you do not have to pay any income tax when you withdraw money (at a later date) from the Roth IRA.

Suppose we consider converting our conventional IRA that has grown to $40,000. Suppose that after we convert it (or do not convert it) it grows by another factor of 10.

1. If we did not convert it, it is now worth $400,000. If we take that $400.000 out of the IRA, we have to pay $100,000 in taxes, so we are left with $300,000

2. If we convert it, we sell the $40,000, pay $10,000 in taxes, and invest $30,000 in the Roth IRA. It then grows by a factor of 10 to $300,000. We can then take that $300,000 out tax-free.

Again we end up with the same amount of money in either case.

However, if you think the tax rate it going to increase over the coming years, you can convert your conventional IRA into a Roth IRA now while the tax rate is low. On the other hand, if you think your income and hence your tax rate will be lower when you retire and withdraw money from your IRA, then you might not want to convert.

Also, as we discuss below, you do not have to take money out of a Roth IRA when you are 70 1 / 2 as you do with a conventional IRA, so the money has an opportunity to grow tax free for a longer period of time. This might be a significant advantage of converting a conventional IRA into a Roth IRA.

Taking Money Out of Roth IRAs and Conventional IRAs

Since you have already paid income tax on your investment in a Roth IRA, the rules for withdrawing money are much more lenient.

1. You do not have to start taking money out of a Roth IRA when you are 70 1 /2 (as you do with a conventional IRA) so your money can grow tax-free for a longer period of time. This is a significant advantage.

2. There are more situations in which you can take money out of your Roth IRA without any penalty before you are 59 1 / 2. than there are for a conventional IRA. The rules are complicated and we do not discuss them

3. If, when you die, you leave the Roth IRA to some beneficiary, that beneficiary does not have to pay any income tax when taking distributions out of the IRA. In the conventional IRA, beneficiaries have to pay income tax on all distributions. Again, the rules are complicated and we do not discuss them.

Summary of the Main Advantages of a Roth IRA Compared with a Conventional IRA

Based on our previous discussion, there are two significant advantages of Roth IRAs compared with conventional IRAs.

1. You are allowed to “invest” more money in a Roth IRA than in a conventional IRA, and hence its final value is higher

.

2. You do not have to start withdrawing money from a Roth IRA when you get to be 70 1 / 2 as you do with a conventional IRA so the money can grow tax free for a longer period of time.