Aurora G.

asked • 21d# In perspective, if you won a $25 million lottery, would you take the lump sum prize of $17 million immediately or the annual annuity of $1 million dollars for the next 25 years?

In perspective, if you won a $25 million lottery, would you take the lump sum prize of $17 million immediately or the annual annuity of $1 million dollars for the next 25 years?

## 1 Expert Answer

Dave M. answered • 21d

Business Teacher & 1st Place DECA Advisor

This is a "time value of money" question which requires you to determine the net present value of both options (one of which is an annuity). While philosophically / personally (at age 36) I wouldn't hesitate to take the lump sum today (without even calculating the cash flow streams), the problem becomes harder in situations where Option 2 seems much larger (what if it were $2m per year? $5m?). Besides, the question asker doesn't want your opinion, it wants the financial breakdown. We want to get the question "right", so let's take a look at the math...

Option 1: At face value, you get $17M and are free to spend it, give it away, burn it, invest it, etc. however you like.

Option 2: Although $1M per year for each of the next 25 years in theory is $25M, which is $8M more than Option 1, it's not so simple when you consider inflation and the opportunity cost of having to wait for the rest of your money. Inflation is easiest defined as the reduction in the purchasing power of your dollar due to economic growth ("when I was your age, a soda was a nickel"), typically 2-4%, and opportunity cost is like FOMO (fear of missing out) in that you are giving up something to choose this option (you can't both stay in and go to the party, so you give up one of those as your opportunity cost. Financially, you can't take the $17M from option 1 and invest it in the stock market hoping for 10% gains, nor can you buy something for more than your Option 2 amount and enjoy the intangibles that go with it).

When you receive a sum of money again and again over time, this is called an annuity. There are two types, which the problem doesn't clarify-- whether you get the money at the start of the period (here, year) or the end (you'd rather pay your rent on the 30th but I'm sure your landlord would prefer it due on thirty days earlier on the 1st). This affects the compounding potential as you need to know if we're considering 25 years or 24. When payments are made at the end of the period (here, year), it's called an ordinary annuity. When made at the beginning, it's called an annuity due (eg, rent due on the 1st-- whether it's you paying it or the landlord receiving it). Even if the two options were $25M now or $1M for 25 years, because of the investment potential and opportunity to "buy soda for a nickel" of today's stronger dollar / higher purchasing power compared to money 25 years from now, the lump sum $25M would be worth more than the annuity.

The formula used for calculating the present value of an annuity due (we're making the assumption that you can get $17M today OR $1M today, and therefore the assumption that it's an annuity due) is below. In it, PV stands for "present value" (what would this cash flow stream be worth today-- to compare to Option 1); PMT stands for "payment" (here, the amount of the annual annuity payment of $1M); r stands for "interest rate per period" (the rate we will use is called a "discount rate" and is whatever rate you want to use that best represents your realistic expectations for investing the $17M from Option 1. If you think you can do 10% every year, go for it. If you think you can only invest in US Treasury bills at 4%, use that; note, a T-Bill is very "safe" and considered a "risk free rate", and is a good starting point for our problem because of course we hope to get 10%, but we don't want greed and starry eyes to cloud our financial decision making judgement); n is the number of periods we will be compounding this interest rate. If your rate is annual, just count the years. If it's a monthly rate compounding (like in a typical checking account), your number of periods for 25 years would be 25 x 12 months, and your rate would be the annual rate divided by 12 (eg, 12% per year / 12 months = 1% per month). Ready?

If you received one payment one year from now, you would discount it back by the chosen rate using the following formula:

PV = PMT / ((1+r)^n) and use n = 1 year (see below for more on this)

You would then consider the payment received two years from now, and change your n to 2 years, then add it to the 1 year, plus the "zero year" $1M received today. (Because you get paid $1M today, it does not need to be discounted back any for our present value calculation-- it's already today.) For the third year, n = 3, and so on. Adding all of these together is inefficient, so use the following formula that can be used for any number of periods. That formula is:

PV of annuity due = PMT * ((1-(1+r)^ (-n)) / r) * (1+r)

Ok, let's list what we know:

PV= ? (goal)

PMT = $1,000,000

n= 25 years

r = You need to consult a rate table to find the current treasury bill rate (or pick your own if prompted or have some other reason for doing so). Today 9/30/19 shows a 1.94% annual yield (rate) for a 20-year term and 2.12% for a 30-year term. Let's just pick 2% because it's an easy number in the middle, and is typical of historic inflation. Note well that if you were to invest your money at 2% over time, and inflation reduced your dollar's spending power by 2% over that same time, you have erased your investment gains. This is why you want to have a higher rate of return than inflation, and why 99% of the time just leaving your money in a low-yield checking or money market account-- or worse, under the mattress (does anyone actually do that?)-- you are actually losing money the longer it sits there and inflation takes its "worth" / ability to buy goods away 2% per year. So, our rate (r) is 2%, or as a decimal 0.02 . Since our "n" is annual, and our rate is annual, we don't need to divide by 12 months or some other time period (quarterly, etc.).

Then it's just a matter of plugging in the values to our formula, simplifying, and calculating (remember PEMDAS!). Also of importance, try to avoid rounding until the very end, as this isn't "just a math problem", it's money, and lost decimals means lost dollars! Use the memory function (M+) on your calculator to store denominators, then recall them back (MR). Better yet, use a spreadsheet and link to your list of variables. Then you can change the rate and periods to see what difference time, money, and rates can make on your decisions. You can even use a Solver or Goal Seek function (or "guess and check") to actually determine the breakeven point between the two options (how long, at what rate, how much per month, etc.) to see what it would take to make you change your mind. Remember, the question asker doesn't want your "what would you do with a million dollars" personal opinion, they want the math.

PV= PMT * ((1-(1+r)^ (-n)) / r) * (1+r)

PV= 1,000,000 * ((1-(1.02)^(-25)) / 0.02) * (1.02)

PV= $19,913,925.60

(round up to $19,913,926)

So......

This means that using a 2% "discount rate", your Option 2 is actually not "worth" $25M in today's dollars, but only $19,913,926 (a difference of $5,086,074).

However, compared to Option 1's $17M lump sum, Option 2 is worth $2,913,926 MORE.

So, quantitatively, you have to ask yourself if you think you can achieve a better return on the $17M lump sum than the treasury bill rate (very likely considering the stock market's historic rate of return can be closer to 10% than 2%), if you could invest the money into a business and really multiply it, etc.. Qualitatively, you would need to consider factors like your age/mortality, immediate needs, and all the rest. Is it worth waiting 25 years to get that "extra" $2.9M, excluding any better investment opportunities? For me, not at all. I'd take the $17M and run. For a millionaire who won't notice this "extra" lump sum in their back account? They'd likely take Option 2 and sign over the annuity to their trust/heirs because it is worth more in hands-free investing terms.

Hope this helps!

## Still looking for help? Get the right answer, fast.

Get a free answer to a quick problem.

Most questions answered within 4 hours.

#### OR

Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.

Rick R.

21d