Notes for homework Marie Holmes, a twenty seven year old mother of four children, won $188 million in the North Carolina Powerball As a winner she had two choices take the $188 million in thirty annual installments of roughly $6 27 million or a lump sum payment of $127 million Ms Holmes took the $127 million lump sum payment Why If you have $100 today, you can earn interest on that money And so, one year from now you'll have more than $100 Therefore, $100 one year from now is worth less to you than $100 today Perhaps Ms Holmes believed that investing $127 million today would, with interest, produce more than $6 27 million per year in income over the thirty years We'll see if she's right further on Interest Rates Decimals and Percentages Jessie borrowed $850 from his sister She asked that he repay her the $850 in one year plus give her $51 in interest What interest rate is she charging Jessie Interest Rate Interest Payment Amount Borrowed (or the principal ) Interest Rate $51 $850 0 06 Calculated interest will result in a decimal solution You should become comfortable converting back and forth between percentages and decimals In this case, 0 06 corresponds to 6 0 17 would be 17 And 1 50 would be 150 Percentage ( ) Decimal Value x 100 Future Value Simple Interest Kane has $1,000 in his savings account It will pay him 3 interest over one year How much will he earn in interest over the one year Interest Earned Principal Interest Rate (expressed as a decimal) Interest Earned $1,000 0 03 $30 How much will he have on hand after one year The thirty dollars in interest plus the original $1000, or $1,030 Future Value Principal (Principal Interest Rate) Rearranging Future Value (Principal 1) (Principal Interest Rate) Future Value Principal (1 Interest Rate) Future Value $1,000 (1 03) Future Value $1,000 1 03 $1,030 Future Value Compound Interest What if Kane left his $1,000 in the account for two years Both his principal and interest from the first year would earn interest the second year Future Value After Two Years $1,030 (1 03) $1,060 90 Future Value After Two Years Future Value After One Year (1 Interest Rate) Future Value After Two Years Principal (1 Interest Rate) (1 Interest Rate) Future Value After Two Years Principal (1 Interest Rate) 2 Future Value After Two Years $1,000 (1 03) 2 Future Value After Two Years $1,000 1 0609 Future Value After Two Years $1,060 90 Let's take a moment to review order of operation First, whatever is in parentheses takes place before what is outside parentheses And so, we added the 1 and the 0 03 before doing anything else Second, if a term is raised to a power (the exponent), next Our next step was to square 1 03 1 03 1 03 1 0609 Third, multiplication or division takes place before addition or subtraction Our last step was to multiply $1,000 by the 1 0609 There was no addition outside the parentheses and so we were done And finally, when faced with the same operation, move from left to right For example, 6 2 5 15, not 3 5 Future Value General Formula You should be able to figure out how the problem would change if Kane left his money in the account for three years, four years, and so on The following is a general formula that applies to any length of time, where the letter n represents the number of periods (years in our case) Future Value After n Years Principal (1 Interest Rate) n Chi puts $1,000 in an account paying 6 for five years After six years she'll have Future Value After 5 Years $1,000 (1 06) 5 $1,338 As a final note, we are using annual interest rates and assuming that interest is paid annually But this isn't usually true Interest is often compounded monthly or quarterly The conversion isn't hard In the case of monthly compounding, you'd take the annual interest rate and divide by twelve Then use this for the Interest Rate in the problem And you'd use the number of months instead of number of years for n So, if Chi's 6 annual interest is paid monthly, we'd use 0 005 instead of 0 06 (check this) and n 60 instead of five Future Value After 5 Years $1,000 (1 005) 60 $1,348 Present Value Discounting Some investments pay a fixed lump sum on a designated date in the future As an investor, you earn interest by purchasing this investment for less than that lump sum (or at discount ) The difference is your interest For example, General Electric might issue commercial paper that promises to pay $1,000 in exactly one year You would need to decide how much you're willing to pay for it For sake of example, let's say you could earn 5 in an equally risky bank account To solve this problem, we'll start by setting it up as a simple Future Value problem Future Value Principal (1 Interest Rate) n In our case, we know the Future Value is $1,000 and n 1 If you don't buy this investment, you can take the same money and invest it at 5 in the bank account So, we want to earn at least 5 We'll use this as our Interest Rate What remains is the principal We'll substitute the term price for principal since this will be how much we'll pay Let's solve for this $1,000 Price 1 05 Price $1,000 1 05 $952 38 If I take $952 38 and put in the bank account and earn 5 interest over the year, I'll receive back $1,000 And so, what am I willing to pay for the General Electric commercial paper that will pay me $1,000 in one year $952 38 The hardest thing about this problem is determining the appropriate interest rate referred to as the discount rate for present value problems For example, what if you had two choices for what to do with some money you have on hand Your first choice is to invest it in virtually risk free commercial paper that promises to pay $1,000 in one year Your second choice is to lend it to your friend's brother who's opening a comic bookstore and who also promises to pay you $1,000 in one year Seems you might not be willing to pay as much for the second opportunity as for the first Or equivalently, you want a higher rate of return from the comic bookstore to compensate you for the added risk And so, in determining the discount rate to use, you need to decide first what rate of return seems sufficient to compensate you for both the hassle of waiting for your money plus the risk of never seeing it again Now, let's set up a general formula based on our example Present Value Future Value (1 Interest Rate) But we already know what to do if we're looking at receiving the money two, three, or n years later instead Present Value Future Value (1 Interest Rate) n The lottery commission promises to pay you $6 27 million one year from now How much would you be willing to pay for that today We can, but first we need to choose a discount rate You believe that you could invest your proceeds and earn 5 per year We'll use that as the discount rate Present Value $6 27 million 1 05 $5 98 million What about a second payment that would take place in two years Present Value $6 27 million 1 01 2 $5 69 million If you continued this for each of the thirty years and then added them together, you'd come up with approximately $96 4 million Present Value $6 27 million 1 05 1 $6 27 million 1 05 2 $6 27 million 1 05 30 Present Value $96 4 million approximately So, the string of thirty annual payments is worth only $96 4 million today By this we mean that investing $96 4 million at five percent would yield more than $6 27 million per year over the thirty years Perhaps now we understand why Ms Holmes accepted the $127 million lump sum payment She was convinced that she would end up with more than the $188 million prize amount Similarly, what am I willing to pay for an apartment complex that will generate rent each month What about a cargo ship that'll generate freight charges Or a solar array for my roof that will save me utility charges each month These are all Present Value problems and you now know how to solve them Future Value of an Annuity An annuity is an identical Dollar amount that is either invested or received each of multiple periods Your car payment is an annuity So also, hopefully, is your retirement savings Had Ms Holmes from section 6 1 accepted the lottery commission's offer to receive $6 27 million every year for thirty years, this too would have been an annuity It's easy enough to calculate the value of annuity with the tools already at hand You just need patience or to know how to use a computer spreadsheet An annuity is simply a series of Future Value (or Present Value) problems added together For example, if I save $6,000 per year for retirement every year for 30 years, how much will I have Well, first we need an interest rate Some suggest the one time market average of 10 , but let's be realistic and cut that in half and assume 5 The first payment will compound for 30 years The second for only 29 years And so on Payment 1 FV $6,000 (1 05) 30 $25,931 65 Payment 2 FV $6,000 (1 05) 29 $24,696 81 Payment 30 FV $6,000 (1 05) 1 $6,300 Grand Total FVA $418,564 Of course, you don't need to solve this the hard way There's a simple formula you can use But before that, an important caveat There are two types of annuities normal annuities and annuities due The difference is when the first payment is made Is at the beginning of the first year or the end of that year Above I added interest to every payment That's because I assumed that each was paid at the beginning of the year This is called an annuity due Otherwise, the money would compound one less period and the first exponent would have been 29 instead of 30 And the last 0 instead of one Here is the Annuity Due formula Think Order of Operation FVA Annuity Due Payment (1 Interest Rate) n 1 (1 Interest Rate) Interest Rate FVA $6,000 (1 05 30 1) (1 05) 0 05 $418,564 If you delay making your first contribution to the end of the first year, you have a normal annuity You'll earn a little less because your money will compound one less year FVA Normal Annuity Payment (1 Interest Rate) n 1 Interest Rate FVA $6,000 (1 05 30 1) 0 05 $398,633 08 Present Value of an Annuity So, the Lottery Commission offered Ms Holmes an annual annuity as one option How much would $6 27 million per year for thirty years be worth today Or equivalently, how much invested today would yield $6 27 million over thirty years Again, we already have the necessary tools But we do need to make one assumption and one decision First, is this an Annuity Due or a Normal Annuity I haven't won Powerball, but my guess is that the winner doesn't have to wait one year for their first payment So, it's also an Annuity Due The decision is what interest rate to use This time let's try a very conservative one percent Payment 1 Present Value $6 27 million (1 01) 0 $6 27 million Payment 2 Present Value $6 27 million (1 01) 1 $6 21 million Payment 30 Present Value $6 27 million (1 01) 29 $4 71 million Grand Total Present Value of the Annuity $163 43 million With the more conservative one percent, we would be indifferent between $6 27 million per year for thirty years or a lump sum of $163 43 million up front At one percent, the $127 million she did accept doesn't seem such a good deal This is why the discount rate you select is so relevant You can easily enough with a spreadsheet But there is always a formula We'll use i for the Interest Rate Again, order of operation PVA Annuity Due Payment (1 i) (1 (i (1 i) n )) (1 i) PVA $6 27 million (1 0 01 (1 (0 01 (1 01) 30 )) (1 01) $163 43 million That covers the Annuity Due And what about a Normal Annuity PVA Normal Annuity Payment (1 i) (1 (i (1 i) n )) Your brother in law asks you to lend him $10,000 so he can open a pawn shop He promises to pay you $1,000 at the end of each year for the next twenty years, effectively doubling your money he says Should you make the investment Your experience with your brother in law leads you to decide a discount (interest) rate of 15 is in order PVA Normal Annuity $1,000 (1 0 15) (1 (0 15 (1 0 15) 20 )) $6,259 You politely tell him that you're a bit short on cash, but thanks so much for the opportunity Here's another useful application of the formula You plan to buy a $250,000 house with $50,000 down and $200,000 mortgage The annual interest rate is 6 and you'll take out a thirty year mortgage First, payments are due monthly rather than annually So, we need to convert to a monthly interest rate and then determine the number of periods That'd be 6 12 or 0 5 (0 005 as a decimal) and 360 months In the case of a mortgage, the first payment isn't due until month end So, this is a Normal Annuity $200,000 Payment (1 0 005) 1 (0 005 1 005 360 ) $200,000 Payment 166 8 Payment $200,000 166 8 $1,199 Annuity Formulas Personally, I still do these problems using a spreadsheet Mostly because I want to easily do what if scenarios What if I contribute an extra $1,000 for the next three months But the formulas, once you're reliable with order of operation, are much faster Here are all four together Homework Q1 You (hypothetically) plan to retire at the age of 72 Assume you'll live another 25 years beyond that just to be safe How much will you need to have saved at 72 if you want to spend $100,000 per year for the next 25 years I want you to set the problem up as you would in a spreadsheet and provide calculations for the first three years You'll need the money right away So, assume an annuity due And for simplicity, use 5 as your discount rate Show all your work Q2 Now use the Present Value of an Annuity formula to provide the complete calculation to the problem presented in Question 1 Again, assume an annuity due and a 5 discount rate Q3 You have forty work years before you hit 72, hypothetically How much will you need to save MONTHLY to reach the saving target above This time simply use the Future Value of an Annuity formula In this case, you'll have to save up the money first So, assume an ordinary annuity Again use 5 as your compounding rate

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Notes for homework: Marie Holmes, a twenty-seven-year-old mother of four children, won $188 million in the North Carolina Powerball. As a winner she had two

Notes for homework:

Marie Holmes, a twenty-seven-year-old mother of four children, won $188 million in the North Carolina Powerball. As a winner she had two choices: take the $188 million in thirty annual installments of roughly $6.27 million or a lump sum payment of $127 million. Ms. Holmes took the $127 million lump sum payment. Why?

If you have $100 today, you can earn interest on that money. And so, one year from now you'll have more than $100. Therefore, $100 one year from now is worth less to you than $100 today. Perhaps Ms. Holmes believed that investing $127 million today would, with interest, produce more than $6.27 million per year in income over the thirty years. We'll see if she's right further on.

Interest Rates: Decimals and Percentages

Jessie borrowed $850 from his sister. She asked that he repay her the $850 in one year plus give her $51 in interest. What interest rate is she charging Jessie?

Interest Rate = Interest Payment / Amount Borrowed (or the "principal")

Interest Rate = $51 / $850 = 0.06

Calculated interest will result in a decimal solution. You should become comfortable converting back and forth between percentages and decimals. In this case, 0.06 corresponds to 6%. 0.17 would be 17%. And 1.50 would be 150%.

Percentage (%) = Decimal Value x 100

Future Value - Simple Interest

Kane has $1,000 in his savings account. It will pay him 3% interest over one year. How much will he earn in interest over the one year?

Interest Earned = Principal * Interest Rate (expressed as a decimal)

Interest Earned = $1,000 * 0.03 = $30

How much will he have on hand after one year? The thirty dollars in interest plus the original $1000, or $1,030.

Future Value = Principal + (Principal * Interest Rate)

Rearranging...

Future Value = (Principal * 1) + (Principal * Interest Rate)

Future Value = Principal * (1 + Interest Rate)

Future Value = $1,000 * (1 + .03)

Future Value = $1,000 * 1.03 = $1,030

Future Value - Compound Interest

What if Kane left his $1,000 in the account for two years? Both his principal and interest from the first year would earn interest the second year.

Future Value _{After Two Years} = $1,030 * (1.03) = $1,060.90

Future Value _{After Two Years} = Future Value _{After One Year} * (1 + Interest Rate)

Future Value _{After Two Years} = Principal * (1 + Interest Rate) * (1 + Interest Rate)

Future Value _{After Two Years} = Principal * (1 + Interest Rate)²

Future Value _{After Two Years} = $1,000 * (1.03)²

Future Value _{After Two Years} = $1,000 * 1.0609

Future Value _{After Two Years} = $1,060.90

Let's take a moment to review order of operation.

First, whatever is in parentheses takes place before what is outside parentheses. And so, we added the 1 and the 0.03 before doing anything else.

Second, if a term is raised to a power (the exponent), next. Our next step was to square 1.03. 1.03 * 1.03 = 1.0609

Third, multiplication or division takes place before addition or subtraction. Our last step was to multiply $1,000 by the 1.0609. There was no addition outside the parentheses and so we were done.

And finally, when faced with the same operation, move from left to right. For example, 6 / 2 * 5 = 15, not 3/5.

Future Value - General Formula

You should be able to figure out how the problem would change if Kane left his money in the account for three years, four years, and so on. The following is a general formula that applies to any length of time, where the letter n represents the number of periods (years in our case).

Future Value _{After n Years} = Principal * (1 + Interest Rate)ⁿ

Chi puts $1,000 in an account paying 6% for five years. After six years she'll have

Future Value _{After 5 Years} = $1,000 * (1 + .06)⁵ = $1,338

As a final note, we are using annual interest rates and assuming that interest is paid annually. But this isn't usually true. Interest is often compounded monthly or quarterly. The conversion isn't hard. In the case of monthly compounding, you'd take the annual interest rate and divide by twelve. Then use this for the Interest Rate in the problem. And you'd use the number of months instead of number of years for n.

So, if Chi's 6% annual interest is paid monthly, we'd use 0.005 instead of 0.06 (check this) and n = 60 instead of five.

Future Value _{After 5 Years} = $1,000 * (1 + .005)⁶⁰ = $1,348

Present Value - Discounting

Some investments pay a fixed lump sum on a designated date in the future. As an investor, you earn interest by "purchasing" this investment for less than that lump sum (or "at discount"). The difference is your interest. For example, General Electric might issue "commercial paper" that promises to pay $1,000 in exactly one year. You would need to decide how much you're willing to pay for it. For sake of example, let's say you could earn 5% in an equally risky bank account.

To solve this problem, we'll start by setting it up as a simple Future Value problem:

Future Value = Principal * (1 + Interest Rate)ⁿ

In our case, we know the Future Value is $1,000 and n = 1. If you don't buy this investment, you can take the same money and invest it at 5% in the bank account. So, we want to earn at least 5%. We'll use this as our Interest Rate. What remains is the principal. We'll substitute the term "price" for "principal" since this will be how much we'll pay. Let's solve for this:

$1,000 = Price * 1.05

Price = $1,000 / 1.05 = $952.38

If I take $952.38 and put in the bank account and earn 5% interest over the year, I'll receive back $1,000. And so, what am I willing to pay for the General Electric commercial paper that will pay me $1,000 in one year? $952.38

The hardest thing about this problem is determining the appropriate interest rate - referred to as the discount rate for present value problems. For example, what if you had two choices for what to do with some money you have on hand. Your first choice is to invest it in virtually risk-free commercial paper that promises to pay $1,000 in one year. Your second choice is to lend it to your friend's brother who's opening a comic bookstore and who also promises to pay you $1,000 in one year. Seems you might not be willing to pay as much for the second opportunity as for the first. Or equivalently, you want a higher rate of return from the comic bookstore to compensate you for the added risk. And so, in determining the discount rate to use, you need to decide first what rate of return seems sufficient to compensate you for both the hassle of waiting for your money plus the risk of never seeing it again.

Now, let's set up a general formula based on our example:

Present Value = Future Value / (1 + Interest Rate)

But we already know what to do if we're looking at receiving the money two, three, or n years later instead.

Present Value = Future Value / (1 + Interest Rate)ⁿ

The lottery commission promises to pay you $6.27 million one year from now. How much would you be willing to pay for that today? We can, but first we need to choose a discount rate. You believe that you could invest your proceeds and earn 5% per year. We'll use that as the discount rate.

Present Value = $6.27 million / 1.05 = $5.98 million

What about a second payment that would take place in two years?

Present Value = $6.27 million / 1.01² = $5.69 million

If you continued this for each of the thirty years and then added them together, you'd come up with approximately $96.4 million.

Present Value = $6.27 million/1.05¹ + $6.27 million/1.05² +...+ $6.27 million/1.05³⁰

Present Value = $96.4 million approximately

So, the string of thirty annual payments is worth only $96.4 million today. By this we mean that investing $96.4 million at five percent would yield more than $6.27 million per year over the thirty years. Perhaps now we understand why Ms. Holmes accepted the $127 million lump sum payment. She was convinced that she would end up with more than the $188 million prize amount.

Similarly, what am I willing to pay for an apartment complex that will generate rent each month? What about a cargo ship that'll generate freight charges? Or a solar array for my roof that will save me utility charges each month? These are all Present Value problems and you now know how to solve them.

Future Value of an Annuity

An annuity is an identical Dollar amount that is either invested or received each of multiple periods. Your car payment is an annuity. So also, hopefully, is your retirement savings. Had Ms. Holmes from section 6 1 accepted the lottery commission's offer to receive $6.27 million every year for thirty years, this too would have been an annuity.

It's easy enough to calculate the value of annuity with the tools already at hand. You just need patience or to know how to use a computer spreadsheet. An annuity is simply a series of Future Value (or Present Value) problems added together.

For example, if I save $6,000 per year for retirement every year for 30 years, how much will I have? Well, first we need an interest rate. Some suggest the one-time market average of 10%, but let's be realistic and cut that in half and assume 5%. The first payment will compound for 30 years. The second for only 29 years. And so on.

Payment 1 FV = $6,000 (1.05)³⁰= $25,931.65 + Payment 2 FV = $6,000 (1.05)²⁹= $24,696.81 .... + Payment 30 FV = $6,000 (1.05)¹= $6,300 Grand Total FVA = $418,564

Of course, you don't need to solve this the hard way. There's a "simple" formula you can use. But before that, an important caveat. There are two types of annuities:normal annuitiesandannuities due. The difference is when the first payment is made. Is at the beginning of the first year or the end of that year? Above I added interest to every payment. That's because I assumed that each was paid at the beginning of the year. This is called an annuity due. Otherwise, the money would compound one less period and the first exponent would have been 29 instead of 30. And the last 0 instead of one. Here is the Annuity Due formula. ThinkOrder of Operation!

FVA_{Annuity Due}= Payment * [(1 + Interest Rate)ⁿ- 1]*(1 + Interest Rate) / Interest Rate FVA = $6,000 *(1.05³⁰- 1) * (1.05) / 0.05 = $418,564

If you delay making your first contribution to the end of the first year, you have anormal annuity. You'll earn a little less because your money will compound one less year.

FVA_{Normal Annuity}= Payment * [(1 + Interest Rate)ⁿ- 1] / Interest Rate FVA = $6,000 * (1.05³⁰- 1) / 0.05 = $398,633.08

Present Value of an Annuity

So, the Lottery Commission offered Ms. Holmes an annual annuity as one option. How much would $6.27 million per year for thirty years be worth today? Or equivalently, how much invested today would yield $6.27 million over thirty years? Again, we already have the necessary tools. But we do need to make one assumption and one decision. First, is this an Annuity Due or a Normal Annuity? I haven't won Powerball, but my guess is that the winner doesn't have to wait one year for their first payment. So, it's also an Annuity Due. The decision is what interest rate to use. This time let's try a very conservative one percent.

Payment 1 Present Value = $6.27 million / (1.01)⁰= $6.27 million Payment 2 Present Value = $6.27 million / (1.01)¹= $6.21 million ...+ Payment 30 Present Value = $6.27 million / (1.01)²⁹= $4.71 million Grand Total Present Value of the Annuity = $163.43 million

With the more conservative one percent, we would be indifferent between $6.27 million per year for thirty years or a lump sum of $163.43 million up front. At one percent, the $127 million she did accept doesn't seem such a good deal. This is why the discount rate you select is so relevant.

You can easily enough with a spreadsheet. But there is always a formula. We'll use "i" for the Interest Rate. Again,order of operation!

PVA_{Annuity Due}= Payment * [(1/i) - (1/(i*(1 + i)ⁿ))] * (1 + i)

PVA = $6.27 million * (1/0.01 - (1 / (0.01*(1.01)³⁰)) * (1.01) = $163.43 million

That covers the Annuity Due. And what about a Normal Annuity?

PVA_{Normal Annuity}= Payment * [(1/i) - (1/(i*(1 + i)ⁿ))]

Your brother-in-law asks you to lend him $10,000 so he can open a pawn shop. He promises to pay you $1,000 at the end of each year for the next twenty years, effectively "doubling your money" he says. Should you make the investment?

Your experience with your brother-in-law leads you to decide a discount (interest) rate of 15% is in order....

PVA_{Normal Annuity}= $1,000 * [(1/0.15) - (1/(0.15*(1 + 0.15)²⁰))] = $6,259

You politely tell him that "you're a bit short on cash, but thanks so much for the opportunity".

Here's another useful application of the formula. You plan to buy a $250,000 house with $50,000 down and $200,000 mortgage. The annual interest rate is 6% and you'll take out a thirty-year mortgage. First, payments are due monthly rather than annually. So, we need to convert to a monthly interest rate and then determine the number of periods. That'd be 6%/12 or 0.5% (0.005 as a decimal) and 360 months. In the case of a mortgage, the first payment isn't due until month end. So, this is a Normal Annuity.

$200,000 = Payment * [(1/0.005) - 1/(0.005*1.005³⁶⁰)] $200,000 = Payment * 166.8

Payment = $200,000 / 166.8 = $1,199

Annuity Formulas

Personally, I still do these problems using a spreadsheet. Mostly because I want to easily do what-if scenarios. What if I contribute an extra $1,000 for the next three months? But the formulas, once you're reliable with order of operation, are much faster. Here are all four together...

Homework:

Q1 - You (hypothetically) plan to retire at the age of 72. Assume you'll live another 25 years beyond that just to be safe. How much will you need to have saved at 72 if you want to spend $100,000 per year for the next 25 years? I want you to set the problem up as you would in a spreadsheet and provide calculations for the first three years. You'll need the money right away. So, assume an annuity due. And for simplicity, use 5% as your discount rate. Show all your work!

Q2 - Now use the Present Value of an Annuity formula to provide the complete calculation to the problem presented in Question 1. Again, assume an annuity due and a 5% discount rate.

Q3 - You have forty work years before you hit 72, hypothetically. How much will you need to save MONTHLY to reach the saving target above. This time simply use the Future Value of an Annuity formula. In this case, you'll have to save up the money first. So, assume an ordinary annuity. Again use 5% as your compounding rate.