Most the TVM evaluation on your Financial Calculator have the right to be done v the 5-key approach. The five keys are as follows

N ⇒ This key refers come the variety of periods

I% ⇒ This vital refers to the interest rate (do not go into as a decimal ⇒ 10% would be 10 no 0.10). Periodically this interest price is referred to as a discount rate or rate of return.

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PV ⇒ This crucial refers to the present Value

PMT ⇒ This an essential refers to the Annuity Payment

FV ⇒ This key refers come the Future Value

In the TI-83 and TI-84, the 5-key technique application is described as the TVM Solver. To accessibility this, choose “APPS” and also then “Finance”. Once you’ve done this friend will watch a display screen with several finance applications, pick the TVM Solver. Once entering values right into your financial calculator you get in a value for each of the “keys” that you know. Because that example, if we want to put in 10 periods, we would go to the N= line and also put in 10. Periodically you will require to enter a an unfavorable value. To execute this, you have to use the “(-)“ an essential on her calculator rather of the “–“ key. The order the you go into the variables doesn’t issue as lengthy as you enter the 4 that you understand first, and then settle for the fifth. Come solve, you just move the cursor come the heat of the change you space trying come find, then press the “alpha” change and “solve”. Now let’s walk through several examples.

EXAMPLE ONE – Future worth of a single Cash FlowYou room investing $10,000 today and want to know how much friend will have actually after 6 year if you knife a 7% rate of return end the 6-year time frame. Due to the fact that you are starting with $10,000, that is your existing value. You have actually 6 years, so the number of time periods is 6. The 7% price of return means you have a 7% interest rate. In this example we are not utilizing an annuity, therefore we room going to set the Annuity Payment come 0. Note that I/Y and C/Y need to be collection to 1 due to the fact that we room dealing with annual periods.

N= 6I%= 7PV= 10000PMT= 0**FV= -15,007.30**I/Y= 1C/Y= 1

Thus, friend will have $15,007.30 at the end of the 6th year. An alert that the answer came out negative instead of positive. This is because of the method the calculator “thinks” as soon as it is resolving TVM problems. The calculator needs to save track not just of the dissension amounts, however which means the money is flowing. Due to the fact that you gotten in the current Value (PV) as $10,000, the calculator suspect you were receiving $10,000. If you get $10,000 today, the only means for the difficulty to “balance out” is for you come give earlier $15,007.30 at the end of the sixth year. In a problem like this you have the right to just neglect the negative sign in prior of the $15,007.30. However, there are particular problems whereby this is important. Especially – IF YOU enter NON-ZERO worths FOR 2 OR more OF THE CASH circulation KEYS (THE CASH flow KEYS IN THE 5-KEY strategy ARE THE PV, PMT, and FV KEYS), YOU must BE mindful OF CASH circulation SIGNS. In ours example, we only entered 1 non-zero value for a cash circulation (the $10,000 PV), for this reason the authorize doesn’t matter. We will certainly reintroduce this in a little bit.

### PRACTICE problem ONE

You room investing $400 today and want come know just how much you will have actually after 45 year if you knife a 9.5% rate of return end the 45-year time period. The systems to this and also other practice troubles can be uncovered at the finish of this tutorial.

EXAMPLE 2 – present Value that a solitary Cash Flow

You room going to obtain $6000 in 5 years. Assuming a 9% discount rate, what is this worth to girlfriend today?

N= 5I%= 9PMT= 0FV= 6000**PV= -3899.59**I/Y= 1C/Y= 1

Again, we can ignore the negative sign in the prize (since the only non-zero cash flow that we entered was the $6000 Future Value). Thus, $6000 received in 5 year is only worth $3899.59 this particular day (assuming a 9% discount rate). In various other words, we space indifferent between receiving $3899.59 today and also receiving $6000 in 5 year – lock both room worth the exact same to us. Alternatively, we would certainly be willing to pay $3899.59 or less to get $6000 in 5 years, however we would NOT be ready to pay any more than $3899.59. The factor for this is the if us invested $3899.59 today and let it link for 5 year at 9%, that would thrive to $6000 at the end of the 5th year. Existing value will be critical concept in valuation since most investments are structured in a way that us pay a collection amount now to get cash operation in the future. When we recognize what those future cash flows space worth to us today, we have the right to evaluate the investment.

### PRACTICE difficulty TWO

You are offered an chance to make an investment this day that will certainly pay friend $100,000 in 20-years. Presume a 5% discount rate, what is the most you would be ready to pay because that that investment today?

ANNUITIES

An annuity is a sequence of equal regular cash flows. Countless financial instances can be modeled together an annuity. For instance, calculating a mortgage payment on a residence is one annuity. Straightforward retirement evaluation can it is in structured together an annuity. Also, link valuation is partially modeled as an annuity because we obtain a fixed coupon payment every year. With annuities, we assume cash operation come at the finish of each period. Note that over there is a variation described as one “Annuity Due” the assumes cash operation come in ~ the beginning of the period. We will not work-related with Annuity Due situations in this text, however it is reasonably simple to perform so by do a simple adjustment to her calculator. Us will incorporate a sample example to show this, however, in this textbook we will assume cash operation come in ~ the finish of each duration in every one of our annuity problems.

EXAMPLE three – Lottery JackpotAssume you have just won a $10 Million Lottery Jackpot. However, rather of paying you the $10 Million up front, you have the an option of receiving $5 Million today or $400,000 every year in ~ the finish of annually for the following 25 years. Presume a 6% discount rate, which would you prefer? In order come answer this, you need to find the PV of the $400,000 per year because that 25 years. This is done together follows:

N= 25I%= 6PMT= 400000FV= 0**PV= 5,113,342.46 (Note that us dropped the negative sign)**P/Y= 1C/Y= 1

Since the annuity is worth much more than $5 Million to us, we would prefer to take the $400,000 every year because that the next 25 years.

EXAMPLE 3 A – Lottery Jackpot Annuity early out (OPTIONAL)Since numerous lotteries actually offer you your an initial installment today if you take it the installment plan, we could make the example more realistic by presume the prize to be paid together $400,000 per year at the beginning of each year for the following 25 years and leave whatever else the same. To readjust your calculator because that this, you simply need to on slide the highlight over to begin at the bottom the the screen. Currently repeat the calculation indigenous the example Three

N= 25I%= 6PMT= 400000FV= 0**PV 5,420,143.01 (Note that us dropped the an adverse sign)**P/Y= 1C/Y= 1PMT: finish BEGIN

To set it back, just toggle the on slide the highlight back to finish when you carry out the next trouble on the bottom line. Remember, in class we will nearly always use finish of duration payments.

### PRACTICE difficulty THREE

You are offered an invest that pays you $1000 every year because that the following 30 years. Presume a 10% discount rate, what is this invest worth to you today?

EXAMPLE 4 – I desire to it is in a Millionaire

You want to come to be a millionaire and plan to perform so with a savings/investment plan. Assuming you desire to reach your goal in two decades and anticipate earning a 10% rate of return, how much have to you save at the finish of every year in order come reach your goal?

N= 20I%= 10PV= 0FV= 1000000**PMT $17,459.62**I/Y= 1C/Y= 1

This means you will should save $17,459.62 every year in stimulate to achieve your goal.

### PRACTICE difficulty FOUR

Since saving $17,459.62 per year is no realistic for many of us, let’s shot some adjustments. Calculate just how much you would have to save under the complying with conditions

30 year at 10%40 year at 10%30 years at 7.5%30 year at 5%Note the big difference that time and also rate the return make on savings. Having actually a short savings horizon or earning a low rate of return mean you must save considerably more each year to reach the very same goal. This is particularly important for retirement planning.

EXAMPLE 5 – changing Periods every Year

Now assume that you want to accumulate $1 million in 30 years, however instead of conserving each year, you room going to conserve every 2 weeks (we will earn a 10% annual rate of return). There space 26 2-week periods in each year, so currently you have actually to adjust your calculator to job-related with 26 durations per year. You deserve to do this as follows by simply putting 26 right into the P/Y heat (note the the C/Y line will automatically change to 26 as well after you put in the 26 because that the P/Y line).

Now your calculator will identify that you space not making annual contributions to your savings plan, yet instead make a contribution every other week. Another issue once you change the variety of periods per year is to identify that the N crucial stands because that periods and also not have to years. Since all of our previous instances were done making use of 1 P/YR, the number of periods and years were the same. However, now 1 year will have actually 26 periods. Therefore, 30 year is indistinguishable to 780 durations (calculated by taking 26 times 30). Currently that ours adjustments have actually been made, we are prepared to enter the trouble into ours calculator.

N= 780I%= 10PV= 0FV= 1000000**PMT $202.75**P/Y= 26C/Y= 26

If we save $202.75 every 2 weeks because that the next 30 years and earn a 10% price of return, us will have actually $1,000,000 in ~ the end of the 30th year.

### PRACTICE difficulty FIVE

Repeat the over example, yet now assume weekly payments (52 weeks per year) rather of payment every 2 weeks. Once you are done, number out how much friend are saving per year under both the once per week and also once every various other week alternatives and to compare this to the price we got in Practice trouble Four-1. Why are the answer different?

EXAMPLE six – addressing for attention Rates

Let’s save working with the score of coming to be a millionaire. However, instead of calculating exactly how much you have to save, fine assume you have the right to save $3000 per year and also want to find the price of return girlfriend will should earn to reach her goal. This time us will offer ourselves 35 year of conserving $3000 every year.

N= 35PV= 0PMT= -3000FV= 1000000**I%= 10.89%**P/Y= 1C/Y= 1

If we deserve to save $3000 per year at the end of each year for the following 35 years, us will have to earn a 10.89% price of return in order to end up being a millionaire. There is a very important action in this that must be excellent in stimulate to get the best answer. Note that us made the annuity payment equal negative 3000 rather of 3000. This is because we are currently entering 2 non-zero values into our cash circulation keys (PV, PMT, FV). When go into 2 or 3 non-zero values right into our cash circulation keys, we should be careful with the indicators of the cash flows. The signs suggest the direction of the cash flow. A negative sign suggests that the cash circulation is flowing away from us. In this case, we are saving $3000 every year so us are giving up that amount and making the negative. In ~ the finish of the 35 years, we will receive earlier $1,000,000 so the is positive.

### PRACTICE difficulty SIX

What rate of return would you need to earn if you to be able to conserve $4500 per year every year for the next 35 years in order to end up being a millionaire?

EXAMPLE 7 – combine PMT, FV, and PV

Here is one critical variation on our millionaire example. This time, rather of beginning with nothing, stop assume the we currently have $40,000 and also plan come save an additional $3000 per year end the following 35 years. Now, what price of return must we earn in order come accumulate $1,000,000 at the finish of the 35th year?

N= 35PV= -40000PMT= -3000FV= 1000000**I%= 7.63%**P/Y= 1C/Y= 1

Note the here, we need to make both the present Value and also the Annuity Payment an unfavorable as castle both room flowing away from us into the to save plan. The Future value will flow back to us at the end of the time period so it is positive.

### PRACTICE difficulty SEVEN

You want to retiree a millionaire and also have built up $20,000 i beg your pardon you are putting into your retirement plan. In addition, you plan to knife a 9% rate of return. Exactly how much must you save PER MONTH over the following 35 years in order to reach her goal?

**Video: arrival and 5-Key strategy (TI-83 or TI-84)**

Assuming a 6.5% discount rate, fix for the current value the the complying with cash flow stream.

Here we can no much longer use the 5-key technique (technically, we CAN…it would simply be a lot an ext tedious). Rather we want to relocate to the cash flow worksheets on our financial calculator. The cash flow worksheet for the TI-83/TI-84 is actually a finance application called the Net existing Value (NPV). You can access it buy walking to your finance applications and also scrolling down until you check out the one labeling npv(. Once you pick this, it is an essential that you get in your data in the exactly format. The layout is npv(k, CF0, CFLIST, CFFREQUENCIES and then you use the settle (which is an Alpha shift) to acquire the solution. So, in this example, your k is 6.5%; CF0 is 0 (there is no worth for the year 0 cash flow – keep in mind that some difficulties WILL have a value there and it have the right to be either hopeful or negative); CFLIST is 400, 600, 1000, 1300 (there must be a worth for every cash circulation sequence); and CFFREQUENCY is 5, 4, 1, 6. This gives us: npv(6.5, 0, 400, 600, 1000, 1300, 5, 4, 1,6 and when we resolve we should gain a value of $7047.87. If you acquire an error message, make sure that (A) whatever is gone into in the appropriate order, (B) you’ve acquired all her commas in place, and also (C) the you use the right brackets . Note that the frequencies include up to 16 i beg your pardon matches the size of the moment line. If her frequencies execute not include up to the length of the timeline, girlfriend miscounted.

EXAMPLE ripe – Uneven Cash circulation Stream – Future ValueWe can use a similar process to deal with for the future value of one uneven cash circulation stream. However, we will begin by act the precise same actions we go to get the current value. The reason is that to acquire the future value of one uneven cash flow stream we very first (A) fix for the current value the the cash circulation stream and then (B) figure out what that worth will thrive to over the moment horizon. So, if the problem would have provided you the very same cash flow stream as above, yet instead request what it would be worth together of year 16 (the finish of the time horizon). Together we found above, the present value of the cash circulation stream (what is precious today) is $7047.87. So, if we want to know what the cash circulation stream is precious in year 16, us just lug the existing value ($7047.87) forward 16 year at the 6.5% rate of return utilizing the 5 crucial approach as follows:

N= 16I%= 6.5PV= 7047.87PMT= 0**FV= -19,304.19**I/Y= 1C/Y= 1

This tells us that the value of the cash flows will flourish to $19,304.19 at the finish of the 16 year time horizon if we deserve to invest lock to knife a 6.5% price of return. Keep in mind that the PV of an uneven cash circulation stream will constantly be much less than the sum of all the individual cash operation ($13,200 in this example) and the FV of one uneven cash circulation stream will always be an ext than the sum of every the separation, personal, instance cash flows.

EXAMPLE TEN – Uneven Cash circulation Stream – rate of ReturnAssume you might buy the cash flow stream in this instance for $6000 today. Based on this, what would your price of return be? To do this, we will usage the IRR applications on the calculator. Go to the Applications ⇒ Finance and also then scroll down till you discover the irr( application. The style is irr(CF0, CFLIST, CFFREQUENCIES and then you use the deal with (which is one Alpha shift) to acquire the solution. In any problem favor this, her CF0 need to be an unfavorable (you space paying something today to to buy the cash flow stream) – if you get in the CF0 as positive, you will get an error. For this problem, we space paying $6000 this day to buy the cash circulation stream, for this reason our CF0 is -$6000 and also we would collection it up together follows:

irr(-6000, 400, 600, 1000, 1300, 5, 4, 1,6 and when we solve we should get a worth of 8.39%.

This is the typical annualized price of return we earn over the 16 year time horizon on ours $6000 investment.

### PRACTICE difficulties EIGHT, NINE and TEN

Assuming a 12.5% discount rate, fix for the present value and also future value of the following time line. Also, assuming you can buy the cash flow stream for $80,000, what would your rate of return be?

**Video: Uneven Cash flow Streams (TI-83 or TI-84)**

You are offered the an option of 7.8% compounded quarterly or 7.6% compounded daily. I m sorry is a better investment (assuming both have actually the exact same risk)? In bespeak to attend to whether us are far better with the higher interest price compounded less frequently or the reduced interest rate compounded much more frequently, we have to make them secure comparison by converting both come their yearly compounding equivalent. We do this through the effective annual rate. It can be done v a formula or her financial calculator. If we use the formula, it looks prefer this:

where

keff to represent the yearly equivalentknom represents the in the name of or stated interest ratem to represent the number of compounding periods per year

Plugging in our worths for the 7.8% compounded quarterly we would get:

And for the 7.6% compounded day-to-day we would get:

In this case, the 7.8% compounded quarterly is better. If using the formulas, be certain to (A) bring out your calculations to number of decimal areas (or far better yet, don’t round in ~ all until you space done), (B) plug in rates right into the formula together decimals, and (C) round your final answer to 2 decimal areas in percent terms. Girlfriend can likewise do this v the jae won calculator as follows:

Go to your Applications, choose Finance and then scroll down until you acquire to the efficient interest price app ⇒ eff(.

To go into values right into this app, you usage the style eff(NOM, C) whereby NOM is the nominal rate (as a percent, no decimal) and also C is the compounding durations per year. This would provide us:

eff(7.8, 4) resolve ⇒ 8.03% and also eff(7.6,365) deal with ⇒ 7.90%

### PRACTICE trouble ELEVEN

You are readily available investments of 12% compounded annually, 11.75% compounded quarterly, or 11.5% compounded weekly. Suspect the same risk, which would certainly you prefer?

### Practice trouble Solutions

## Practice trouble 1

N= 45I%= 9.5PV= 400PMT= 0**FV= $23,751.74**P/Y= 1C/Y= 1

## Practice trouble 2

N= 20I%= 5PMT= 0FV= 100000**PV= $37,688.95**P/Y= 1C/Y= 1

## Practice difficulty 3

N= 30I%= 10PMT= 1000FV= 0**PV= $9,426.91**P/Y= 1C/Y= 1

## Practice problem 4A

N= 30I%= 10PV= 0FV= 1000000**PMT= $6,079.25**P/Y= 1C/Y= 1

## Practice difficulty 4B

N= 40I%= 10PV= 0FV= 1000000**PMT= $2,259.41**P/Y= 1C/Y= 1

## Practice trouble 4C

N= 30I%= 7.5PV= 0FV= 1000000**PMT= $9,671.24**P/Y= 1C/Y= 1

## Practice difficulty 4D

N= 30I%= 5PV= 0FV= 1000000**PMT= $15,051.44**P/Y= 1C/Y= 1

## Practice trouble 5

Calculate variety of Periods ⇒ 52 x 30 = 1560N= 1560I%= 10PV= 0FV= 1000000**PMT= $101.07**P/Y= 52C/Y= 52

Annual Savings required to Accumulate $1,000,000 in 30 year at 10%

A) conserving at the end of yearly ⇒ $6,079.25B) saving at finish of every 2 main ⇒ $202.75 x 26 = $5,271.50C) conserving at finish of every week ⇒ $101.07 x 52 = $5,255.64

The much more frequently we make contributions, the much less we need to save each year. This is because of the compounding effect. Once we make annual contributions, we earn no return the very first year. Through weekly contribute we start earning a return during the 2nd week.

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## Practice trouble 6

N= 35PV= 0PMT= -4500FV= 1,000,000**I/YR= 9.13%**P/Y= 1C/Y= 1

## Practice trouble 7

N= 420I/Y= 9PV= -20,000FV= 1,000,000**PMT= $183.13**P/Y= 1C/Y= 1

## Practice trouble 8

npv(12.5, 0, 10000, 12000, 5000, 8000, 10000, 3, 5, 2, 1, 2 solve ⇒ $63,878.58

## Practice difficulty 9

Step 1: resolve for current Value (See equipment to 8) ⇒ $63,878.58Step 2: carry forward to end of year 13N= 13I%= 12.5PV= 63,878.58PMT= 0**FV= $295,350.73**P/Y= 1C/Y= 1

## Practice difficulty 10

irr(-80000, 10000, 12000, 5000, 8000, 10000, 3, 5, 2, 1, 2 deal with ⇒ 7.93%

## Practice trouble 11

Since the 12% compounded yearly is currently annual, there is no require for one effective yearly rate. The 11.75% compounded quarterly is eff(11.75, 4) resolve ⇒ 12.28% and also the 11.5% compounded weekly is eff(11.5, 52) settle ⇒ 12.17%, making the 11.75% quarterly the finest deal.