Lesson 6 + 7

Remember, inflation affects not only the (1) of the boat but also (2)

1. final price 2. each serial payment.

Lisa deposits $2,500 into an account at the end of each year. If she earns 6% compounded annually, how much will the account be worth in 7 years?

7 [N] 6 [I/YR] 0 [PV] 2,500 [+/-] [PMT] [FV] Answer: 20,984.59 the end of each year.

AMORT key strokes important to note

Amortize the 20th payment of the loan.: 20[INPUT] [orange key][Amort] Amortize the 1st through 24th loan payments., 1 [INPUT] 12 [orange key][Amort] 13 [INPUT] 24 [orange key][Amort]

*****Oliver and Rosalind would like to plan for their son's college education. They would like their son, who was born today, to attend a private university for 4 years beginning at age 18. Tuition is currently $70,000 per year and has increased at an annual rate of 6%, while inflation has only increased at 3% per year. They can earn an after-tax rate of return of 8%. How much must they save at the end of each year if they would like to make the last payment at the beginning of their son's first year of college?

BEGIN MODE N = 4 i = 1.8868 = [(1.08 ÷ 1.06) − 1] × 100 PMT = <70,000> FV = 0 This will result in a PV at age 18 of $272,317.77. Step 2: Determine the present value today: FV = <272,317.77> N = 18I = 1.8868 PMT = 0 This will result in a PV today of $194,515.13. Step 3: Use the following figures to determine the annual payments needed to fund college tuition costs. END MODE PV = <194,515.13> N = 18i = 8 FV = 0 This will result in a PMT of $20,755.

What is the present value of the cost of college education for a 1-year-old child assuming the following fact pattern? The current cost of college is $25,000 She will begin college at age 18 She will be in college for 4 years Education inflation is expected to be 6%, The parents' portfolio rate of return is 8%

BEGIN MODE N = 4 i = [(1.08/1.06) - 1] x 100 PMTAD = $25,000 Cost in real (inflation adjusted) dollars @ 18 = $97,256.35694 Age: 1N = 17i = [(1.08/1.06) - 1] x 100FV = $97,256.35694PV = $70,780.54

CFj = Nj =

CFj = represents each cash inflow or (outflow); j represents each period of cash flows Nj = represents the number of consecutive times cash flow occurs in an even amount

Jan wants to plan for her daughter's education. Her daughter, Rachel, was born today and will go to college at age 18 for five years. Tuition is currently $15,000 per year, in today's dollars. Jan anticipates tuition inflation of 7% and believes she can earn an 11% return on her investment. How much must Jan save at the end of each year, if she wants to make her last payment at the beginning of her daughter's first year of college?

Determine the Annual Savings It is important to determine two items: How long does the client intend to save? When will the savings payments be made? Jan's daughter was born today. Jan saves until her daughter's first year of college (18 years). She saves "at the end of each year" (END mode). We'll effectively pay down the PV of the cost of her daughter's education over the next 18 years using the keystrokes below. 18 (N) 11 (I) 36,046.41 (PV) 0 (FV) PMT =$4,680.37

Calculating the present value of a perpetuity using a formula

Divide the payment per period by the interest rate per period. In our example, the payment is $1,000 per year and the interest rate is 9% annually. Therefore, if that was a perpetuity, the present value would be: $11,111.11 = 1,000 ÷ 0.09

To pay off a mortgage in less than 360 months, homeowners can:

Double the monthly mortgage payment Pay the mortgage payment plus 10% Make an extra mortgage payment each year Pay an extra $100 every month Each of these strategies allows the homeowner to allocate more money to principal earlier on, which results in less interest being paid over time.

Suppose that you have $1,250 today and you would like to know how long it will take you double your money to $2,500. Assume that you can earn 9% per year on your investment.

Enter 9 into I/YR, -1250 into PV, and 2500 into FV. Now solve for N and you will see that it will take 8.04 years for your money to double. One important thing to note is that you absolutely must enter your numbers according to the cash flow sign convention. If you don't make either the PV or FV a negative number (and the other one positive), then you will get No Solution on the screen instead of the answer.

Imagine that you have just retired, and that you have a nest egg of $1,000,000. This is the amount that you will be drawing down for the rest of your life. If you expect to earn 6% per year on average and withdraw $70,000 per year, how long will it take to burn through your nest egg (in other words, for how long can you afford to live)? Assume that your first withdrawal will occur one year from today (End Mode).

Enter the data as follows: 6 into I/YR, -1,000,000 into PV (negative because you are investing this amount), and 70,000 into PMT. Now, press N and you will see that you can make 33.40 withdrawals. Assuming that you can live for about a year on the last withdrawal, then you can afford to live for about another 34.40 years.

(T/F) If the internal rate of return for a project is less than the required rate of return, the investor should accept the investment.

FALSE a project should be accepted if the IRR is greater than the required rate of return.

(T/F) The internal rate of return is the discount rate that equates the future value of an investment's expected costs to the future value of the expected cash inflows.

FALSE because the IRR is the discount rate that equates the present value of an investment's expected costs to the present value of the expected cash inflows.

(T/F) Net present value assumes that the cash flows generated from the project are spent or otherwise discarded immediately.

FALSE. NPV assumes the cash flows generated from the project are reinvested at the assumed discount rate.

Olivia purchased a new house for $450,000. She put 20 percent down and planned to finance the rest over 30 years at 3.5 percent. After 10 years, she is surprised by how much she still owes. You tell her that she still owes so much because so much of her monthly payment has been going toward interest. How much interest has she paid over this 10-year period?

FV = 0 i = 0.2917 (that is, 3.5 ÷ 12) N = 360 (that is, 15 x 12) This will result in a payment of $1,617. The interest paid can be determined using the AMORT function and the following keystrokes: 1 INPUT 120 SHIFT AMORT = = The interest paid over the first 10 years will be shown as $112,724.

Lisa anticipates making the following tuition payments for her son, Carson, who is starting his first year of college today. Year 1: $10,000 payment Year 2: $10,000 payment Year 3: $15,000 payment Year 4: $15,000 payment How much must Lisa have currently saved to make the tuition payments, assuming she can earn 5% each year on her investments?

Financial Calculator Inputs -10,000 CFj -10,000 CFj -15,000 CFj -15,000 CFj 5 i Orange Key + NPV 46,086.82 Lisa needs to have already saved $46,086.82 in order to be able to make the annual tuition payments over the next four years.

David purchased a new printer for his book publishing company. The printer was purchased for $10,000 and is expected to help the company generate the following cash flows for the next 4 years at the end of each year: Year 1: $3,000 Year 2: $4,000 Year 3: $2,500 Year 4: $1,000 Assuming the printer can be sold for $2,000 at the end of year 4 and David's required rate of return is 8%. What is the net present value of the printer?

Financial Calculator Inputs -10,000 CFj 3,000 CFj 4,000 CFj 2,500 CFj 1,000 + 2,000 CFj 8 I Orange Key + NPV 396.80 Since the NPV is greater than zero, David should consider purchasing the printer. $396.80 represents the difference between the present value of the future cash flows discounted at his required return of 8% less his initial investment.

David purchased a new printer for his book publishing company. The printer was purchased for $10,000 and is expected to generate the following cash flows for the next four years: Year 1: $3,000 Year 2: $4,000 Year 3: $2,500 Year 4: $1,000 Assume the printer can be sold for $2,000 at the end of Year 4 and David's required rate of return is 8 percent. What is David's internal rate of return if he purchases the printer?

Financial Calculator Inputs -10,000 CFj 3,000 CFj 4,000 CFj 2,500 CFj 1,000 + 2,000 CFj 8 I [Orange] [IRR] 9.81% As we can see, the IRR of 9.81% exceeds David's required rate of return of 8%.

Three years ago, Sydney purchased a stock for $75. Over the past three years, the stock has paid the following dividends. Year 1: $2.25 Year 2: $2.50 Year 3: $2.75 At the end of the third year, the stock was selling for $85. What was Sydney's internal rate of return (IRR) on this investment?

Financial Calculator Inputs -75 [CFj] 2.25 [CFj] 2.50 [CFj] 2.75 + 85 [CFj] [Orange] [IRR] 7.45%

Your client, Bob, won $50 million in the New York lottery this year. He can elect to receive the payout in one of two ways: Option 1: A single lump-sum payout of $22 million after taxes. Option 2: An annuity of $1,500,000 after tax, at the end of each year for the next 20 years. What rate of return would he need to earn to make the lump-sum payout equivalent to the annuity payment? Should he take the lump-sum payment or the annuity?

Financial Calculator Inputs N = 20 PV = (22,000,000) PMT = 1,500,000 FV = 0 I/YR = 3.15 If Bob can invest the lump-sum payout and earn a return greater than 3.15% tax-free and (without taking on additional risk), he should take the lump-sum payout (Option 1). If Bob is uncertain that he can earn a riskless 3.15% return, he should consider electing the annuity payment option (Option 2).

what should the financial planner recommend when... If IRR is equal to or greater than the required rate of return If IRR is less than the required rate of return

If IRR is equal to or greater than the required rate of return =Accept InvestmentIf IRR is less than the required rate of return = Reject Investment

'trick' for present value of perpetuity for financial calculator

If you can't remember that formula, you can "trick" the calculator into getting the correct answer. The trick involves the fact that the present value of a cash flow far enough into the future (way into the future) is going to be approximately $0. Therefore, beyond some future point in time the cash flows no longer add anything to the present value. So, if we specify a suitably large number of payments, we can get a very close approximation (in the limit it will be exact) to a perpetuity. Let's try this with our perpetuity. Enter 500 into N (that will always be a large enough number of periods), 9 into I/YR, and 1000 into PMT. Now solve for PV and you will get $11,111.11 as your answer.

Suppose that you are offered an investment which will pay you $1,000 per year for 10 years. If you can earn a rate of 9% per year on similar investments, how much should you be willing to pay for this annuity?

In this case we need to solve for the present value of this annuity since that is the amount that you would be willing to pay today. Press Shift C to clear the financial keys. Enter the numbers into the appropriate keys: 10 into N, 9 into I/YR, and 1000 (cash inflow) into PMT. Now press PV to solve for the present value. The answer is -6,417.6577. Again, this is negative because it represents the amount you would have to pay (cash outflow) to purchase this annuity.

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest today as a lump sum to achieve your goal?

In this case, we already know the future value ($100,000), the number of periods (18 years), and the per period interest rate (8% per year). We want to find the present value. Enter the data as follows: 18 into N, 8 into I/YR, and 100,000 into FV. Note that we enter the $100,000 as a positive number because you will be withdrawing that amount in 18 years (it will be a cash inflow). Now press PV and you will see that you need to invest $25,024.90 today in order to meet your goal. That is a lot of money to invest all at once, but we'll see on the next page that you can lessen the pain by investing smaller amounts each year.

Suppose that you have $100 to invest for a period of 5 years at an interest rate of 10% per year. How much will you have accumulated at the end of this time period?

In this problem, the $100 is the present value (PV), N is 5, and i is 10%. The answer you get should be 161.05.

Internal rate of return (IRR)

Internal rate of return (IRR) is the interest rate that causes the sum of the discounted PV of all future cash flows to equal the cost of the investment (NPV = 0). General IRR assumptions: IRR is a compounded annual rate of return. All cash flows are reinvested at the IRR.

is closing cost based on the loan amount or purchase price?

LOAN

Pete and Pam are retiring today and want to receive $35,000 per year in today's dollars at the beginning of each year for the next 30 years. If they earn 6 percent per year and inflation will be 2.5 percent each year, how large does their investment portfolio need to be today to meet this goal?

Make sure that your calculator is in BEGIN mode and then use the following figures to solve for PV: FV = 0 PMT = 35,000 I/YR = [(1.06 ÷ 1.025) - 1] x 100 = 3.4146 N = 30 This will result in a PV of <$672,882>.

Brett and Neil want to save for their 5-year-old son's education. They have already determined this will cost $50,000 in today's dollars. If they can earn a 7% return on their investments, how much must they save at the beginning of each year if they want to make their last savings payment at the beginning of their son's first year of college?

N = 13 i = 7 PV = 50,0000 FV = 0 Solve for PMT= $5,591.1611

Danny buys a house for $500,000, putting 20% down. His loan is for 30 years at 6% and he includes closing costs of 3% into his mortgage. How much is his monthly payment (rounded to whole dollars)?

N = 30 x 12 = 360 i = 6/12 PV = 500,000 x 0.80 x 1.03 = 412,000 PMT = ? FV = 0 Key: Add 3% closing cost to the amount borrowed <2,470.1482>

what type of question is this: Will the savings that will result from using the software be enough to cover the cost of the software?

Net Present Value (NPV) When the NPV is positive, it makes sense to move forward with purchasing the software. When the NPV is negative, it does not make sense from a financial perspective to move forward

Jan wants to plan for her daughter's education. Her daughter, Rachel, was born today and will go to college at age 18 for five years. Tuition is currently $15,000 per year, in today's dollars. Jan anticipates tuition inflation of 7% and believes she can earn an 11% return on her investment.What is the present value cost of her daughter's education?

Note I/YR is the client's investment returns divided by the inflation rate BEGIN 5 (N) 1.11/1.07 -1 x 100 (I) 15,000 (PMT) (0) FV PV= $69,785.90 Jan's daughter's education will cost a total of $69,785.90 in real (inflation adjusted) dollars. Now, we need to determine the present value of that cost today, when Jan's daughter is a newborn BEGIN MODE 18 (N) ****1 year old child (18-7) 1.11/1.07-1 x 100=I 0 (PMT) 69,785.90 (FV) PV -36,046.41

Chinyere purchased a new house for $200,000 and has a $180,000, 30-year mortgage with a 4 percent interest rate. Based on this information, how much will she pay in interest in her first year of payments?

PV = $180,000 FV = 0 i = 0.3333 (that is, 4 ÷ 12) N = 360 (that is, 15 x 12) This will result in a payment of $859.35. The interest paid can be determined using the AMORT function and the following keystrokes: 1 INPUT 12 SHIFT AMORT = = The interest paid over the first year will be shown as $7,142.

What does the following mean... Positive NPV? Negative NPV? NPV Equal to Zero

Positive NPV -Investment is providing excess cash flows -Investment should be considered Negative NPV -Investment is generating a shortfall of present value cash values -Investment should not be made without other compelling reasons NPV Equal to Zero -Investment is generating cash flow equal to required rate of return -Investment should be considered if investor is satisfied with rate of return being equal to the discount rate

Suppose that for a cost of $800, you are offered an investment which will pay the following cash flows at the end of each of the next five years PeriodCash Flow 0>0 1>100 2>200 3>300 4>400 5>500 What is the MRR?

The present value of the cash flows can be found as in Example 3. Clear the TVM keys and then enter the cash flows (remember that we are ignoring the cost of the investment at this point): press Shift C to clear the cash flow keys. Now, press 0 then CFj, 100 CFj, 200 CFj, 300 CFj, 400 CFj, and finally 500 CFj. Now, enter 10 into the I/YR key and then press Shift NPV. We find that the present value is $1,065.26. To find the future value of the cash flows, enter -1,065.26 into PV, 5 into N, and 10 into I/YR. Now press FV and see that the future value is $1,715.61. At this point our problem has been transformed into an $800 investment with a lump sum cash flow of $1,715.61 at period 5. The MIRR is the discount rate (I/YR) that equates these two numbers. Enter -800 into PV and then press I/YR. The MIRR is 16.48% per year. So, we have determined that our project is acceptable at a cost of $800. It has a positive NPV, the IRR is greater than our 12% required return, and the MIRR is also greater than our 12% required return. Please continue on to the next page to learn how to solve problems involving non-annual periods.

Cindy, age 45, is currently making $78,000 and has been with her employer for 20 years. She is expecting to receive an annual pension of $23,400 at her normal retirement age of 65. The pension formula is 1.5% per year times her final salary of employment

The present value of the pension today is calculated as follows:N = 20 (life expectancy at 65 to 85)i = 4 (the riskless rate if a strong company)PMTAD = $23,400 (20 years x 1.5% x $78,000)FV = $0 (a single life annuity)PVAD@65 = $330,734 (from 85 to age 65) (N = 20)3 PVAD@45 = $150,943 (from 65 to age 45) (N = 20)

uneven cash flow method for education funding

This education funding method uses cash flow and IRR calculations. Is a good approach for education funding calculations because: It is only two steps It works for any type of education funding situation. Other methods may not work if a client continues saving while the child is attending college and will only work if the client stops saving when the child starts going to college. The uneven cash flow method has two steps: 1. Determine the lump-sum amount needed today to fund the college education. Be sure to use an inflation-adjusted rate of return. 2. Determine how long the client intends to save and whether the savings payments are at the beginning or end of the year.

Using traditional Time Value of Money (TMV) calculations, an 8% return until retirement at age 70 will leave the client with a retirement portfolio of about $1,870,000. If that portfolio continues to grow at 6% in retirement, it will generate $90K/year in today's dollars for 30 years and leave the client's beneficiaries with almost $700,000 at his passing. Time Value of Money A client, age 62, who has $1M in retirement savings and wants to predict retirement income.

Using traditional Time Value of Money (TMV) calculations, an 8% return until retirement at age 70 will leave the client with a retirement portfolio of about $1,870,000. If that portfolio continues to grow at 6% in retirement, it will generate $90K/year in today's dollars for 30 years and leave the client's beneficiaries with almost $700,000 at his passing.

Another definition of IRR is the interest rate or discount rate where the NPV equals

ZERO Another way of saying that: the IRR is the rate at which the present value of the outflows of the investment is equal to the present value of the inflows

In the examples above, we assumed that the first payment would be made at the end of the year, which is typical. However, what if you plan to make (or receive) the first payment today? This changes the cash flow from from

a regular annuity into an annuity due.

a loan that is paid over time is known as a

amortized loan think: no future value because you're diminishing it over time

A timeline is a useful tool that illustrates the ____________,__________________, and ________________-of cash flows for a TVM calculation.

amount, timing, and direction (inflows versus outflows)

The present value of an ordinary annuity of $1 is today's value of

an even cash flow stream received or paid over time.

The annuity due payments from a lump-sum deposit are the payments that can be generated ____________________ This calculation is useful in determining:

at the beginning of each period, based on a lump-sum amount deposited today. • Amount of retirement income payments that can be generated from a lump-sum amount • Amount of periodic income payments that can be generated from a lump-sum amount

The present value of $1 is used when The future value of $1 is used when

calculating how much should be deposited today to meet a financial goal in the future. determining a future amount based on today's lump-sum deposit that will be earning interest (e.g., a certificate of deposit)

Present Value of Uneven Cash Flows strokes

cash flow key (CFj). Net Present Value (shift PRC)

The simplest TVM calculation is the ______________________________ after earning interest over a period of time. This calculation is useful when we already have some money today and want to know how much we will have in the future.

future value of a present lump-sum deposit

IRR flaws

implicitly assumes that the cash flows will be reinvested for the life of the project at a rate that equals the IRR. A good project may have an IRR that is considerably greater than any reasonable reinvestment assumption. Therefore, the IRR can be misleadingly high at times.

Solving for the present value of a lump sum is nearly identical to solving for the future value. One important thing to remember is that the present value will always (unless the interest rate is negative) be ____________________________________

less than the future value

Another type of financing decision a client may consider is whether to pay points on a mortgage to reduce the interest rate. The higher the points paid, the ____________________ on the loan, since paying points is essentially prepaying a portion of the interest.

lower the interest rate

Time Value of Money (TVM) definition

mathematical concept that determines the value of money, at a point or over a period of time, at a given rate of interest.

Remember, most debt repayments are____________________________--, so repayment calculations are in END mode.

ordinary annuities (they are made in arrears) Even though most mortgage payments are made at the beginning of the month, the repayment is still an ordinary annuity (because each payment includes a portion of principal repayment and interest expense incurred from the loan being outstanding for the previous month

Occasionally, we have to deal with annuities that pay forever (at least theoretically) instead of for a finite period of time. This type of cash flow is known as a __________________

perpetuity (perpetual annuity, sometimes called an infinite annuity). The problem is that the HP 10BII has no way to specify an infinite number of periods using the N key.

Calculating the net present value (NPV) and/or internal rate of return (IRR) is virtually identical to finding the

present value of an uneven cash flow stream

Periods of Compounding Impact to Period

semi annually = IR * 2 quarterly = IR* 4 monthly= IR* 12

Periods of Compounding Impact to Interest Rate

semi annually = IR / 2 quarterly = IR/ 4 monthly= IR/ 12

examples of a regular/ordinary annuity

• Most debtor payments (car loans, student loans, or mortgages) • Many savings contributions to an IRA or 401(k) if regular and recurring and made at month, quarter, or year end

Investments or projects that generate periodic cash flows with uneven dollar amounts cannot be solved with the following keys you have familiarized yourself with on your financial calculator: PV, N, I, PMT, FV. There are separate uneven cash flow keys that must be used to solve for (1) and (2)

1. Net Present Value (NPV) 2. Internal Rate of Return (IRR),

Solving for N answers the question,

"How long will it take..."

Earnest has an investment portfolio worth $500,000 that earns a fixed 4 percent each year. He hopes this portfolio will last him at least 10 more years. How much can he pull from the portfolio at the beginning of each year?

$59,274 Make sure that your calculator is in BEGIN mode and then use the following factors to solve for FV: PV =<500,000> FV = 0 i = 4 N = 10 This will result in a PMT of $59,274.

You are considering the purchase of a new home for $250,000. Your banker has informed you that they are willing to offer you a 30-year, fixed rate loan at 7% with monthly payments. If you borrow the entire $250,000, what is the required monthly payment?

***because the payments must be made every month, the length of a period is one month, and you must convert the variables to a monthly basis in order to get the correct answer. N=12 months per year x 30 years= 360 months IR= dividing the 7% annual rate by 12 to get 0.5833% per month PV= stays at 250k because it occurs once In this problem, then, we would solve for the payment amount by entering 360 in N, 0.5833 into I/YR, and 250,000 into PV. When you press PMT you will find that the monthly payment is $1,663.26.

Examples of solving for annuity payments

-how much a mortgage or auto loan payment will be. -how much you will need to save each year in order to reach a particular goal (saving for college or retirement perhaps).

1. Ken and Amy bought a house for $400,000 on August 1st. They made a down payment of 20% and financed the balance over 15 years at 5% annual interest. What is their monthly payment? (Note that debt payments are paid at the end of the period. So, in this example, Ken and Amy make their first payment on September 1st.) 2. Ken and Amy have made four mortgage payments (in September, October, November, and December). How much interest have they paid so far?

1) N= 15 x 12 = 180 I/YR = 5/12 PV = 400,000 x 0.80 = 320,000 FV = 0 PMT = (2,530.54) Ken and Amy's monthly mortgage payment would be $2,530.54, of which a portion of the payment would be allocated to interest (higher amount earlier over the life of the loan) and the remainder to principal (higher amount later over the life of the life). 2) Without clearing your calculator, enter: 1 [INPUT] 4 [orange key][Amort][=] 4,818.84 (the principal paid) [=] (5,303.32) $5,303.32 represents the amount of interest they have paid over the four month period.

For an annuity due, the financial calculator should be in "1" mode, to signify the timing of the first payment is at the 1 of the period. For an ordinary annuity, the calculator should be in "2" mode.

1. BEGIN 2. END

Normally, the calculator is working in (1) Mode. It assumes that cash flows occur at the end of the period. In this case, though, the payments occur at the beginning of the period. Therefore, we need to put the calculator into (2) Mode. To change to (2) Mode, press

1. End 2. Begin Shift MAR (note that the key says BEG/END in orange). The screen will now show BEGIN at the bottom. Note that nothing will change about how you enter the numbers. The calculator will simply shift the cash flows for you. Obviously, you will get a different answer.

4 step method of TVM calculations

1. Start with a timeline of cash flows. 2. Write down the TVM variables. 3. Clear all registers in the financial calculator. 4. Populate the TVM variables in the calculator.

Cash inflows are entered as (1) numbers and cash outflows are entered as (2) numbers. In this problem, the $100 was an investment (i.e., a cash (3) ) and the future value of $161.05 would be a cash (4) in five years.

1. positive 2. negative 3. outflow 4. inflow

Jan has an investment account with a balance of $200,000. She intends to make withdrawals each year for the next 10 years from this account. If the investment account earns 7%, compounded annually, how much can Jan receive at the end of each year?

10 [N] 7 [I/YR] 200,000 [+/-] [PV] 0 [FV] [PMT] Answer: 28,475.5

William wants to withdraw $12,000 at the end of each year (ordinary annuity) from a savings account for the next 5 years. How much must he deposit today, if the account earns 6%, compounded annually? (Note: The calculator is set to end mode for this question.)

5 [N] 6 [I/YR] 12,000 [PMT] 0 [FV] [PV] Answer: <50,548.37>

The more often the periods of compounding,

the larger the future account balance because the interest rate is being compounded (or calculated on previous interest earnings) more frequently.

Annuity Regular Annuity Annuity Due

An annuity is a series of equal cash flows paid at equal time intervals for a finite number of periods. A lease that calls for payments of $1000 each month for a year would be referred to as a "12-period, $1000 annuity." Note that, strictly speaking, in order for a series of cash flows to be considered an annuity, each cash flow must be identical and the amount of time between each cash flow must be the same in all cases. There are two types of annuities that vary only in the timing of the first cash flow: Regular Annuity - The first payment is made one period in the future (at period 1). Annuity Due - The first payment is made immediately (at period 0).

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at the beginning of each year (starting today) to achieve your goal?

As before, enter the data: 18 into N, 8 into I/YR, and 100,000 into FV. The only thing that has changed is that we are now treating this as an annuity due. So, once you have changed to Begin Mode, just press PMT. You will find that, if you make the first investment today, you only need to invest $2,472.42. That is about $200 per year less than if you make the first payment a year from now because of the extra time for your investments to compound. Be sure to switch back to End Mode after solving the problem. Since you almost always want to be in End Mode, it is a good idea to get in the habit of switching back. Press Shift MAR . When in End Mode, the bottom of the screen will be blank.

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you have $20,000 to invest today, what compound average annual rate of return do you need to earn in order to reach your goal?

As before, we need to be careful when entering the PV and FV into the calculator. In this case, you are going to invest $20,000 today (a cash outflow) and receive $100,000 in 18 years (a cash inflow). Therefore, we will enter -20,000 into PV, and 100,000 into FV. Type 18 into N, and then press I/YR to find that you need to earn an average of 9.35% per year. Again, if you get No Solution instead of an answer, it is because you didn't follow the cash flow sign convention.

modified internal rate of return (MIRR) calculation

Calculate the total present value of each of the cash flows, starting from period 1 (leave out the initial outlay). Use the calculator's NPV function just like we did in Example 3, above. Use the reinvestment rate as your discount rate to find the present value. Calculate the future value as of the end of the project life of the present value from step 1. The interest rate that you will use to find the future value is the reinvestment rate. Finally, find the discount rate that equates the initial cost of the investment with the future value of the cash flows. This discount rate is the MIRR, and it can be interpreted as the compound average annual rate of return that you will earn on an investment if you reinvest the cash flows at the reinvestment rate.

(T/F) There is no such thing as the future value of a perpetuity because the cash flows never end (period infinity never arrives).

True

If you get confused with ordinary annuity or annuity due terms, remember to ask yourself, "does the first payment occur NOW or LATER?"

Use annuity due if now ordinary annuity if later.

Annual compounding assumes that the interest earned is calculated and applied to

the beginning of the year balance, only once each year, at the end of the year.

how do you enter interest rate in financial calculator

When we entered the interest rate, we input 10 rather than 0.10. This is because the calculator automatically divides any number entered into I/YR by 100. Had you entered 0.10, the future value would have come out to 100.501 — obviously incorrect.

Suppose that you will be borrowing $1000 each year for 10 years at a rate of 9%, and then paying back the loan immediate after receiving the last payment. How much would you have to repay?

Enter the numbers into the appropriate keys: 10 into N, 9 into I/YR, and 1000 (cash inflow) into PMT. then press FV to find that the answer is -15,192.92972 ( a cash outflow).

Karen and Dave are planning to retire today. They would like to receive $25,000 per year in today's dollars, at the beginning of each year, for the next 25 years. Karen and Dave expect their investments to earn 8% per year and inflation to be 2% each year. How much must Karen and Dave have today, to meet their retirement funding goal?

Financial Calculator Inputs (BEGIN MODE) N = 25 I/YR = 1.08 / 1.02 - 1 x 100 = 5.8824 PMT = 25,000 FV = 0 PV = (342,198.97) Karen and Dave must have already accumulated $342,198.97 in order to meet their retirement funding goals as it is assumed they will no longer be working and will only rely on the principal and earnings of their investments to meet their annual spending goals in retirement. If we did not use the inflation-adjusted rate of return, the advisor might suggest that Karen and Dave would be able to meet their retirement funding goal with fewer assets, which would not likely be the case.

Frank and Stephanie are planning for their son's education. Tuition currently costs $15,000 per year and is paid in advance (annuity due) and they expect their son to attend college for 4 years. They expect their investments to earn 7.5% per year and for tuition inflation to be 5% each year. How much must Frank and Stephanie invest today, to meet their goal?

Financial Calculator Inputs (BEGIN MODE) N: 4 I/YR: 1.075 / 1.05 - 1 x 100 = 2.3810 PMT: 15,000 FV: 0 PV: (57,939.24) Assuming Frank and Stephanie's son starts college this year, they would need to invest $57,939.24 today to meet their education funding goal.

Sheryl makes the following deposits into a savings account that earns 12% interest. Year 0 - 4: $200 (a total of 5 deposits) Year 5 - 10: $300 (a total of 6 deposits) Year 11 - 15: $400 (a total of 5 deposits) Financial Calculator Inputs using CFj, the Nj shortcut:

Financial Calculator Inputs using CFj, the Nj shortcut: 200 [+/-] [CFj] 200 [+/-] [CFj] * 4 [ORANGE] [Nj] 300 [+/-] [CFj] 6 [ORANGE] [Nj] 400 [+/-] [CFj] 5 [ORANGE] [Nj] It is important to note that the first time we use the Nj shortcut key, we only input 4 years even though there have been a total of 5 deposits of $200. The reason for this is because the first $200 figure that is inputted in the calculator is treated as the first deposit, so the computation is correctly accounting for 5 deposits of $200.

30 year mortgage $250,000 4% interest How much is paid off in 10 years?

First you need to calculate the payment 12 x 30= 360 [N] 4/12= [I/YR] 250,000 [PV] 0 [FV] [PMT] = -1,193.54 12x10=120 payments 120 + INPUT + SHIFT+ AMORT= 120-120 (meaning it has calculated the values for the 120th payment) Press the equal key to scroll through these values -$535.2211 is the amount of the 120th payment that went towards paying down the principle -$658.31 is the amount paid towards interest $196,959.90 is balance of current loan after 120th payment $250,000-$196,959.90= $53,040.10 paid off in 10 years

The difference between the present value of an ordinary annuity and present value of an annuity due of $1, is the timing of the first payment.

For the ordinary annuity, the timing of the first payment is at the end of the period, whereas for an annuity due the timing of the first payment is at the beginning of a time period (today) representing today's value of that even cash flow stream.

calculation of NPV

NPV = PV of the Future Cash Flows - Cost of the Investment

What is Net Present Value (NPV) used to measure?

NPV is used in capital budgeting by managers and investors to evaluate investment alternatives. NPV measures the excess or shortfall of cash flows based on the discounted PV of the future cash flows, less the investment cost. NPV uses the investor's required rate of return for similar projects as the discount rate. NPV assumes that the cash flows generated from the project are reinvested at the required rate of return or discount rate. NPV = PV of the Future Cash Flows - Cost of the Investment

Suppose that you are offered an investment that will cost $925 and will pay you interest of $80 per year for the next 20 years. Furthermore, at the end of the 20 years, the investment will pay $1,000. If you purchase this investment, what is your compound average annual rate of return?

Note that in this problem we have a present value ($925), a future value ($1,000), and an annuity payment ($80 per year). As mentioned above, you need to be especially careful to get the signs right. In this case, both the annuity payment and the future value will be cash inflows, so they should be entered as positive numbers. The present value is the cost of the investment, a cash outflow, so it should be entered as a negative number. If you were to make a mistake and, say, enter the payment as a negative number, then you will get the wrong answer. On the other hand, if you were to enter all three with the same sign, then you will get an error message, Let's enter the numbers: Type 20 into N, -925 into PV, 80 into PMT, and 1000 into FV. Now, press I/YR and you will find that the investment will return an average of 8.81% per year. This particular problem is an example of solving for the yield to maturity (YTM) of a bond.

(T/F) you can always solve a TMV problem annually and then convert to monthly

You might be tempted to think that you could treat the problem as an annual one, and then adjust your answer to be monthly. Don't do that! The math simply doesn't work that way. Always adjust your variables before solving the problem. The reason for the difference is the compounding of interest.

Important to note when solving for number of periods (years, months)

One important thing to note is that you absolutely must enter your numbers according to the cash flow sign convention. If you don't make either the PV or FV a negative number (and the other one positive), then you will get No Solution on the screen instead of the answer. That is because, if both numbers are positive, the calculator thinks that you are getting a benefit without making any investment. If you get this error, just press C to clear it and then fix the problem by changing the sign of either PV or FV.

Future Value of Uneven Cash Flows

One way to find the future value of any set of cash flows is to first find the present value. Next, find the future value of that present value and you have your solution. In this case, we've already determined that the present value is $1,000.17922. Clear the financial keys (Shift C) then enter -1000.17922 into PV. N is 5 and I/YR is 12. Now press FV and you'll see that the future value is $1,762.65753.

Jan is considering purchasing a new car for $40,000. Jan has two options on the purchase: Cash Rebate or 0% Financing. Option 1: Cash Rebate Receive a $5,000 rebate on the price of the car and finance the balance over 5 years at 4% interest. Option 2: 0% Financing Finance the vehicle for 6 years at 0% interest, but no rebate ~Solve for PMT~

Option 1: ($5,000 Rebate and 4% Loan) Financial Calculator Inputs N = 5 x 12 = 60 I/YR = 4/12 PV = 40,000 - 5,000 = 35,000 FV = 0 PMT = (644.58) Option 2: (No Rebate and 0% Loan) Financial Calculator Inputs N = 6 x 12 = 72 I/YR = 0 PV = 40,000 FV = 0 PMT = (555.56)

Sylvia is considering purchasing a new home for $500,000. She intends to put 20% down and finance $400,000. She is presented with the following financing options: Option 1: Fixed rate mortgage over 30 years at 5.5% interest, zero points. Option 2: Fixed rate mortgage over 30 years at 5% interest plus two discount points ($8,000).

Option 1: Fixed rate mortgage over 30 years at 5.5% interest, zero points. Financial Calculator Inputs N = 30 x 12 = 360 I/YR = 5.5/12 PV = 400,000 FV = 0 PMT = (2,271.16) Option 2: Fixed rate mortgage over 30 years at 5% interest, two discount points ($8,000). Financial Calculator Inputs N = 30 x 12 = 360 I/YR = 5/12 PV = 400,000 FV = 0 PMT = (2,147.29) How long would her financial planner recommend that she live in the house to justify Option 2? calculate the cost savings and determine the amount of time it would take for her to pay back the $8,000 paid in discounted points up front. In order to determine the cost savings, we would need to divide the amount paid in points by the difference between the two monthly payments. Savings Per Monthly Payment: $123.87 ($2,271.16 - $2,147.29) $8,000 / $123.87 = 64.6 months or 5.4 years (64.6/12) Therefore, if Sylvia intends to live in the new house for more than 5.4 years, she would be better off with Option 2 and paying the discount points now.

Annuities are reflected on a financial calculator as a

PMT (payment).

Assume $5 is invested for one year earning 6% (Rn) and the inflation rate (i) during that one year is 3%. What is the real (inflation-adjusted) rate of return?

Real Rate of Return = [(1 + Rn) ÷ (1 + i) - 1] x 100 Real Rate of Return = [(1.06) ÷ (1.03) - 1] x 100 Real Rate of Return = [ 1.0291 - 1] x 100 Real Rate of Return = 0.0291 x 100 Real Rate of Return = 2.91%

The future value of $1 is the value of

a present lump-sum deposit after earning interest over a period of time.

Notice that the difference in the future values between the annuity due and the ordinary annuity is simply the difference between

the first (period 0) payment for the annuity due and the last (period 3) payment for the ordinary annuity compounded

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at the end of each year to achieve your goal?

Recall that we previously determined that if you were to make a lump sum investment today, you would have to invest $25,024.90. That is quite a chunk of change. In this case, saving for college will be easier because we are going to spread the investment over 18 years, rather than all at once. (Note that, for now, we are assuming that the first investment will be made one year from now. In other words, it is a regular annuity.) Let's enter the data: Type 18 into N, 8 into I/YR, and 100,000 into FV. Now, press PMT and you will find that you need to invest $2,670.21 per year for the next 18 years to meet your goal of having $100,000.

PMT

Represents "Payments" on the timeline, which can be debt payments, savings contributions, income payments received, or any other type of periodic cash flow. The PMT register is only used for even cash flows or cash flows of an equal amount.

The time value of money principles covered in this chapter can also be applied to every day decisions facing clients, such as

Should a client take a cash rebate or 0% financing on a new car? Should a client pay points to reduce their mortgage payment when purchasing a new home? Should a lottery winner receive a lump-sum payment or an annuity over 20 years?

Solving for the Number of Periods

Sometimes you know how much money you have now, and how much you need to have at an undetermined future time period. If you know the interest rate, then we can solve for the amount of time that it will take for the present value to grow to the future value by solving for N

In 4 years, Joe wants to purchase a boat that costs $50,000 in today's dollars. He can earn 8% on his investments and expects inflation to be 2.5% per year. What serial payment should Joe make at the end of the first, second, third, and fourth years to be able to purchase the boat in four years?

Step 1: Real (Inflation-Adjusted) Rate of Return = (1.08 / 1.025) - 1 x 100 = 5.3659 Step 2: How much will the boat cost at the end of this year? We'll need to multiply the boat's current price by 1 plus the inflation rate:Boat price at end of the first year = $50,000 x 1.025 = $51,250 Step 3: Calculate Joe's first payment at the end of this year: Financial Calculator Inputs: N= 4 I = 1.08 / 1.025 - 1 x 100 = 5.3659 PV = 0 FV = 51,250 PMT = (11,826.13) Step 4: Now that we have the first year's payment, we need to inflate it 3 more times to get the final 3 year's payments using the following formula: Current Year's Payment x (1 + Inflation) = Next Year's Payment Second Year Payment: $11,826.13 x (1.025) = $12,121.79 Third Year Payment: $12,121.79 x (1.025) = $12,424.83 Fourth Year Payment: $12,424.83 x (1.025) = $12,735.45

Internal Rate of Return

The compound average annual rate of return that is expected to be earned on an investment, assuming that the investment is held for its entire life and that the cash flows are reinvested at the same rate as the IRR. Investments that have an IRR that is greater than or equal to the cost of funds (WACC) should be accepted.

real rate of return

The nominal rate of return adjusted for inflation. An ETF performance is 13%. Assuming a 2% inflation rate, the real(inflation-adjusted) rate of return for the ETF is 10.78%, which is calculated as (1 + .13) / (1 + .02) - 1 x 100.

Time value of money concepts begin with two key values:

The present value is the value today of one or more future cash flows discounted to today at an appropriate interest rate. The future value is the value at some point in the future of a present amount or amounts after earning a rate of return, for a period of time.

net present value (NPV)

The present value of the future cash flows less the cost of the investment. (investment's market value-cost) The NPV is a direct measure of "cost versus benefit." It represents the economic profit to be earned by making an investment. Rational investors will take all investment opportunities that have an expected NPV greater than or equal to zero.

Cash Flow Sign Convention

This convention, used by financial calculators and spreadsheet functions, specifies that the sign (i.e., positive or negative numbers) indicates the direction of the cash flow. Cash inflows are entered as positive numbers, and cash outflows are entered as negative numbers. Failure to properly adhere to this convention will usually result in incorrect answers from your calculator or spreadsheet. Please note that whether a cash flow is an inflow (+) or outflow (-) depends on the part that you play in a transaction. For example, loan payments are an outflow (-) for the borrower, but an inflow (+) for the lender.

Suppose that for a cost of $800, you are offered an investment which will pay the following cash flows at the end of each of the next five years PeriodCash Flow 0>0 1>100 2>200 3>300 4>400 5>500 What is the NPV? IRR?

To solve this problem we must not only tell the calculator about the annual cash flows, but also the cost. Generally speaking, you'll pay for an investment before you can receive its benefits so the cost (initial outlay) is said to occur at time period 0 (i.e., today). To find the NPV or IRR, first clear the financial keys and then enter -800 into CFj, then enter the remaining cash flows exactly as before. For the NPV we must supply a discount rate, so enter 12 into I/YR and then press Shift PRC (note that below the PRC key is NPV in orange). You'll find that the NPV is $200.17922. Solving for the IRR is done exactly the same way, except that the discount rate is not necessary because that is the variable for which we are solving. This time, you'll press Shift CST and find that the IRR is 19.5382%.

The ordinary annuity payments from a lump-sum deposit are the payments that can be generated ______________ This calculation is useful in determining an:

at the end of each period, based on a lump-sum amount deposited today. • Amount of payment required to repay a loan • Amount of income payments that can be generated from a lump-sum amount

While annuity payments are an equal dollar amount throughout the payment period, __________________ are adjusted upward periodically throughout the payment period at a constant rate, usually in order to adjust for inflation's impact.

serial payments

modified internal rate of return (MIRR) definition

solves this problem that IRR can be misleadingly high by using an explicit reinvestment rate. Unfortunately, financial calculators don't have an MIRR key like they have an IRR key. That means that we have to use a little ingenuity to calculate the MIRR.

Time value of money concepts can calculate the monthly, quarterly, or annual payment necessary to retire a debt obligation. Debt repayment calculations can be used for any type of debt including

student loans, credit cards, mortgages or car loans.

the future value of an annuity due will always be greater than the future value of an ordinary annuity by exactly

the interest earned on the first payment of the annuity due over the total term.

The first, and most important, thing to think about when dealing with non-annual periods is

the number of periods in a year. The reason that this is so important is because you must be consistent when entering data into the HP 10BII. The numbers entered into the N, I/YR and PMT keys must agree as to the length of the time periods being used. So, if you are working on a problem with monthly compounding, then N should be the total number of months, I/YR should be the monthly interest rate, and PMT should be the monthly annuity payment.

The decision to pay (or not pay) points on a mortgage is primarily a function of

the time of ownership of the property, so the borrower can recoup the points paid through savings on a lower interest rate (interest expense).

Remember that the present value of an annuity due will always be greater than the present value of an ordinary annuity because of ________________________________________. Likewise, the future value of an annuity due will also always be greater than the future value of an ordinary annuity because of _____________________________.

the timing of the first withdrawal and compounded interest the additional periods of compounding

Suppose that you are offered an investment which will pay the following cash flows at the end of each of the next five years: PeriodCash Flow 0>0 1>100 2>200 3>300 4>400 5>500 How much would you be willing to pay for this investment if your required rate of return is 12% per year?

use the cash flow key (CFj). All we need to do is enter the cash flows exactly as shown in the table. Again, clear the financial keys first. Now, press 0 CFj, 100 CFj, 200 CFj, 300 CFj, 400 CFj, and finally 500 CFj. Now, enter 12 into I/YR and then press Shift NPV. We find that the present value is $1,000.17922.

Examples of cash inflows, which are positive amounts:

• A client is receiving annuity payments each month during retirement. • A client takes out a loan to purchase a house. • The lump-sum amount that is accumulated after a period of savings. • Any type of income received during retirement, inheritance, or distribution of savings

Examples of cash outflows, which are negative amounts

• A client makes tuition payments. • Any type of periodic savings or a lump-sum amount contributed / deposited to a savings account. • Periodic repayment of any type of debt. • The purchase of a piece of equipment or investment

Term calculations provide the amount of time required to accomplish a financial goal. This calculation is useful in determining an:

• Amount of time required to attain an account balance given a certain rate of return • Amount of time to retire a debt

TVM questions

• If I invest a certain sum of money into my IRA each year beginning now, and assuming a fixed interest rate and identified time period, how much will I have at the end of the period? • If my goal is to pay for my children's college education, how much do I need to save each year beginning today or at some time in the future? • If I borrow money to buy a house or car, how much is my monthly payment? • If I want to retire debt early, how much in additional principal payments would be required? • Should I purchase a piece of equipment for my business or rent it? • What is my annual rate of return on an investment?

Examples of questions that may be answered using the present value of an ordinary annuity may include:

• If a client needs x dollars each year while in retirement, how much should be deposited today? • Given the amount of debt repayment, how much was originally borrowed? • How much would a client be willing to pay today for an annuity or income stream that begins at the end of the year?

Examples of an annuity due:

• Rents (usually paid in advance) • Tuition payments (usually paid at the beginning of the term in advance) • Retirement income (usually paid at the beginning of the month or year in advance)