Present Value, Future Value, Annuities, Amortization, Net Present Value

Amortized loans with equal payments (like mortgage payment) total interest of the loan formula

# of payments * PMT - Amount Borrowed

Future value of a deferred annuity: Sutton Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow problems, Sutton budgets deposits of $80,000 on which it expects to earn 5% annually, only at the end of the fourth, fifth, and sixth periods. What future value will Sutton have accumulated at the end of the sixth year?

Computing the future value simply ignores the deferred period so you would use n = 3 https://imgur.com/a/9DSIIck

How to calculate amortization schedule using financial calculator

Calculate PMT using PV, FV of 0, I/Y and N. CPT PMT then choose 2nd PV for AMORT

Effective yield

the yield actually earned by the investor ; includes the interest that is earned compounded on interest

You will receive $73,466 in 5 years. What is the value today if the appropriate rate is 8% with annual compounding?

$73,466 x 1/(1.08)⁵ = $50K (rounded up)

Formula for converting APR into EAR when it's compounding but NOT continuously . This also can be used to convert EAR to interest to EAR to use for your financial calculator

(1 + APR/m)^m - 1 Ex: APR is 5.9% and the term of the loan is 48 months. What is the EAR? (1 + 0.059/48)^48 - 1 = 6.07%

Future value factor formula

(1+i)^n

Continuous compounding to find APR and EAR is given

1 + interest rate in decimal form, LN

Two different ways of calculating future value of $2,000 invested at the end of each year for 5 years

1) Compound the accumulated balance forward one year at a time as below: https://imgur.com/E6qzS4l 2) Calculate the future value of each cash flow and add them up as below: https://imgur.com/oJbZGpG

Present Value of a Deferred Annuity: Bob Boyd has developed and copyrighted tutorial software for students in advanced accounting. He agrees to sell the copyright to Campus Learning Systems for six annual payments of $5,000 each. The payments will begin 5 years from today. Given an annual interest rate of 8%, what is the present value of the six payments?

1) Find the present value of the annuity as if it wasn't deferred so then n = 6. 2) Find the present value of the annuity for the periods that were deferred so this is 4 (since payments begin year 5 and end in year 6) 3) Subtract the difference. There are two options for solving. Option 1 using one table: https://imgur.com/a/zGLMku8 Option 2 using two tables: https://imgur.com/a/ahvoCdr

Present value factor formula

1/(1+i)^n

Your uncle is giving you $2,000 for a trip to Europe when you graduate from college 3 years from now. He proposes to finance the trip by investing a sum of money now that will provide you with $2,000 upon your graduation. The only conditions are that you graduate and that you tell him how much to invest now. How much should your uncle invest today, earning 8% compound annual interest that will provide you with $2,000 upon your graduation?

1/(1+i)^n = 1/(1.08)^3 x 2,000 = 1,587.66

Discount rate formula or Present value factor

1/(1+i)ⁿ Ex: You need $1K in 3 years and you earn 15% on your money. How much do you have to invest today? 1/(1+0.15)^3 = 1/1.5209 = .6575 ; $1000 x .6575 = $657.52 is the amount you must invest today. We call this the present or discounted value

PV for a perpetuity

= C/r Ex: What is the value of a perpetuity that provides $500 annually forever and the rate of return is 8%? $500/0.08 = $6,250

Ordinary annuity definition

An annuity that pays at the end of each period.

Interest rate per compound period formula

Annual Interest Rate / Number of Compounding Periods

Annuity present value formula or present value interest factor for annuities PVIFA(r,t)

Annuity present value = C * (1 - [1/(1+r)^t])/r OR C * (1 - present value factor/r) C = amount of annuity (equal future cash flows) r = rate of return t = Number of years

Present Value Formula for single cash flow

CF/(1+i)ⁿ Ex: You need $1K in 3 years and you earn 15% on your money. How much do you have to invest today? 1/(1+0.15)^3 = 1/1.5209 = .6575 ; $1000 x .6575 = $657.52 is the amount you must invest today. We call this the present or term-discounted value

Hightown Electronics deposits $75,000 at the end of each 6-month period for the next 3 years, to accumulate enough money to meet debts that mature in 3 years. What is the future value that Hightown will have on deposit at the end of 3 years if the annual interest rate is 10%?

FVF-OAₙ,ᵢ = (1 + i)ⁿ - 1/i r = 75,000 , n= 6 , i = .1/2 or 0.05 (1+ 0.05)⁶ - 1/0.05 $510,143.25

Joe Morgan is investing $9,069 at the end of each year in a fund that earns 5% interest. In how many years will the fund be at $100,000?

First, find the FVF-OA factor as here: https://imgur.com/a/9xMMAmG Second, use the Future Value of an Ordinary Annuity of 1 table and scroll down the 5% column. n = 9 as found here: https://imgur.com/a/2ltoi31

To create his retirement fund, Walter Goodwrench, a mechanic, now works weekends. Mr. Goodwrench deposits $2,500 today in a savings account that earns 9% interest. He plans to deposit $2,500 every year for a total of 30 years. How much cash will Mr. Goodwrench accumulate in his retirement savings account, when he retires in 30 years?

First, find the FVF-OA which is (1 + 0.09)³⁰ - 1 = 136.30754 Second, multiply the FVF-OA by 1 + i. 136.30754 x 1.09 = 148.57522 Third, multiply the FVF-AD by R or the principal rent. 2,500 x 148.57522 = $371,438

Sue Lotadough plans to deposit $800 a year on each birthday of her son Howard. She makes the first deposit on his tenth birthday, at 6% interest compounded annually. What amount will Sue have accumulated for college expenses by her son's eighteenth birthday?

First, find the FVF-OA which is (1+ 0.06)⁸ - 1/i = 9.89747 Second, multiply the FVF-OA by 1 + i. 9.89747 x 1.06 = 10.49132 Third, multiply the FVF-AD by R or the principal rent. 800 x 10.49132 = $8,393.06

Future value of an annuity due formula

First, find the ordinary annuity future value factor. FVF-OAₙ,ᵢ = (1 + i)ⁿ - 1/i ; Second, multiply the FVF-OAₙ,ᵢ by 1 + i to get the FVF-ADₙ,ᵢ Third, once you found the FVF-ADₙ,ᵢ, multiply it by R or the periodic rent

Present value of an annuity due formula

First, we need to determine the PVF-OA. Second, multiply that by 1 + i Third, multiply the PVF-AD by R which is the principal rent

Future value of an ordinary annuity formula

Future value factor: FVF-OAₙ,ᵢ = (1 + i)ⁿ - 1/i ; once you found the FVF-OA, multiply it by R or the periodic rent

Continuously compounding with a single cash flow how to calculate in financial calculator

Interest rate (in decimal form), 2nd LN/e^x * Principal

Continuous compounding to find EAR and APR is given

Interest rate (in decimal form), 2nd LN/e^x, -1

basic present value equation

PV = FVt x [1/(1 + i)^n] or you can write it as PV = FVt / (1 + i)^n Ex: To find out the present value of $400 in 3 years $400 x [1/(1 + i)^n] = $400/1.1^3 = $400/1.331 = $300.53

How to calculate the payment of an annuity formula

PV/[(1 - 1/(1 + r)^n)/0.10]

Annuity due definition

Payment occurs at the beginning of each payment period

When payments and compound periods are different, which should you convert to the other?

Payments. "Payments drive everything." - Professor T Ex: payments are monthly and compounding periods are weekly, you convert the compounding periods to monthly using "Formula for converting APR into EAR when it's compounding but NOT continuously". You also make the periods monthly. If there are 26 years, you multiply by 12 = 312

Present value (PV) vs Net Present Value (NPV)

Present value (PV) is the current value of a future sum of money or stream of cash flow given a specified rate of return. Meanwhile, net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

What determines present value

Present value has an inverse relationship with the interest rate (r) and the length of time until future payment. The greater the interest rate and length of time until future payment, the smaller the present value is The less the interest rate and length of time until future payment, the greater the present value is Here is a graph showing this: https://imgur.com/UBe1x2D

Interest rate increases. What happens to the present value?

Present values are less when discount rates are high as compared to when they are low. This is because the interest owed or the discount reported is proportionate to the interest rate. That is, a company cannot borrow as much money on high-interest-paying loans as on low-interest-paying loans and yet could be required to pay the same periodic amount to satisfy the terms of the loan.

Present Value Formula (used for multiple cash flows)

See pic

Calculating present value with multiple cash flows

Suppose you need $1,000 in one year and $2,000 more in 2 years, how much do you have to invest a rate of 9 percent return? Present value of $2,000 in two years at 9 percent: $2,000/1.09^2 = $1,683.36 Present value of $1000 in one year at 9 percent: $1,000/1.09 = $917.43 Add these two amounts up for the present value of $2,600.79

The present value is always smaller or bigger than the future value?

The present value is always a smaller amount than the known future value, due to earned and accumulated interest.

When do you need to use the EAR formula?

When the compounding period is different than the payment period

Long-Term Bond Valuation: Alltech Corporation on January 1, 2025, issues $100,000 of 5% bonds due in 5 years with interest payable annually at year-end ($100,000 × .05 = $5,000). The current market rate of interest for bonds of similar risk is 6%. What will investors pay for this bond issue?

You need to find the present value of the annuity, then find the present value of the face value and add them up https://imgur.com/a/dUH252K

Norm and Jackie Remmers have saved $36,000 to finance their daughter Dawna's college education. They deposited the money in their local bank, where it earns 4% interest compounded semiannually. What equal amounts can their daughter withdraw at the end of every 6 months during her 4 college years, without exhausting the fund?

You use the present value of an ordinary annuity formula to get the PVF-OA and then divide the PV of $36,000 by the PVF-OA. https://imgur.com/a/atEijA8

Growing annuity formula

https://imgur.com/UUDBFjK r = interest rate, g = growth rate

Steve Malpezzi needs $250,000 in 10 years. How much must he invest at the end of each year, at 5% interest, to meet his needs?

https://imgur.com/a/47ld74K

Lucky Louie has just won a state lottery prize totaling $4,000,000. He learns that he will receive a check in the amount of $200,000 at the end of each of the next 20 years. What amount has Louie really won? That is, what is the present value of the $200,000 checks he will receive over the next 20 years? Assume an interest rate of 10%.

https://imgur.com/a/KHAaDYg

You wish to determine the value today of receiving rental receipts of $6,000 each at the end of each of the next 5 years when discounted at 6%? What is the present value of these future receipts?

https://imgur.com/a/Q10E3uH

You plan to accumulate $14,000 for a down payment on a condominium 5 years from now. For the next 5 years, you earn an annual return of 8% compounded semiannually. How much should you deposit at the end of each 6-month period?

https://imgur.com/a/RA6Wflp

Present value of an ordinary annuity formula

https://imgur.com/a/npIa4XM

Space Odyssey, Inc., leases a communications satellite for 4 years with annual rental payments of $4.8 million to be made at the beginning of each year. If the relevant annual interest rate is 5%, what is the present value of the rental obligations?

https://imgur.com/a/pDbdn2H

Number of Compounding Periods formula

number of years x number of compounding periods

Amortized loans with equal payments (like mortgage payment) Interest payment formula

periodic interest rate * beginning balance

Net Present Value (NPV) definition

the difference between an investment's market value and its cost