Unit 6 Seminar: Compound Interest, Future Value, and Present Value

Please have the lecture note tables handy.

Find the future value and compound interest by compounding manually.

Find the future value and compound interest by using a $1.00 future value table.

Future value formula for any number of periods, useful for a high number of periods to calculate.

Find the effective interest rate.

Find the interest compounded daily using a table.

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In some loans, interest is computed once during the life of the loan, using the simple interest formula.

In other loans, interest is computed more than once during the life of the loan or investment.

The interest is added to the principal and that amount becomes the principal for the next calculation of interest.

This process is called compounding interest.

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Interest period: the amount of time which interest is calculated and added to the principal.

Compound interest: the total interest that accumulated after more than one interest period.

Future value, maturity value, compound amount: the accumulated principal and interest after one or more interest periods.

Period interest rate: the rate for calculating interest for one interest period-the annual interest rate is divided by the number of periods per year.

Period interest rate = annual interest rate ÷ number of interest periods per year

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Period interest rate = annual interest rate ÷ number of interest periods per yr

A 12% annual interest rate with 4 interest periods per year.

3%

An 18% annual rate with 12 interest periods per year.

1 ½ %

An 8% annual rate with 4 interest periods per year. 2%

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Using the simple interest formula method:1. Find the end of period principal:

Multiply the original principal by the sum of 1 and the period interest rate. You may need to calculate the period interest rate first. P1 * (1 + period interest rate) = P2

2. For each remaining period in turn, find the next end of period principal: Multiply by the previous end of period principal by the sum of 1 and the period interest rate.P2 * (1 + period interest rate) = P3

3. Identify the last end-of-period principal as the future value.

PN * (1 + period interest rate) = PN-1

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Find the future value of a loan of $800 at 13% for three years.

The period interest rate is 13% since it is calculated annually.

First end-of-year = $800 x (1 + 13%) = $904

Second end-of-year =$904 x 1.13 = $1021.52

Third end-of-year = $1021.52 x 1.13 = $1,154.32

The FV of this loan is $1,154.32

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Compound interest = future value – original principal

In the previous example…

FV = $1,154.32

Principal = $800

CI = $1,154.32 - $800 = $354.32 The compound interest = $354.32

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From the example problem CI = $354.32 Simple interest for the same loan would be:

I = PRT = Principle * Rate * TimeI = $800 x 0.13 x 3 = $312.00Simple interest would be $312.00

The difference between compound interest and simple interest for this loan = $354.32 - $312 = $43.32The difference is $43.32

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Principal = $10,000 8% annual interest rate, compounded semi-

annually Find the FV at the end of three years.

Answer:Start by finding the interest and the periods Find the period interest rate: 8% ÷ 2 = 4%

8% annual rate compounded twice a year (semi-annually)

Determine number of periods: 3 x 2 = 6 Time is 3 years but compounded twice per year

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Then calculate each end-of-period principal for 6 periods.

Period 1 = Principle x (1+rate)Period 1 = 10,000 x 1.04 = $10,400

Period N = Period N-1 value x (1+rate)

Period 2 = $10,400 x 1.04 = $10,816

Period 3 = $10,816 x 1.04 = $11,248.64

Period 4 = $11,248.64 x 1.04 = $11,698.59

Period 5 = $11,698.59 x 1.04 = $12,166.53

Period 6 = $12,166.53 x 1.04 = $12,653.19 = FV

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Since it would be tedious and time-consuming to calculate a large number of periods with the previous method, we can use Table 10-1, which is the future value or compound amount of $1.00.

Find the number of periods and the rate per period to identify the value by which the principal is multiplied.

Tip: Using a FV table, the table value will always be greater than 1. This is logical too. If you have money now and you invest it you should have more later so it will be more than 100% of your initial investment.

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Using Table 10-1, find the FV, compound interest, and simple interest on $500 for six years compounded annually at 8%.

Determine the number of periods: 6 years Determine the interest rate per period: 8% annually Locate the value in the intersecting cell: 1.58687 Multiply the principal, $500 x 1.58687 = $793.44 The FV of the loan is $793.44. Compound interest = Future Value - Principal

$793.44- $500 = $293.44 Simple interest for the same loan = I = P*R*T

$500 x 0.08 x 6 = $240

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The following formula can be used to find the future value (FV) over any number of periods manually. This formula is extremely helpful when you have a large number of periods to calculate.

FV = P(1 + R)N

Where: FV is the future valueR is interest rate per periodN is the total number of periods

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Barry purchases a $2,000 Roth IRA when he is 25 years old and expects to earn 6% a year, compounded annually until he is 60. How much will accumulate in the investment.

Solution:FV Factor = 1(1+R)N We are given R = 6% N = 60 – 25 = 35 yearsFV Factor = 1(1+0.06)35 = (1.06)35 = 7.68609We are also given that he has $2,000 to invest today = PVFV = PV * FV Factor = $2,000 x 7.68609 = $15,372.17He will have $15,372.17 in the account in 35 years at 6% interest.

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Effective interest rate is also called the annual percentage yield or APY when identifying rate of earning on an investment.

It is called APR, annual percentage rate, when identifying the rate of interest on a loan.

Effective rate: the equivalent simple interest rate that is equivalent to a compound rate. This is the rate you will actually be charged. As a check, this rate will always be equal to or higher than the compound interest rate.

Effective annual interest rate Manually: = compound interest rate for one year * 100%

Table: = [ (FV of $1.00 after 1 yr - $1.00) ÷ $1.00 ] * 100%

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Marcia borrowed $600 at 10% compounded semiannually. What is the effective interest rate?

Using the manual compound interest method: Period rate interest = 10% ÷ 2 = 5% = 0.05 First end-of-period principal = $600 * 1.05 = $630 Second end-of-principal = $630 * 1.05 = $661.50 Compound interest after first year = $661.50 - $600 = $61.50

Annual effective interest rate = Compound interest for 1 yr = $61.50 = 0.1025

Principal $600 Multiply by 100%. 0.1025 x 100% = 10.25% effective

interest rate Check: 10.25% > 10% or effective is greater than compound

rate

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Table 10-2 gives compound interest for $100 compounded daily (using 365 days as a year.)

Pay attention to the table value given. Table 10-2 uses $100 as the principal amount; other tables may use $1, $10 or other amounts.

Using Table 10-2 is exactly like using Table 10-1.

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Find the interest on $800 at 7.5% annually, compounded daily for 28 days.

Divide the principal by $100 as you are using Table 2. [$800 ÷ 100 = 8]

Find the corresponding value by intersecting the number of days (28) and annual interest rate (7.5%) = 0.576941

Multiply this value by the number of principal increments in $100 8 x 0.576941 = $4.62The compounded interest is $4.62

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Find the interest on $1,000 for 30 days compounded at a 6% annual rate.

Answer:Divide $1,000 ÷ 100 = 10

Locate the cell where 30 days and 6% intersect to determine the value: 0.494328

Multiply by 10. 0.494328 x 10 = 4.94

The interest is $4.94

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Find the present value based on annual compounding for one year.

Find the present value using a $1.00 present value table.

Present value formula for any number of periods, useful for a high number of periods to calculate

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Suppose you want to go on a long vacation in a couple of years…or pay for your child’s college education. How much money would you have to invest right now to be able to pay for it?

Using the concepts of compound interest, you can determine amounts needed now to cover expenses in the future.

The amount of money you set aside now is called present value.

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The simplest case would be annual compounding interest for one year: the number of interest periods is 1 and the period interest rate is the annual interest rate.

Principal (present value) = future value1 + annual interest

rate*

* denotes decimal equivalent

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Find the amount of money that The 7th Inning needs to set aside today to ensure that $10,000 will be available to buy a new large screen plasma television in one year if the annual interest rate is 4% compounded annually.

PV = FV1+rate

PV = 10,000 1 + 0.04 = $9,615.38

An investment of $9,615.38 at 4% would have a value of $10,000 in one year.

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Calculate the amount of money needed now to purchase a laptop computer and accessories valued at $2,000 in a year if you invest the money at 6%.

PV = FV1+rate

PV = 2,000 1 + 0.06 = $1,886.79

An investment of $1,886.79 at 6% would have a value of $2,000 in one year.

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Using a present value table is the most efficient way to calculate the money needed now for a future expense or investment.

Table 10-3 shows the present value of $1.00 at different interest rates for different periods.

Tip: Using a PV table, the table value will always be less than 1. This is logical too. If you want more money in the future, the amount you invest now will be less than what you will have in the future. It will be less than 100% of your final value in the future.

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1. Find the number of interest periods: multiply the time period in years by number of interest periods per year.Interest periods =

(# of years) * (# of interest periods per year)

2. Find the interest rate: divide the annual interest rate by the number of interest periods per year.Period interest rate = annual interest rate

# of interest periods per yr

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3. Select the periods row corresponding to the number of interest periods.

4. Select the rate per period column corresponding to the period interest rate.

5. Locate the value in the cell where the periods row intersects the rate-per-period column.

6. Multiply the future value by value from step

Present Value = Future Value * Table rate

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The 7th Inning needs $35,000 in 4 years to buy new framing equipment. How much should be invested at 4% interest compounded annually?

4 years x 1 period per year = 4 periods 4% divided by 1 period per year = 4% interest 4 periods at 4% shows a value of 0.85480 FV = $35,000 PV = FV x table rate = $35,000 x 0.85480 The result is $29,918 They must invest $29,918 at 4% compounded

annually for four years to have $35,000

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How much money would you have to invest for 5 years at 6% paid semi-annually to make a down payment of $20,000 on a house?

2 periods per year x 5 years = 10 periods 6% per year / 2 periods = 3% interest per period Table value for 10 periods and 3% interest =

0.74409 PV = FV x table value = $20,000 x 0.74409 =

$14,881.80 $14,881.80

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The following formula can be used to find the present value (PV) needed to invest now to get a predetermined future value (FV) in any number of periods manually. This formula is extremely helpful when you have a large number of periods to calculate.

PV = FV ÷ [(1 + R)N] 

Where: FV is the future valueR is interest rate per periodN is the total number of periods

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Hotel Victoria would like to put away some of the holiday profits to save for a planned expansion. A total of $8,000 is needed in three years. How much money in a 5.2% three-year certificate of deposit that is compounded monthly must be invested now to have the $8,000 in three years?

 

Solution:5.2% = 0.052 AnnuallyInterest rate per period = 0.052 / 12 = 0.00433 MonthlyNumber of periods = 3 years * 12 months = 36PV = FV ÷ (1 + R)N = $8,000 ÷ (1 + 0.00433)36 PV = $8,000 ÷ (1.00433)36 PV = $8,000 ÷ 1.16829 = $6,847.61

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Reminder of what to complete for Unit 6 by Tuesday at Midnight: Discussion = initial

response to one question + 2 reply posts

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