Chapter 5Chapter 5

The Time Value of MoneyThe Time Value of Money

Chapter ObjectivesChapter Objectives Understand and calculate compound interest Understand the relationship between compounding

and bringing money back to the present Annuity and future value Annuity Due Future value and present value of a sum with non-

annual compounding Determine the present value of an uneven stream

of payments Perpetuity Understand how the international setting

complicates time value of money

Compound InterestCompound Interest

When interest paid on an investment is added to the principal, then during the next period, interest is earned on the new sum

Simple InterestSimple Interest

Interest is earned on principal$100 invested at 6% per year1st year interest is $6.002nd year interest is $6.003rd year interest is $6.00Total interest earned:$18

Compound InterestCompound Interest

Interest is earned on previously earned interest $100 invested at 6% with annual compounding 1st year interest is $6.00 Principal is $106 2nd year interest is $6.36 Principal is $112.36 3rd year interest is $6.74 Principal is $119.11 Total interest earned: $19.10

Future ValueFuture Value

How much a sum will grow in a certain number of years when compounded at a specific rate.

Future ValueFuture Value

What will an investment be worth in a year?$100 invested at 7%FV = PV(1+i)$100 (1+.07)$100 (1.07) = $107

Future ValueFuture Value

Future Value can be increased by:– Increasing number of years of compounding– Increasing the interest or discount rate

Future ValueFuture Value

What is the future value of $1,000 invested at 12% for 3 years? (Assume annual compounding)

Using the tables, look at 12% column, 3 time periods. What is the factor?

$1,000 X 1.4049 = 1,404.90

Present ValuePresent Value

What is the value in today’s dollars of a sum of money to be received in the future ?

or

The current value of a future payment

Present ValuePresent Value

What is the present value of $1,000 to be received in 5 years if the discount rate is 10%?

Using the present value of $1 table, 10% column, 5 time periods

$1,000 X .621 = $621

$621 today equals $1,000 in 5 years

AnnuityAnnuity

Series of equal dollar payments for a specified number of years.

Ordinary annuity payments occur at the end of each period

Compound AnnuityCompound Annuity

Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.

Compounding AnnuityCompounding Annuity

What will $1,000 deposited every year for eight years at 10% be worth?

Use the future value of an annuity table, 10% column, eight time periods

$1,000 X 11.436 = $11,436

Future Value of an AnnuityFuture Value of an Annuity

If we need $8,000 in 6 years (and the discount rate is 10%), how much should be deposited each year?

Use the Future Value of an Annuity table, 10% column, six time periods.

$8,000 / 7.716 = $1036.81 per year

Present Value of an AnnuityPresent Value of an Annuity

Pensions, insurance obligations, and interest received from bonds are all annuities. These items all have a present value.

Calculate the present value of an annuity using the present value of annuity table.

Present Value of an AnnuityPresent Value of an Annuity

Calculate the present value of a $100 annuity received annually for 10 years when the discount rate is 6%.

$100 X 7.360 = $736

Present Value of an AnnuityPresent Value of an Annuity

Would you rather receive $450 dollars today or $100 a year for the next five years?

Discount rate is 6%.To compare these options, use present

value.The present value of $450 today is $450.The present value of a $100 annuity for 5

years at 6% is XXX?

Present Value table, five time periods, 6% column factor is 4.2124

$100 X 4.2124 = 421.24

Which option will you choose?$450 today or $100 a year for five years

Annuities DueAnnuities Due

Ordinary annuities in which all payments have been shifted forward by one time period.

Amortized LoansAmortized Loans

Loans paid off in equal installments over time– Typically Home Mortgages

Payments and AnnuitiesPayments and Annuities

If you want to finance a new motorcycle with a purchase price of $25,000 at an interest rate of 8% over 5 years, what will your payments be?

Use the present value of an annuity table, five time periods, 8% column – factor is 3.993

$25,000 / 3.993 = 6,260.96 Five annual payments of $6,260.96

Amortization of a LoanAmortization of a Loan

Reducing the balance of a loan via annuity payments is called amortizing.

A typical amortization schedule looks at payment, interest, principal payment and balance.

Amortization ScheduleAmortization Schedule

Amortize the payments on a 5-year loan for $10,000 at 6% interest.

N Payment Interest Prin. Pay New Balance(PxRxT) (Payment - (Principal – Prin Pay)

Interest)

1 $2,373.96 $600 $1,773.96 $8,226.04

2 $2,373.96 $493.56 $1,880.40 $6,345.64

3 $2,373.96 $380.74 $1,993.22 $4352.42

4 $2,373.96 $261.15 $2,112.81 $2,239.61

5 $2,373.99 $134.38 $2239.61 -----------

Mortgage PaymentsMortgage PaymentsHow much principal is paid on the first

payment of a $70,000 mortgage with 10% interest, on a 30 year loan (with monthly payments)

Payment is $614How much of this payment goes to

principal and how much goes to interest?$70,000 x .10 x 1/12 = $583Payment of $614, $583 is interest, $31 is

applied toward principal

Compounding Interest with Compounding Interest with Non-annual periodsNon-annual periods

If using the tables, divide the percentage by the number of compounding periods in a year, and multiply the time periods by the number of compounding periods in a year.

Example: 10% a year, with semi annual compounding for 5

years. 10% / 2 = 5% column on the tables N = 5 years, with semi annual compounding or 10 Use 10 for Number of periods, 5% each

Non-annual CompoundingNon-annual Compounding

What factors should be used to calculate 5 years at 12% compounded quarterly

N = 5 x 4 = 20% = 12% / 4 = 3%Use 3% column, 20 time periods

PerpetuityPerpetuity

An annuity that continues forever is called perpetuity

The present value of a perpetuity is PV = PP/iPV = present valuePP = Constant dollar amount of perpetuity i = Annuity discount rate

Future Value of $1 TableFuture Value of $1 Table

N 6% 8% 10% 12%1 1.06 1.0800 1.1000 1.12002 1.1236 1.1664 1.2100 1.25443 1.1910 1.2597 1.3310 1.40494 1.2625 1.3605 1.4641 1.57355 1.3382 1.4693 1.6105 1.76236 1.4185 1.5869 1.7716 1.97387 1.5036 1.7138 1.9487 2.21078 1.5938 1.8509 2.1436 2.47609 1.6895 1.9990 2.3579 2.773110 1.7908 2.1589 2.5937 3.1058

Present Value of $1Present Value of $1N 6% 8% 10% 12%1 .9434 .9259 .9091 .89292 .8900 .8573 .8264 .79723 .8396 .7938 .7513 .71184 .7921 .7350 .6830 .63555 .7473 .6806 .6209 .56746 .7050 .6302 .5645 .50667 .6651 .5835 .5132 .45238 .6274 .5403 .4665 .40399 .5919 .5002 .4241 .360610 .5584 .4632 .3855 .3220

Future Value of AnnuityFuture Value of AnnuityN 6% 8% 10% 12% 1 1.000 1.0000 1.000 1.0002 2.060 2.0800 2.100 2.12003 3.1836 3.2464 3.310 3.37444 4.3746 4.5061 4.6410 4.77935 5.6371 5.8666 6.1051 6.35286 6.9753 7.3359 7.7156 8.11527 8.3938 8.9228 9.4872 10.80908 9.8975 10.6366 11.4359 12.29979 11.4913 12.4876 13.5795 14.775710 13.1808 14.4866 15.9374 17.5487

Present Value of an AnnuityPresent Value of an AnnuityN 6% 8% 10% 12%1 .9434 .9259 .9091 .89292 1.8334 1.7833 1.7355 1.69013 2.6730 2.5771 2.4869 2.40184 3.4651 3.3121 3.1699 3.03735 4.2124 3.9927 3.7908 3.60486 4.9173 4.6229 4.3553 4.11147 5.5824 5.2064 4.8684 4.56388 6.2098 5.7466 5.3349 4.96769 6.8017 6.2469 5.7590 5.328210 7.3601 6.7101 6.1446 5.6502