Time Value of Money Notes

Time Value of Money - Appendix B

Learning Objectives

• C1: Describe the earning of interest and the concepts of present and future values.

• P1: Apply present value concepts to a single amount using interest tables.

• P2: Apply future value concepts to a single amount using interest tables.

• P3: Apply present value concepts to an annuity using interest tables.

• P4: Apply future value concepts to an annuity using interest tables.

Present and Future Value Concepts (C1)

• The value of assets and liabilities changes over time due to interest.

• Present and future value computations help measure interest over time.

• Present value is used to determine the value of future-day assets today.

• Future value is used to determine the value of present-day assets at a future date.

Present Value (PV) of a Single Amount (P1)

• Formula: p = \frac{f}{(1 + i)^n}

  • p = present value (PV)

  • f = future value (FV)

  • i = interest rate per period

  • n = number of periods

Illustration of PV of a Single Amount for One Period:

• Example:

  • p = $200

  • f = $220

  • i = 10\%

  • n = 1

Illustration of PV of a Single Amount for Multiple Periods:

• Example:

  • p = $200

  • f = $242

  • i = 10\%

  • n = 2

Using Present Value Table (Table B.1) to Compute PV of 1 (P1)

• The present value table provides present values (factors) for a variety of interest rates and periods.

• Present value assumes that the future value equals 1.

• Formula to compute present value of a single amount: p = present \ value

  • i = interest \ rate \ per \ period

  • n = number \ of \ periods

• Case 1: solve for p of $220 when knowing i = 10\% and n = 1.

  • Table B.1: row for one period and column for 10%

  • p = 0.9091

  • p = $220 × 0.9091 = $200

• Case 2: solve for n when knowing i = 12\% and p =$13,000; Want $100,000 in n years and have $13,000 today.

  • \$13,000 / \$100,000 = 0.1300

  • Table B.1: find value nearest to 0.1300 in 12% column

  • n = 18 periods or 18 years

• Case 3: solve for i when knowing p = $60,000 and n = 9; Want $120,000 in 9 years and have $60,000 today.

  • \$60,000 / \$120,00 = 0.5000

  • Table B.1: find nearest value to 0.5000 in row 9

  • i = 8\%.

Future Value (FV) of a Single Amount (P2)

• Formula to compute future value of a single amount: f = p × (1 + i)^n

  • p = present value (PV)

  • f = future value (FV)

  • i = interest rate per period

  • n = number of periods

Illustration of FV of a Single Amount for One Period:

• Example:

  • p = $200

  • f = $220

  • i = 10\%

  • n = 1

Illustration of FV of a Single Amount for Multiple Periods:

• Example:

  • p = $200

  • f = $266.20

  • i = 10\%

  • n = 3

Using Future Value Table (Table B.2) to Compute FV of 1 (P2)

• The future value table provides future values (factors) for a variety of interest rates and periods.

• Future value assumes that the present value equals 1.

• Formula to compute future value of a single amount: f = future \ value

  • i = interest \ rate \ per \ period

  • n = number \ of \ periods

• Case 1: solve for f when knowing i = 12\% and n = 5; Invest $100 for five years at 12%.

  • Table B.2: row for five periods and column for 12%.

  • f = 1.7623

  • f = $100 × 1.7623 = $176.23 ($176.24 rounded)

• Case 2: solve for n when knowing i = 7\% and f =$3,000; Have $2,000 and want $3,000 in n years at 7%.

  • \$3,000 / \$2,000 = 1.500

  • Table B.2: look in 7% column and find value nearest to 1.500

  • n = 6 periods or 6 years

• Case 3: solve for i when knowing f = $4,000 and n = 9; Have $2,001 and want $4,000 in 9 years.

  • \$4,000 / \$2,001 = 1.9990

  • Table B.2: row 9 find nearest value to 1.9990

  • i = 8\%.

Present Value (PV) of an Annuity (P3)

• Annuity – series of equal payments occurring at equal intervals.

• Ordinary annuity – equal end-of-period payments at equal intervals.

• Can compute PV of ordinary annuity using Table B.1.

• Direct way use PV of annuity table – Table B.3.

Future Value (FV) of an Annuity (P4)

• Can compute FV of ordinary annuity using Table B.2.

• Direct way use FV of annuity table – Table B.4.

Key Terms

• Present Value – this is the value of an amount right now

• Future Value – this is the value of an amount some time in the future

• Interest Rate – this is the “cost” of the money over the time it’s borrowed or invested

• Periods/Term – also known as “n,” this is how many compounding periods exist

• Lump Sum – this means one sum

• Annuity – this means equal payments over equal intervals

• Ordinary Annuity – this is an annuity paid at the END of a period

• Annuity Due – this is an annuity paid at the BEGINNING of a period

Examples

Example 1

• Bill needs $200,000 to cover his daughter’s college expenses in the next few years.

• The account earns 8% interest, compounded annually.

• Question: How much does Bill have to invest today to make this possible in 10 years?

Example 2

• Carly wants to buy a car and has two options:

  1. Pay $30,000 today.

  2. Pay monthly installments of $400 at a 3.5% interest rate, compounded monthly, for 3 years.

• Question: Which is a better deal for Carly?

Example 3

• Ted wants to start saving for college.

• He is 15 and his parents will match his contributions to the account.

• Ted contributes $200 to his college account on the last day of every month, matched by his parents.

• The account has an 8% interest rate that compounds monthly.

• Question: How much money will be in the account at the end of 3 years?