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=f(1+i)np = \frac{f}{(1 + i)^n}

    • pp = present value (PV)

    • ff = future value (FV)

    • ii = interest rate per period

    • nn = number of periods

Illustration of PV of a Single Amount for One Period:

  • Example:

    • p = $200

    • f = $220

    • i=10%i = 10\%

    • n=1n = 1

Illustration of PV of a Single Amount for Multiple Periods:

  • Example:

    • p = $200

    • f = $242

    • i=10%i = 10\%

    • n=2n = 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 valuep = present \ value

    • i=interest rate per periodi = interest \ rate \ per \ period

    • n=number of periodsn = number \ of \ periods

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

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

    • p=0.9091p = 0.9091

    • p = $220 × 0.9091 = $200

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

    • $13,000/$100,000=0.1300\$13,000 / \$100,000 = 0.1300

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

    • n=18n = 18 periods or 18 years

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

    • $60,000/$120,00=0.5000\$60,000 / \$120,00 = 0.5000

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

    • i=8%i = 8\%.

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

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

    • pp = present value (PV)

    • ff = future value (FV)

    • ii = interest rate per period

    • nn = number of periods

Illustration of FV of a Single Amount for One Period:

  • Example:

    • p = $200

    • f = $220

    • i=10%i = 10\%

    • n=1n = 1

Illustration of FV of a Single Amount for Multiple Periods:

  • Example:

    • p = $200

    • f = $266.20

    • i=10%i = 10\%

    • n=3n = 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 valuef = future \ value

    • i=interest rate per periodi = interest \ rate \ per \ period

    • n=number of periodsn = number \ of \ periods

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

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

    • f=1.7623f = 1.7623

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

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

    • $3,000/$2,000=1.500\$3,000 / \$2,000 = 1.500

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

    • n=6n = 6 periods or 6 years

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

    • $4,000/$2,001=1.9990\$4,000 / \$2,001 = 1.9990

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

    • i=8%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?