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

  • $p$$$p$$ = present value (PV)

  • $f$$$f$$ = future value (FV)

  • $i$$$i$$ = interest rate per period

  • $n$$$n$$ = number of periods

Illustration of PV of a Single Amount for One Period:

• Example:

  • p = $200$$p = $200$$

  • f = $220$$f = $220$$

  • $i = 10\%$$$i = 10\%$$

  • $n = 1$$$n = 1$$

Illustration of PV of a Single Amount for Multiple Periods:

• Example:

  • p = $200$$p = $200$$

  • f = $242$$f = $242$$

  • $i = 10\%$$$i = 10\%$$

  • $n = 2$$$n = 2$$

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$$$p = present \ value$$

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

  • $n = number \ of \ periods$$$n = number \ of \ periods$$

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

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

  • $p = 0.9091$$$p = 0.9091$$

  • p = $220 × 0.9091 = $200$$p = $220 × 0.9091 = $200$$

• Case 2: solve for n when knowing $i = 12\%$$$i = 12\%$$ and p =$13,000$$p =$13,000$$; 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 = 18$$$n = 18$$ periods or 18 years

• Case 3: solve for i when knowing p = $60,000$$p = $60,000$$ and $n = 9$$$n = 9$$; 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)^n$$$f = p × (1 + i)^n$$

  • $p$$$p$$ = present value (PV)

  • $f$$$f$$ = future value (FV)

  • $i$$$i$$ = interest rate per period

  • $n$$$n$$ = number of periods

Illustration of FV of a Single Amount for One Period:

• Example:

  • p = $200$$p = $200$$

  • f = $220$$f = $220$$

  • $i = 10\%$$$i = 10\%$$

  • $n = 1$$$n = 1$$

Illustration of FV of a Single Amount for Multiple Periods:

• Example:

  • p = $200$$p = $200$$

  • f = $266.20$$f = $266.20$$

  • $i = 10\%$$$i = 10\%$$

  • $n = 3$$$n = 3$$

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$$$f = future \ value$$

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

  • $n = number \ of \ periods$$$n = number \ of \ periods$$

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

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

  • $f = 1.7623$$$f = 1.7623$$

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

• Case 2: solve for n when knowing $i = 7\%$$$i = 7\%$$ and f =$3,000$$f =$3,000$$; 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 = 6$$$n = 6$$ periods or 6 years

• Case 3: solve for i when knowing f = $4,000$$f = $4,000$$ and $n = 9$$$n = 9$$; 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?