Finance - Time Value of Money

Compounding Interest: Cathy is going to save her $5,000 bonus in a CD offering an APR of 6.5% with quarterly compounding and 4 years to maturity. How much money will she have when the CD matures?

$6,471.11 Formula: FV = 5,000 (1 + (.065 / 4)) ^16 FV = 6,471.11

Future Value of a Single Cash Flow

FVn = PV ( 1 + r ) ^n

Compound Interest: How much would $4,550.00 grow to in 200 years at an interest rate of 5% (or 0.05) per year?

$78,681,242 Formula: FVn=CF0(1+r)n FVn = $4,550.00(1+0.05)200= $78,681,242

PV: What is the present value of $127.63 to be received in 5 years if the annual interest rate is 8%?

$86.86 Formula: 𝑃𝑉 = 127.63 / (1.08)^5 𝑃𝑉 = 86.86

Solving for Rate: An investment of $100 in 3-month Treasury Bills at the beginning of 1928 would have grown to $1,977.91 at the beginning of 2016. (Treasury Bills are debt obligations backed by the U.S.G. with maturities of 1 year or less.) What annual rate of return did 3-month Treasury Bills earn over this 88-year period?

3.45% Formula: $1,977.91 = $100(1 + 𝑟)88 𝑟 = .0345 𝑜𝑟 3.45%

Solving for Rate: An investment of $100 in 10-year Treasury Bonds at the beginning of 1928 would have grown to $7,061.89 at the beginning of 2016. (Treasury Bonds are debt obligations backed by the U.S.Gov with maturities of more than 1 year.) What annual rate of return did 10-year Treasury Bonds earn over this 88-year period?

4.96% Formula: $7,061.89 = $100(1 + 𝑟)88 𝑟 = .0496 𝑜𝑟 4.96%

Solving for Rate: Arlene loaned her brother-in-law $1,500 five years ago. Last night, after winning big at Bingo, he paid her back $2,000 on the spot. What annual interest rate did she earn on her money?

5.92% Formula: $2,000 = $1,500(1 + 𝑟)5 𝑟 = .0592 𝑜𝑟 5.92%

Interest Rates: Which of the following interest rate offers is the best if you are thinking of opening a savings account? Bank A: 5.00% compounded annually Bank B: 4.95% compounded quarterly Bank C: 4.85% compounded monthly

Answer: Bank B offers the best EAR of 5.04%. Formula: EAR-a= (1 + (.0500 / 1) - 1 = 0.0500 or 5.00% EAR-b = (1 + (.0495 / 4) - 1 = 0.0504 or 5.04% EAR-c = (1 + (.0485 / 12) - 1 = 0.0496 or 4.96%

Future Value of a Single Cash Flow

FV = PV (1 + r)n The value of an investment at a future date depending on a rate of growth is known as future value. The FV of a single cash flow is the amount to which a cash flow today will grow in the future after earning interest. The term "time value of money" refers to the fact that the value of money received today differs from the value of money received one year later.

Future Value of a Single Cash Flow

FV=PV * (1 + r)^t To find the annual rate of return, we will use the formula for the future value of a single cash flow, which is: Where: • FV is the future value (the selling price in 2012) • PV is the present value (the original purchase price in 1974) • r is the annual rate of return • t is the number of periods (years in this case) Rearrange the formula to solve for r: r = (FV /PV^(1 /t) - 1

You just had a daughter. Assume you will need $250,000 in 18 years to pay for her college education. How much would you need to deposit today to have $250,000 in 18 years, if you invest the money at 8% per year?

PV = $62,562.26 Formula: PV = 250,000 / (1.08)^18

Present Value of a Single Cash Flow

PV = FVn / ( 1 + r ) n

Present Value of a Perpetuity with Constant Growth

PV = PMT / (r - g) To calculate the present value of a perpetuity with constant growth, you can use the growing perpetuity formula. A growing perpetuity is an infinite series of cash flows that grow at a constant rate over time. The formula for the present value of a growing perpetuity is: Where: • PV is the present value of the growing perpetuity • C1 is the cash flow received in the first period • ris the discount rate (or interest rate) • g is the constant growth rate of the cash flows This formula assumes that the growth rate (g) is less than the discount rate (r). If the growth rate is equal to or greater than the discount rate, the present value of the growing perpetuity would be infinite or undefined, which is not a realistic scenario in financial analysis.

Present Value (PV)

Present value is the value today of a future cash flow. It is calculated by discounting the future flows at a specified interest rate.