Time Value of Money: Future and Present Value

Key Concepts and Skills

• Students should be able to compute the future/present value of cash flows
• Students should be able to compute return on investment
• Students should be able to apply financial calculators and spreadsheets for time value problems
• Students should understand perpetuities and annuities

Valuation: The One-Period Case

• If you invest $10,000 at 12% interest for one year, it grows to $11,200
• $1,200 is the interest ($10,000 x 0.12), with $10,000 as the principal repayment
• Formula to calculate the total due: $11,200 = $10,000 x (1.12)
• The total amount due at the end of the investment is the Future Value (FV)

One-Period Case Future Value

• Formula to calculate future value: FV = PV x (1 + r)
• PV is the present value
• r is the appropriate interest rate

Present Value

• If promised $11,424 in one year with a 12% interest rate, the investment is worth $10,200 today
• The amount needed to set aside today to meet the $11,424 payment is the Present Value (PV)
• Formula: $10,220 = $11,424 / 1.12
• $11,424 = $10,200 x (1.12)

Present Value Formula

• Formula: PV = C₁ / (1 + r)
• C₁ is cash flow at Date 1
• r is appropriate interest rate
• Alternative formula: PV = FV₁ / (1 + r)

Net Present Value

• Net Present Value (NPV) of an investment is the present value of expected cash flows minus the investment cost
• Example: An investment that promises to pay $10,000 in one year and the interest rate is 5 percent
• $10,000 cost $9,500, therefore NPV = -$9,500 + ($10,000 / 1.05)
• NPV = -$9,500 + $9,523.81 = $23.81
• The investment should be purchased if NPV is positive

NPV Formula

• Formula: NPV = -Cost + PV

Positive NPV Project

• Investing $9,500 elsewhere at 5% will be less than $10,000
• Calculation: $9,500 x 1.05 = $9,975 < $10,000

The Multiperiod Case

• General future value formula: FV = PV x (1 + r)^t
• PV is present value, r is interest rate, t is number of periods

Multiperiod Case Future Value

• A stock pays a $1.10 dividend and grows at 40% per year for five years
• The dividend in five years will be $5.92
• Calculation: FV = $1.10 x (1 + 0.40)^5 = $1.10 x 1.40^5 = $5.92

Future Value and Compounding

• A dividend in year five, $5.92, is higher than the sum of of the original dividend plus 40%
• $5.92 is greater than $1.10 + 5 x [$1.10 x 0.40] = $3.30
• This due to compounding

Present Value and Discounting

• Formula to calculate present value: PV = $20,000 / (1.15)^5 = $9,943.53
• This amount is how much an investor should set aside to have $20,000 in five years with a 15% interest rate

Finding the Number of Periods

• Example calculation to determine how long it takes to grow $5,000 to $10,000 with a 10% interest rate; $10,000 = $5,000 x 1.10^T
• 1.10^T = $10,000 / $5,000 = 2
• ln(1.10)^T = ln(2)
• T = ln(2) / ln(1.10) = 0.6931 / 0.0953 = 7.27 years

What Rate Is Enough

• College education will be $50,000 in 12 years, with $5,000 available to invest today
• The required interest rate must be about 21.15%
• $50,000 = $5,000 x (1 + r)^12
• (1 + r)^12 = $50,000 / $5,000 = 10
• (1 + r) = 10^(1/12)
• r = 10^(1/12) - 1 = 1.2115 - 1 = 0.2115

Calculator Keys

• Texas Instruments BA-II Plus calculator keys
• FV is future value, PV is present value
• I/Y is the periodic interest rate, with P/Y = 1
• Interest shows as a percent, not as a decimal
• N is the number of periods
• Clear registers (CLRTVM) after each problem

Multiple Cash Flows

• An investment that pays $200 one year from now, increasing by $200 per year through Year 4, with a 12% interest rate, its present value should be: $1,432.93
• If the issuer offers this investment for $1,500, it should not be purchased

Valuing "Lumpy" Cash Flows

• First, set calculator to one payment per year
• Then, use the cash flow menu

Compounding Periods

• Compounding an investment m times a year for T years provides the future value of wealth
• TVM Formula relating compounding periods: FV = C₀ x (1 + r/m)^m

Compounding Periods Example

• Investment of $1,000 for one year at 10 percent interest compounded semiannually, the investment will grow to $70.93
• FV = $50 x (1 + 0.12/2)^(2x3) = $50 x 1.06^6 = $70.93

Effective Annual Rates of Interest

• Asking "what is the effective annual rate of interest on that investment" is reasonable
• The effective annual rate (EAR) of interest is the annual rate to give the investment wealth after three years
• EAR Example: FV = $50 x (1 + 0.12/2)^(2x3) = $50 x 1.06^6 = $70.93
• $50 x (1 + EAR)^3 = $70.93

EAR Calculation

• (1 + EAR)^3 = $70.93 / $50
• EAR = ($70.93 / $50)^(1/3) - 1 = 0.1236
• Investing at 12.36% compounded annually equals investing at 12% compounded semiannually

EAR APR Loan

• To find EAR of an 18% APR loan compounded monthly the calculation starts by understanding that there is a 11/2% monthly interest rate
• The equivalent annual interest rate is 19.56%
• [1 + (r/m)]^m = [1 + (.18 / 12)] = 1.015^12 = 1.1956

EAR on a Financial Calculator

• Texas Instruments BAII Plus calculator is needed

Continuous Compounding

• Formula for the future value of a compounded investment; FV = C₀ x e^rt
• C₀ is the initial investment
• r is the APR, t is number of years
• e is a transcendental number approximately equal to 2.718, and e^x is on the calculator

Simplifications

• Perpetuity is a constant stream of cash flows that lasts forever
• Growing perpetuity is a stream of cash flows that grows at a constant rate forever
• Annuity is a stream of constant cash flows that lasts for a fixed number of periods
• Growing annuity is a stream of cash flows that grows at a constant rate for a fixed number of periods

Perpetuity

• A constant stream of cash flows
• PV = C / (1+r) + C / (1+r)² + C / (1+r)³ + ...
• Simplified formula : PV = C / r

Perpetuity Example

• A British consol promises to pay £15 every year forever; the interest rate is 10 percent;
• Value is computed as: PV= £15 / 0.10 =£150

Growing Perpetuity

• Formula to calculate the present value: PV = C / (1+r) + C x (1+g) / (1+r)² + C x (1+g)² / (1+r)³ + ...
• Simplified Formula to calculate growing perpetuity: PV = C / r-g

Growing Perpetuity Example

• The expected dividend next year is expected to equate to $1.30, and dividends are expected to grow at 5 percent forever
• This calculation allows to derive what value of this promised dividend stream is if the Discount rate is 10%
• PV= $1.30 / .10-.05 = $26

Annuity

• Constant stream of cash flows with a fixed maturity
• PV = C / (1+r) + C / (1+r)² + C / (1+r)³ + ... C / (1+r)^T
• Simplified formula = PV = C/r [1 - C / (1+r)^T

Annuity Example

• If you can afford a $400 monthly car payment, how much car can you afford if rates are 7 percent on 36-month loans?
• PV = $400 / (.07/12 [1 - 1 / (1 + .07/12)^36) = $12,954.59

Four-Year Annuity Example

• What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9 percent?
• PV₁ = $100 / 1.09¹ + $100 / 1.09² + $100 / 1.09³ + $100 / 1.09⁴ = $323.97
• $327.97 / 1.09= $297.22

Growing Annuity

• Growing stream of cash flows that has a fixed maturity
• PV = C(1+r) + C x(1+g) / (1=r) + C x(1+g)^2 / (1+r) + ... C x (1+g)^T-1 / (1+r)^T
• Formula to understand growing annuity is as follows: PV = C/ r - g [1- 1+g/ 1+r) ^ T

Growing Annuity - Retirement Plan Scenario

• Retirement plan pays $20,000 per year for 40 years, increasing the annual payment by 3% per year
• To determine the rate needed apply the following formula:
  PV=$20.000 / .10-.03 - 1 / (1.03 / 1.10)^40= $265,121.57

Evaluating an Income-Generating Property

• Rent expected to increase to be $8,500 and increase 7% a year
• Estimated income and discount rate would be 12%
• $8,500 x (1.07) equals $8,500 x (1.07)
• Growing Annuity result: $34,706.26

Loan Amortization Types

• Pure discount loans, simplest form of loan, borrower receives and repays at future time
• Interest-only loans require interest payment/period, full principal due at maturity
• Amortized loans are required principal payments over time, addition to required interest

Pure Discount Loans

• Treasury bills are examples
• The principal amount is repaid in future, without any interest payments
• Determine how much the bill sell for, using PV, if T-bill promises $10,000 in 12 month and the market interest rate is 7 percent; PV = $10,000 / 1.07 = $9,345.79

Interest-Only Loan

• A five-year, interest-only loan with a 7% rate for $10,000
• A calculation of 0.07 x ($10,000) interest from years 1 to years 4 that equals = $700
• Year 5- interest + principal $10,700; cash flow is for corporate bonds; it is similar to what goes on with corporate bonds

Loan with Fixed Principal Payment

• Consider $50,000, 10 year loan at 8 percent interest, the firm pays $5,000 in principal each year, along with interest
• Formulas to apply; Beginning balance, total payment, interest Paid, principal Paid, ending balance

Loan with Fixed Payment

• Each payments covers the interest expense + reduces the principal
• A four-year loan is considered with annual payments as the interest rate is 8% rate
• If a principal amount equates to $5,000 apply 4 N or 8 I/Y, +$5,000 PV as factors
• Then determine CPT PMT is equal to -$1,509.60 which provides a complete breakdown of each payment

What Is a Firm Worth?

• A firm is the present value of cash flows
• Determining size. timing, and those flows determines the level of risk

Explore the core principles of time value of money, including calculating future and present values of cash flows. Learn how to compute return on investment and utilize financial calculators and spreadsheets. Understand perpetuities and annuities within the context of financial valuation.