Time Value of Money

• Lecture 2 is about Financial Arithmetic from the Spring of 2025 series at Lund University.
• Presented by Andreas Ek.

Learning Objectives

• Basic finance concepts and calculation methods will be covered.
• Topics include asset returns, risk-free rate, and time value of money.
• Present value, future value, net present value, and effective interest rate are also covered.
• Calculation methods will cover compounding and discounting.

Asset Returns

• Asset return is a percentage change in an asset's price over time.
• rt = (Pt+1 - Pt) / Pt, where Pt is the asset price today, and Pt+1 is the price in one year.
• For ordinary stock, rt is the stock return.
• Risky assets have uncertain future values (stocks, oil, gold etc.).
• Risk-free assets have known future values (savings accounts).
• The return on risk-free assets is called the risk-free interest rate.

Cash Flow

• A cash flow is a payment expected in the future.
• In firms, cash flow is the net amount of cash and equivalents transferred in and out.
• Cash received are inflows, and money spent are outflows.

Time Value of Money

• Financial decisions compare/combine cash flows at different times.
• Comparing/combining values can only happen at the same point in time.
• A dollar today and a dollar in one year are not equivalent.
• Compounding moves cash flow forward while discounting it moves it back.
• The difference between the value of money today versus in the future is the time value of money.

Future Value

• Reasons to prefer a dollar today over tomorrow:
  • Earning interest (time value of money).
  • Decreased purchasing power due to inflation.
  • Preference for current possession (uncertainty).

Future Value Calculation - Compounding

• Compounding calculates the future value of a cash flow: FV = PV * (1 + r)^n
  • FV = Future Value
  • PV = Present Value
  • r = compounding rate (interest rate)
  • n = future date for calculating future value

Example Future Value Calculation

• Year 1: Cash flow $100, Interest Rate 5%, FV = $105
• Year 2: Cash flow $0, Interest Rate 5%, FV = $110.25
• Year 3: Cash flow $0, Interest Rate 5%, FV = $115.76
• Calculation: FV = 100 * (1 + 0.05)^3 = 100 * 1.1576 = 115.76

Compounding Practice #1

• Year 1: Cash flow $100, Interest Rate 5%, FV = $105
• Year 2: Cash flow $0, Interest Rate 10%, FV = $115.5
• Year 3: Cash flow $0, Interest Rate 15%, FV = $132.825
• Calculation: 100*(1+0.05)(1+0.1)(1+0.15) = 115.5*(1.15) = 132.825

Compounding Practice #2

• $1,000 invested for 5 years:
  • 3% interest: FV = $1,000 * (1 + 0.03)^5 = $1,159.27
  • 10% interest: FV = $1,000 * (1 + 0.10)^5 = $1,610.51
  • 18% interest: FV = $1,000 * (1 + 0.18)^5 = $2,287.76

Present Value Calculation - Discounting

• Discounting calculates the present value of future payments.
• Formula: PV = FV / (1 + r)^n
  • PV is the present value of the future cash flow FV.
  • r is the rate of discounting (interest rate).
  • n is the time to the future cash flow.

Present Value Calculation - Discounting Example

• Cash flow of $10,000 in 10 years, with a 4% interest rate:
• PV = $10,000 / (1 + 0.04)^10 = $6,755.64

Discounted Cash Flow

• Present value of future cash flows from an investment project: PV = CF1/(1+r) + CF2/(1+r)^2 + ... + CFT/(1+r)^T
  • T is the length of the investment project.
  • CFt is the cash flow from the project in period t.

Discounting Practice (Multiple Years)

• Year 1: Cash flow $2,000, Interest Rate 10%, PV = $1,818.18
• Year 2: Cash flow $2,000, Interest Rate 10%, PV = $1,652.89
• Year 3: Cash flow $2,000, Interest Rate 10%, PV = $1,502.63
• Year 4: Cash flow $2,500, Interest Rate 10%, PV = $1,707.53
• Total PV = $6,681.23
• PV formula example: PV = 2,000/1.1 + 2,000/(1.1)^2 + 2,000/(1.1)^3 + 2,500/(1.1)^4 = 6,681.23

Net Present Value (NPV)

• Net Present Value (NPV) is the difference between the present value of benefits and the present value of costs.
• Formula: NPV = PV(Benefits) - PV(Costs).
• Decision Rule:
  • Accept positive NPV investments (equivalent to receiving NPV in cash today).
  • Reject investments with negative NPV.

NPV Example

• A firm needs to buy a new $9,500 copier. The manufacturer offers to pay $10,000 in one year instead. Risk-free rate 7%. Should they accept?
• Benefit = Won't have to pay $9,500 today.
• Cost = $10,000 in one year, with PV(cost) = $10,000 / 1.07 = $9,345.79 today.
• NPV = $9,500 - $9,345.79 = $154.21 today.
• Offer is a good deal (positive NPV).

Effective Annual Interest Rate

• The stated annual interest rate may differ from the actual rate paid.
• EFF is the actually earned/paid interest rate on an investment/loan.
• Formula: EFF = ( 1 + (SIMP / n) )^n - 1, where SIMP is the simple interest rate and n is the compounding for frequency.

Effective Annual Interest Rate Example

• Borrowing $1,000 at 18% annually, compounded monthly.
  • EFF = ( 1 + (0.18 / 12) )^12 - 1 = 0.1956 or 19.56%.
  • Monthly rate = 18/12 = 1.5%
  • Pay after one year = $1,000 * (1 + 0.015)^12 = $1,195.6

Effective Annual Interest Rates Practice

• Should you choose SIMP-Bank (4.9% annual compounding) or EFF-Bank (4.8% annual rate with daily compounding)?
• EFF = ( 1 + (0.048 / 365) )^365 - 1 = 4.92%
• Choose EFF-Bank.

Application in Excel

• A formula is an expression that calculates the value of a cell.
  • Example: =A1+A2+A3 finds the sum of cells A1 to A3.
• Functions are predefined formulas such as =SQRT(A1).
• Excel formulas start with "=". Enter a calculation or function after the symbol.

Returns in Excel

• To calculate returns in excel, use the formula =(C3-C4)/C4
• Note, that you will lose an observation

Linear approximation monthly risk-free rate

• To approximate the a linear monthly risk-free rate, simply use the formula =N3/12 to divide the annual rate by the number of months in a year

Summary

• Asset return is the percentage change in an asset's price over time.
• Assets are divided into risky and risk-free types.
• The time value of money is critical for calculating the future value of a current cash flow or the present value of a future cash flow.
• Effective interest rate captures interest on interest when calculating future loan or savings values.

Explore the core principles of time value of money, including compounding, discounting, present value, and future value calculations. Learn how interest rates, compounding frequency, and investment horizons impact financial decisions. Understand the importance of time value of money in investment and financial planning.