Chapter 5: Valuation of Future Cashflows PDF

CHAPTER 5; VALUATION OF By Stephen Ross, Randolph Westerfield and Bradford Jordan FUTURE CASHFLOWS LEARNING OBJECTIVES: LO1 How to determine the future value of an investment made today. LO2 How to determine the present value of cas...

CHAPTER 5; VALUATION OF By Stephen Ross, Randolph Westerfield and Bradford Jordan FUTURE CASHFLOWS LEARNING OBJECTIVES: LO1 How to determine the future value of an investment made today. LO2 How to determine the present value of cash to be received at a future date. LO3 How to find the return on an investment. LO4 How long it takes for an investment to reach a desired value.  OVERVIEW One of the core problems financial managers face is determining the value today of future cash flows. This concept, known as the Time Value of Money (TVM), is essential because a dollar today is worth more than a dollar in the future due to its earning potential. This chapter introduces key concepts of future value, present value, compounding, and discounting—tools critical for financial decision- making. FUTURE VALUE (FV) Future Value (FV) refers to the amount of money that an investment made today will grow to over time at a given interest rate. It is crucial in investment planning and financial forecasting. The formula for calculating future value is: FV: Future value of investment PV : Present value or initial amount invested. r : Interest rate (expressed as a decimal). t : Number of periods (years, months, etc.). EXAMPLE: If you invest $100 today at a 10% interest rate for one year, the future value will be: This means your $100 investment will grow to $110 in one year.  COMPOUNDING Compounding is the process of earning interest not only on the original investment but also on any interest accumulated in prior periods. It is a powerful concept because it enables investments to grow at an accelerating rate over time. Simple Interest vs. Compound Interest: Simple Interest is calculated only on the initial principal. For example, if you invest $100 at 10% simple interest for two years, you will earn $10 each year, totaling $20 in interest. Compound Interest involves reinvesting interest, allowing you to earn interest on the interest. Using the same example, with compound interest, you would earn $10 in the first year, but in the second year, you would earn $11 because you are now earning interest on $110. This leads to faster growth.  EXAMPLE (INTEREST ON INTEREST): Suppose you invest $325 at 14% interest per year. After one year, your investment will grow to $370.50. By reinvesting this amount, you earn 14% on $370.50 in the second year, resulting in a total value of $422.37. The total interest earned is $97.37, with $91 from simple interest and $6.37 from compounding. MULTIPLE-PERIOD INVESTMENTS Compounding becomes more significant as the number of periods increases. For example, if you invest $100 at 10% interest for five years, the growth accelerates over time. After one year, you have $110, but by the fifth year, your investment will have grown to $161.05. This process can be summarized in the Future Value Interest Factor (FVIF) formula:  EXAMPLE (COMPOUND INTEREST) You invest $400 at 12% for three years. The future value is calculated as: In seven years, the investment grows to $884.27. The total interest earned is $484.27, of which $336 is simple interest and $148.27 is compound interest. The effect of compounding becomes more significant over long periods. For instance, investing $5 at 6% interest for 200 years results in a future value of $575,629.52, primarily due to compounding interest.  PRESENT VALUE (PV) Present Value (PV) is the current value of future cash flows, discounted at the appropriate interest rate. It answers the question: How much would I need to invest today to receive a specific amount in the future? The formula for calculating present value is: PV: Present value or the amount you need to invest today. FV: Future value or the amount you need in the future. r: Interest or discount rate. t: Number of periods. Discounting is the reverse of compounding; it determines the value today of a future sum of money.  EXAMPLE (SINGLE-PERIOD  PV): If you need $400 to buy a textbook next year and can earn 7% on your money, you need to invest: This means you need to invest $373.83 today to have $400 next year.  PRESENT VALUE FOR MULTIPLE PERIODS The present value of an amount to be received in multiple periods is calculated similarly to the single-period case but compounded over the entire time frame. Example (Saving for a Car) You want to buy a car that costs $68,500 in two years. If you can earn 9% interest, how much do you need to invest today? You need to invest $57,645.47 today to have enough to buy the car in two years. DISCOUNT RATE AND DISCOUNT FACTOR The discount rate is the interest rate used to discount future cash flows to the present. The discount factor is the inverse of the future value factor: This is useful for determining the current value of cash to be received in the future and for comparing different investment opportunities.  COMPOUND GROWTH AND ITS APPLICATIONS The concept of compound growth extends beyond investments. For example, population growth, sales forecasts, and dividend growth follow similar mathematical principles. Example (Dividend Growth) If a company pays a dividend of $5 per share and you expect it to grow by 4% per year, the dividend in eight years will be: This means the dividend will increase by $1.84 over eight years.  USING FINANCIAL CALCULATORS Financial calculators can simplify the process of calculating future and present values. Key functions include: PV: Present value. FV: Future value. r: Interest rate. n: Number of periods  FINDING INTEREST RATES AND TIME PERIODS To calculate the interest rate or time period for an investment, use the present value equation: Rearranging this formula allows you to solve for either r or t.  EXAMPLE If you invest $1,250 and receive $1,350 after one year, the interest rate is:  SUMMARY For a given rate of return, we can determine the value at some point in the future of an investment made today by calculating the future value of that investment. We can determine the current worth of a future cash flow or series of cash flows for a given rate of return by calculating the present value of the cash flow(s) involved. The relationship between present value (PV) and future value (FV) for a given rate r and time t is given by the basic present value equation: PV = FVt/(1+r)t