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**Professor Jaime F Zender**

Financial Management Professor Jaime F Zender

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**A Good Place To Start What is finance?**

Finance is a hybrid of economics, statistics, and accounting. It is the science of capital. In other words, it considers the allocation of money across investment opportunities. Necessarily draws on these underlying disciplines in making such decisions. We will concentrate on corporate finance rather than personal finance but the issues and concepts are fundamentally the same and I will often draw parallels.

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Typical Question Three years ago your cousin Ralph opened a brew-pub in downtown Boulder. While it has been operating fairly successfully its survival depends upon some expansion and upgrades in its production equipment. Ralph has come to you as a potential equity investor. The expansion requires $100,000 and the two of you are discussing the ownership stake this would imply for you.

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Ralph’s Position Ralph argues that three years ago he invested $30,000 of his own capital. He also argues that for three years he has been working at a less than competitive wage (in order to reinvest the generated cash). He estimates this amounts to $40,000 in “sweat equity” for each of the three years. Ralph suggests these facts imply your $100,000 will purchase 40% of the equity. Is this argument valid?

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**Valuation Basics – Where We Are Headed**

Assets have value due to the payoffs they generate for those that purchase them. What does past investment have to do with this? The price you should be willing to pay for an asset depends upon the future value you will receive from owning that asset. Another piece of the puzzle is that cash today is more valuable than cash tomorrow – a concept we call the “time value of money.” Most business decisions come down to an evaluation of money today versus money tomorrow.

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Valuation The present value formula is a way to express today’s value of a stream of cash payments to be received in the future. Suppose you expect to receive cash payments of $100, $150, $180, and $210 respectively at year end for the next 4 years if you purchase a particular security. Today’s value of this security can be expressed using a simple formula.

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Valuation Rather or

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**Payoffs and Rates of Return**

For a given investment, the dollar payoff of that investment is simply the amount of cash it returns to the investor (in one period). The rate of return of the investment is the future payoff net of the initial investment (or the net payoff) expressed as a percentage of the initial cash outlay. It’s the net payoff, per dollar invested, for a given period of time.

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Example An investment project that costs $10 to establish today will provide a cash payment of $12 in one year has: time 1 payoff = $12 time 1 payoff = C1 time 1 net payoff = $12 - $10 = $2 time 1 net payoff = C1 – C0 rate of return = ($12 - $10)/$10 = .20 = 20% r0,1 = (C1 – C0)/C0 = C1/C0 -1

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Future Value We can turn this formula around to answer the question: “How much money will I receive in the future if the rate of return is 20% and I invest $100 today?” The answer of course is: $100(1+20%) = $120 or r1 = (C1 – C0)/C0 C0(1+r1) = C1 = FV1(C0) This is referred to as the future value of $100 if the relevant rate of return is 20%.

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Future Value We can use this idea to compute all sorts of future payoffs. For example: an investment requiring $212 today and earning a holding period return (or total return) of 60% from now (time 0) till time 12 would provide a payoff of: $212(1+60%) = $339.20 C0(1+r0,12) = C12 = FV12(C0)

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Problems A project offers a payoff of $1,050 for an investment of $1,000. What is the rate of return? A project has a rate of return of 30%. What is the payoff if the initial investment is $250? A project has a rate of return of 30%. What is the initial investment if the final payoff is $250? **This is called the present value of the future $250.**

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**Compounding Rates of Return**

What is the two-year holding period rate of return if you earn a one-year rate of return of 20% in both years? Note: It is not 20% + 20% = 40%. Why isn’t it? If you invest $100 at 20% for one year you have $120 = $100(1+20%) at the end of the first year. If you then invest the $120 for the second year at 20% you end up with $144 = $120(1+20%). This is a two-year rate of return of: 44% = ($144 - $100)/$100 = r0,2

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Compounding… This two-year rate of return of 44% is more than 40% because you earned an additional $4 in interest the second year as compared to the first. During the second year you earned interest on the $20 of interest earned during the first year.

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Compounding… We can represent this more generally using the compounding (or the “one plus”) formula. This can be expanded to consider an arbitrary number of periods:

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Problems If for the first year the one year interest rate is 20% and for the second year the one year interest rate is 30%, what is the two-year total interest rate? If the per-year interest rate for all years is 5%, what is the two-year interest rate? If the per-year interest rate is 5%, what is the 100-year interest rate?

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**The Yield Curve The term structure of interest rates.**

Today’s average annualized interest rate that investments pay as a function of their maturity. The important message here is that investments of different maturities have different rates of return. This is true even for Treasury securities. The following graph and chart provide a view of a recent yield curve.

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The Yield Curve

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**The Yield Curve Maturity Yield Yesterday Last Week Last Month 3 Month**

4.89 4.91 4.93 4.98 6 Month 4.92 4.97 2 Year 4.73 4.74 4.76 4.86 3 Year 4.60 4.62 4.65 4.78 5 Year 4.54 4.56 4.59 10 Year 30 Year 4.63 4.64 4.68 4.88

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The Yield Curve On November 25th 2006 how much money did an investment of $100,000 in a 2-year U.S. Treasury note promise to payout in two years time? r0,2 = (1+4.73%)(1+4.73%) – 1 = 9.68%* Thus in two years the $100,000 will turn into: (1.0968) $100,000 = $109,680 *This isn’t exactly correct since semi-annual compounding is always presumed in the bond market but lets keep it simple.

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The Yield Curve On November 25th 2006 the 5 year rate is quoted as 4.54%. This means that if you had invested $100 in a 5 year bond it would (at the maturity of the bond) become: $100×(1+r0,5) = $100×(1.0454)5 = $124.86 These values must therefore be equivalent.

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**Evaluating Investments**

Finance views all investments as if they were a series of cash payments received at different times. The actual costs or benefits may not be in cash, however for a proper evaluation of an investment, the costs and benefits must all be assigned a monetary value. Once all incremental cash flows for an investment are listed, finance can take over and evaluate the desirability of the project. The decision of whether to make an investment will always entail a comparison of that investment to an alternative use of the required cash. How do we make such a comparison?

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**Present Value and Net Present Value**

Recall our basic rate of return formula: Just as we can turn this around to determine the future value of the current cash flow we can also use it to determine the present value of the future cash flow:

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Example A project has an annual rate of return of 30%. If we invest $100 for one year what is the future value of our investment? Ans: $100(1 + 30%) = $100(1.3) = $130. If a project has an annual rate of return of 30% and will payoff $130 in one year, what is the initial investment? Ans: $130/(1+ 30%) = $100 = C1/(1 + r0,1) $100 is the present value of $130 to be received in one year if the relevant rate of return is 30%. This is true because if you had $100 today it would become $130 in one year investing at 30%. Alternatively, we “charge” next year’s $130 the 30% rate of return we could receive if we had money now.

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Extension This technique can be used to find the present value of cash at any future date. For example, suppose the annual interest rate is 15% (for years one and two) and you will receive $300 in two years, what is the present value of this future cash flow? r0,2 = (1 + r0,1)(1 + r1,2) = 1.15 = 32.25% The present value of the $300 is: PV(C2) = $300/(1.3225) = C2/(1 + r0,2) = $226.84

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**So What? How does this help us evaluate a project?**

Investment projects have lots of cash flows at different points in time. To make an investment decision we need a way to compare cash flows received at different times. We can’t compare $100 today with $120 next year but we can compare $100 today with the present value (today’s value) of $120 next year. The present values of future cash flows are all in terms of dollars today. Investment decisions are all made by comparison with a comparable alternative. Is it better than an alternative use of the upfront investment?

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Present Value Consider a project that will generate a payoff of $15 in one year’s time and $10 in two years. What is the present value of these payments if the annual interest rate on Treasury bills is 10% for both years? The present value of the first payoff is: r0,1 = 10% so PV0(C1) = C1/(1 + r0,1) = $15/1.1 = $13.64 ($, today) The present value of the second payoff is: r0,2 = (1.1)(1.1) – 1 = 21% so PV0(C2) = $10/1.21 = $8.26 ($, today) Their sum is $ $8.26 = $21.90 ($, today) Would you undertake this project if it cost $20?

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**Net Present Value This is just the net present value (NPV) rule.**

If the sum of the present values of a project’s future cash flows is greater than its initial cost, then taking it is creating value. This is just like being able to buy $10 bills for $5 or $8 or $9.98. Alternatively, we can think of a positive NPV project as providing a return higher than the available alternative. If the NPV of an investment is negative you are paying $10.05 for the $10 bill.

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**Precision The Net Present Value formula is:**

The discount rate is the expected return from a comparable alternative investment. The net present value rule states that you should accept projects with a positive NPV and reject those with a negative NPV.

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The NPV Decision Rule It is important to note that the timing of the project’s cash flows are irrelevant once you find that the NPV is positive. What if you are 90 years old and find a positive NPV investment that provides payoffs only in 30 years? What if you are saving for your child’s college expenses and there is a positive NPV investment that provides a payoff immediately? All agents agree on the desirability of positive NPV projects. Nice in the corporate world.

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Problem For simplicity assume the relevant interest rate is 5% annually for all years. What is the present value of the future cash flows of a project if it has payoffs of $150 in one year, $200 in two years, $600 in three years, and $100 in four years? What is the most you would pay for such an investment? If it costs $800 to purchase this investment what is its NPV? What is today’s value of being able to invest in this project?

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Incremental Cash Flow Incremental cash flow for a given period is the cash flow we want to estimate for use in the discounted cash flow analysis. Some issues that arise: Sunk costs. Costs, perhaps related to the project, that have already been incurred and cannot be recaptured. Opportunity costs. What else could be done? Side effects. Does the project affect current cash flow? Taxes. Capital expenditures versus depreciation expense. Increased investment in working capital.

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**Sunk Costs vs. Opportunity Costs**

A short time ago you purchased a plot of land for $2.5 million. Currently, its market value is $2.0 million. You are considering placing a new retail outlet on this land. How should the land cost be evaluated for purposes of projecting the cash flows that will be part of the NPV analysis?

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**Sunk Costs vs. Opportunity Costs**

Sunk costs should never be evaluated as part of the incremental cash flow. These are costs faced by the firm, regardless of what the firm may do. They are usually easy to measure as they have already been incurred. Opportunity costs should always be considered as part of incremental cash flow. The most valuable opportunity forgone due to a decision is the lost opportunity. Often these are difficult to measure and sometimes difficult to recognize.

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Side Effects A further difficulty in determining project cash flow comes from affects the proposed project may have on other parts of the firm. The most important side effect is called erosion: cash flow transferred from existing operations to the project.

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**Taxes Typically, Revenues are taxable when accrued.**

Expenses are deductible when accrued. Capital expenditures are not deductible, but depreciation can be deducted as it is accrued. Tax depreciation can differ from that reported on public financial statements. Sale of an asset for a price other than its tax basis (original price less accumulated tax depreciation) leads to a capital gain/loss with tax implications.

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Working Capital Increases in Net Working Capital should typically be viewed as requiring a cash outflow. An increase in inventory (and/or the cash balance) requires an actual use of cash. An increase in receivables/payables means that accrued revenues/expenses exceeded actual cash collections/payments. If you are estimating accrued revenues and expenses you need a correcting adjustment. If you estimate cash revenues/expenses no adjustment is required.

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Handy Short Cuts A growing perpetuity: A growing annuity

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Examples The interest rate is 10%. Your aunt Maude just promised to give you $150 every year for Christmas, forever. If it is now New Year’s eve how generous is she being? What if the promise is for the next 10 years? What if the promise is for 10 years but the amount will grow by 5% after the first year to account for inflation? What if the promise lasts forever and grows by 5% after the first year’s payment?

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Valuation of Cash Flows

Valuation of Cash Flows

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