Question: The equilibrium market price of a financial

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Present Value & Future Value Analysis

Present Value

The equilibrium market price of a financial asset or real asset is its present value, which equals its future value discounted to the current period using the market interest rate as the discount rate, [PV = FV(PVF i,n)]. For example, the equilibrium market price of a bond that matures in 10 years with a face value of $1000, is the present value of that bond; the present value is the price that the market would pay to buy that bond.

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For additional information on Present Value click here.

Fleischer, G. A., Leung, L.C. (1990, Summer). On Future worth and Its relationship to Present Worth as an Investment Criterion. The Engineering Economist. 35(4). 323-332.

Individual market participants may use discount rates that are higher or lower than the market interest rate in calculating the present value of an asset; a lower discount rate creates a higher present value figure. For example, the present value of some future income receipt of $10,000 is higher for a discount rate of 3% than for a discount rate of 8%.

A firm can expect to profit by selling an asset whose market price exceeds the firm’s calculation of its present value and by buying an asset whose market price is less than the firm’s calculation of its present value. In brief, sell if market price exceeds present value; buy if market price is less than present value. For example, a firm can expect to profit by buying an asset whose price is $10,000, if its present value, as calculated by the firm, exceeds $10,000.

Net Present Value

The net present value of a financial or real asset is the present value of its net cash flows, which equals its present value minus its cost (i.e., PV – Cost). For example, using the present value table in the appendix of the textbook, we can calculate that the present value of a $1000 single sum payment to be received in 40 years, discounted at a 12% interest rate is $11.00. If a firm paid more than that today to get that future payment, the net present value of that investment would be negative.

The internal rate of return of a financial or real asset is the discount rate that causes its net present value to be zero. If a firm’s cost of funds for an investment exceeds the internal rate of return on the investment, net present value is negative and the firm would lose money by making the investment.   For example, if the net present value of an investment is zero at a discount rate of 8%, then 8% is the internal rate of return for that asset; borrowing funds at more than 8% would create a loss.

It is profitable for a firm to invest in all financial and real assets whose net present value exceeds zero, which is equivalent to having an internal rate of return higher than the market interest rate on borrowed funds. For example, it’s profitable for a firm to invest in a project whose net present value exceeds $10,000.

In ranking investment projects in the order of most profitable to least profitable, profitability increases with increases in net present value and with increases in the internal rate of return of the project. For example, a firm can profit more by investing in a project whose internal rate of return is 20%, than by investing in a project whose internal rate of return is 10%, at a market interest rate on funds borrowed of 8%; although both are profitable.

“Rule of 72” & Future Value

The “rule of 72” states that the value of an asset compounded annually at a nominal interest rate of i percent will approximately double in nominal value in 72/i years. This rule is an approximation for the future value of a single sum. [FVn = PV(FVFi,n)]. For example, the rule of 72 tell us that a $1000 asset earning 8% interest compounded annually would have a future value of $2000 in approximately 9 years (i.e., 72/8 = 9). By using a future value table and the above equation, we can calculate that its actual value will be $1,999.00. We can also divide 72 by n years to determine the interest rate needed to double the value of an asset in those n years. For example, a $1000 invested for six years would double in value if it earned a compound rate of return of 12% per year (i.e., 72/6 = 12).

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Bond Prices and Interest Rates

Bond prices and interest rates are negatively related. As interest rates rise, bond prices fall. Similarly, as interest rates fall, bond prices rise. For example, the equilibrium market price of a perpetuity bond (i.e., a bond with no maturity date) is its present value; [present value = (coupon payment)/(discount rate)]. Using the market interest rate as the discount rate, this equation shows that a perpetuity bond paying a fixed annual coupon payment indefinitely of $100 with a market interest rate of 5%, has a present value and equilibrium market price of $2000. If the market interest rate rises to 10%, the bond’s present value and equilibrium market price fall to $1000.

Annuity

The future value of an annuity is positively related to the annuity contribution, the interest rate, and the number of years of accumulation. [FVAn = A(FVFAi,n)]. For example, using a future value table for annuities, we can calculate that $1000 annuity contributions annually, earning 5% interest for 20 years, have a future value of $33,066. At 8% interest over 20 years, the future value is $45,762. At 8% interest over 30 years, the future value is $113,280. With annuity contributions of $1500 at 8% over 30 years, the future value is $1500(113.28) = $169,920.

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The present value of an annuity is positively related to the annuity payments and the number of years in which payments will be received, but negatively related to the interest rate at which annuity payments are discounted. [PVA = A(PVFAi,n)]. For example, the present value of $30,000 annuity payments to be received annually for 10 years assuming an interest rate (i.e., discount rate) of 5% is $30,000(7.722) = $231,660, according to the PVFAi,n Table on page 268 in the textbook. If received for 15 years, the present value = $30,000(10.380) = $311,400. If annuity payments were $40,000 for 15 years at 5%, the present value = $40,000(10.380) = $415,200. At an interest rate of 8%, the present value = $40,000(8.560) = $342,400.

The Uniform Annual Series (UAS) is an annuity payment that is equivalent in present value to an irregular cash flow payment series over the same time period.   This enables financial decision makers to compare the present value of potential investments with irregular cash flow streams to the present value of potential investments with constant cash flow streams. [UAS = PVuas/PVFAi,n, where PVuas is the present value of the irregular cash flow stream.] For example, by using the equation and the PVFAi,n Table on p. 268, we can determine that an investment whose irregular cash flow stream over 5 years has a present value of $10,000 when discounted at a rate of 10%, is equivalent in present value terms to a uniform annual series investment of $10,000/3.791 = $2,637.83.

Effective Interest Rate

The more frequently a given interest rate return on an asset is compounded, the higher the effective interest rate, and the faster its future value will grow. [FVn = PV(1 + i/m)exp.mn, where exp.mn means mn is the exponent and m is the number of compounding periods in a year over n years. FVn increases as m increases]. For example, the future value of a $1000 savings account earning 6% interest for five years would be higher when compounded monthly than when compounded annually.

Applications: For each of the following applications, state the financial management principle that appears to be violated and explain how the principle is violated:

(b) A financial manager pays $10,000 for an asset that is expected to provide a cash flow of $1,000 each year for 10 years.

(c) A financial manager chooses to sell an asset whose market price is less then what the manager calculates to be its present value to his or her firm.

(d) A financial manager decides to borrow money to 8% to finance an investment whose internal rate of return is 6%.

(e) A financial manager of a firm with sufficient cash to finance an investment project whose net present value is $10,000, decides not to make the investment because it’s a relatively small one to make.

(f) A financial manager buys a government bond that provides a yield of 6%, given his or expectation that the interest rates will rise sharply in the near future.

(g) A financial manager decides to deposit the firm’s savings in a savings account at one bank that pays 6% interest compounded quarterly, rather than in a comparable savings account at another bank that pays 6% interest compounded daily.

(h) A financial manager calculates that having his or her firm pay off a loan in four annual installments of $2,500 would be equivalent to paying irregular annual installments of $1,000 the first year, $6,000 the 2nd, $2,000 the 3rd, and $1,000 the fourth year, assuming a discount rate of 8%.

Solution:

Ans b The asset can not valued without discounting its cash flows. The present value of cash flow should be arrived by discounting its cash flow by borrowing cost. If suitable discount rate is not available then use ‘Rule 72.’ Net present value = Present Value – Cost.
Ans c Net present value = Present Value – Cost. Calculated Net present value of the asset less than the selling price. So, Rule ‘sell if market price exceeds present value; buy if market price is less than present value’ is violated.
Ans d It is profitable for a firm to invest in assets whose net present value exceeds zero, which is equivalent to having an internal rate of return higher than the market interest rate on borrowed funds. In this case IRR<borrowing cost. So, rule is violated.
Ans e In ranking investment projects in the order of most profitable to least profitable, profitability increases with increases in net present value and with increases in the internal rate of return of the project. So, in this case rule is violated.
Ans f This is speculation. Bond prices and interest rates are negatively related. As interest rates rise, bond prices fall. Similarly, as interest rates fall, bond prices rise. So, this rule is violated.
Ans g The more frequently a given interest rate return on an asset is compounded, the higher the effective interest rate, and the faster its future value will grow. [FVn = PV(1 + i/m)exp.mn, where exp.mn means mn is the exponent and m is the number of compounding periods in a year over n years. FVn increases as m increases]. This rule is violated in this case.
Ans h The Uniform Annual Series (UAS) is an annuity payment that is equivalent in present value to an irregular cash flow payment series over the same time period.   This enables financial decision makers to compare the present value of potential investments with irregular cash flow streams to the present value of potential investments with constant cash flow streams. [UAS = PVuas/PVFAi,n, where PVuas is the present value of the irregular cash flow stream.] This rule is violated in this case.

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