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## Introduction to Time Value of Money

### These concepts play a central role in many practical and vital applications

3. =*NPV(i,x1,x2,…,xN)* returns the net present value of a series of *N* payments and/or receipts, given a required rate of return *i*, but assumes they take place at the end of each period. That is, the first transaction is discounted by 1/(1+i).

“When the U.S. government guarantees a bond, it guarantees that it will make interest payments on the bond on time and that it will pay the principal in full when the bond matures. *There is a misconception that, if a bond is insured or is a U.S. government obligation, the bond will not lose value. In fact, the U.S. government does not guarantee the market price or value of the bond if you sell the bond before it matures. This is because the market price or value of the bond can change over time based on several factors, including market interest rates*.”

Then

We will see that Excel also has a function for net present value, but it requires the income and expenditures to be in separate cells, with one cell per time period. If we wanted to use it on the information in figure 1, we would have to combine the last coupon payment with the bond’s redemption value because the function would otherwise treat the redemption value as taking place in the 19th period. The NPV function will be demonstrated later on.

### Effect of interest rate changes on bond values

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The remaining time to maturity is irrelevant if the available interest rate remains unchanged. Assume that the bond now has five years (10 semiannual periods) left to maturity. The NPV is still $1,000 because, even though fewer coupon payments remain, the principal is closer to repayment, so it is discounted by (1+0.015) to the 10th power rather than the 18th power.

Excel has a built-in function called PMT for the capital recovery factor. The arguments are 1) the interest rate; 2) the number of periods; and 3) the present value of the principal. =*PMT*(0.005,360,340000) returns $2038.47 in red, which means it is a payment rather than income. This function can also return a monthly car loan payment, given the interest rate, number of payments, and price of the car (or price minus down payment).

### Assessment of a project or investment

In this case, there are 18 semiannual periods remaining, each of which has a coupon payment of $15, and the bond’s principal is redeemed at the end of the last period. There is no activity (such as an investment) in period 0. Note that the required *semiannual* rate of return is half the required annual rate of return (2%), so we use 0.01 for *i*.

Financial talk show host Dave Ramsey warns that car dealers take advantage of customers by talking about payments rather than the price of the car.^{4} “When a salesman pops the question, ‘What kind of payment are you looking for?’ before you’ve even talked about the price of the car, that’s a major red flag.” The problem can be phrased in two ways. If the monthly payment and interest rate are known, the car’s actual price is the NPV of the uniform series. If the price and the payments are known, the interest rate can be calculated.

We can check the result with the original formula for the present worth of a uniform series. I used all six significant figures to get the following. =*PV*(0.0115437,75,–600) returns the same result, $29,999.95.

Excel will do the entire problem for us if we include the semiannual interest payments and also the bond’s redemption in the last period: =*PV*(0.015,18,–15,–1000) = $1,000.00. This means we can, given 1) the required rate of return; 2) the remaining periods to maturity; and 3) the semiannual coupon payment, get the current price of a bond with a single function.

Calculating the after-tax cash flow is relatively simple; it is the amount in the left column minus the taxes. The first year has an after-tax cash flow of $20,000 minus $2,000 because the depreciation reduced the taxable income from $20,000 to $10,000. The second through fifth years have income but no taxes at all because the depreciation offsets the income. The entire salvage value of $10,000 is taxable in the last year because the asset has no book value at this point. The column labeled *PV* for *Present Value* divides the after-tax amount by (1+0.15) to the *j*th power, where *j* is the year. As the net present value is negative, the investment doesn’t deliver the organization’s required rate of return and should be rejected.

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Time value of money calculations, including net present value analysis, is important when selecting projects and investments. The calculations are part of the body of knowledge for some of ASQ’s certification exams. They also go a long way toward explaining exactly what happened to Silicon Valley Bank (SVB) just a couple of months ago.

Suppose there are nine years left to maturity, and therefore 18 semiannual coupon payments along with redeeming the bond’s face value in the last period. There is no activity (such as an investment) in period 0. Note that the required *semiannual* rate of return is half the required annual rate of return (3%).

Time value of money calculations play a central role in many practical and vital applications. These include awareness and quantification of the effects of prevailing interest rates and time to maturity of bonds, which played a central role in the failure of Silicon Valley Bank. They are useful in the assessing mortgages and car payments, including revealing the actual cost of a car loan in contrast to what the dealer might tell us. Among their key workplace applications is assessment of projects and investments to determine whether they meet the company’s required rate of return. While the subject is beyond the scope of this article, multiple projects or proposals can be compared in this manner to obtain the best possible allotment of limited resources.

The PV function will also return the present value of a single future payment, in this case the bond’s face value; =PV(0.015,18,0,–1000) returns $764.91, noting that the last argument is the optional one for the future value.

The basic idea is that *future money is not worth as much as today’s money*. If money invested today can accumulate interest to be worth more, say, five years from now, it follows that an amount of money five years from now is worth less in today’s money (e.g., $1,000 five years from now might be worth only $800 in today’s money). Future amounts must be *discounted *as a function of 1) the required rate of return on investments; and 2) time. This is the origin of the phrase “discounted cash flow.”

**References**

1. Rigg, James. L. *Engineering Economics*. McGraw-Hill College, 1977. This book is an excellent reference for time value of money.

2. SEC’s Office of Investor Education and Advocacy. “Interest Rate Risk—When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall.” Public domain.

3. Vanek Smith, Stacey. “Bank fail: How rising interest rates paved the way for Silicon Valley Bank’s collapse.” NPR, March 2023.

4. Ramsey Solutions. “6 Tactics of a Used Car Salesman.” Website, Oct. 2022.

5. Bankrate. Mortgage Calculator.