Introduction to Time Value of Money

SVB invested in 10-year Treasury notes before the Federal Reserve began to raise interest rates sharply to fend off inflation.³ The SEC reference provides two examples we can use to illustrate net present value calculations. The first example is as follows:

Now suppose instead that the interest rate rises from 3% to 4% for this bond. The required semiannual rate of return is therefore 2%. The SEC says the bond is worth only $925; that is, we’ve just lost $75 on a purportedly secure investment. The investment is secure in the sense that, if we wait nine years, we will collect the principal. The face value of the bond is exactly what it says nine years from now, but the bond is worth only $925 today—and this is what happened when SVB needed money quickly to pay depositors. Figure 3 shows the effect of the higher required rate of return when, for example, the second payment is divided by (1+0.02) squared rather than (1+0.015) squared for the original (3%) interest rate.

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.

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.

We can calculate this as shown in equation 1, where the required rate of return is i, there are N periods of activity, and x_k is the cash flow in period k. The reciprocal of (1+i) to the kth power is the present worth factor for a single payment.¹ We can, in fact, use equation 1 in spreadsheet form to solve most if not all present-worth problems.

2. =PMT(i,N,P) is the capital recovery function for a principal of P, to be paid off at interest rate i in N periods.

Lean

Introduction to Time Value of Money

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

This is not particularly suitable for a calculator—who wants to do 18 successive calculations for the coupon payments?—but there is also a present worth factor for a uniform series as shown by equation 2. (P/A,i,N) is shorthand for Present worth given an Annual (or other periodic payment) with a required rate of return i, for N periods.

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.

Bankrate⁵ has an online mortgage payment calculator. Suppose we buy a $425,000 house with a 30-year loan, a 6% interest rate, and an $85,000 down payment. This leaves a $340,000 principal that must be paid off with monthly payments. Bankrate comes up with $2,038 a month (this doesn’t count property taxes and other expenses). We can reproduce these results with the capital recovery factor for a uniform series of payments, which returns the annual (or, in this case, monthly) payment whose net present value is the principal. Equation 3 is the capital recovery factor. The monthly interest rate is 0.5% and there are 360 payments.

Excel’s NPV function can also be used, but with a caveat: It doesn’t treat the transaction in the first row as at time zero. That is, it will divide the ($100,000) payment by (1+0.15) and the subsequent $18,000 after tax income by (1+0.15) squared. Microsoft explains on its website, emphasis is mine:

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).

=PV(0.02,18,–15,–1000) returns the same result of $925.04. This is the most convenient way to do this problem in Excel; the tabulations are useful primarily for illustrating the effect of time on the present value of the money. Note that the present value of the bond redemption is only $700.16 rather than $764.91, the present worth if the required semiannual rate of return is 1.5%. This is the source of most of the damage ($64.75) because the biggest amount involved, $1,000, is paid farthest out, where it is divided by (1+0.02) to the 18th power.

Figure 1: Calculation of bond value, 1.5% semiannual required rate of return

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

 

The series compound amount factor (equation 4) returns the future amount from an annual or other periodic investment, given an interest rate or other rate of return i. This illustrates the benefit of regular investments in a 401k or Roth retirement plan.

“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.”

This article has also covered some useful Excel functions.
1. =PV(i,N,A,(F)), where i is the required rate of return, N the number of periods, A the periodic payment, and F, an optional lump sum payment at the end of the series, can calculate the present value of a bond. The function assumes we are paying money, so we must use negative numbers for A and F if we are receiving payments.
• PV(i,N,A) is the function for the present worth of a uniform series.
• PV(i,N,0,F) is the present worth of a single future amount.

Suppose we pay $1,000 for a 10-year Treasury note with an annual interest of 3%. The bond pays interest semiannually at 1.5%, so the coupon payment is $15. If the required rate of return on our investment is 3%, the net present value (NPV) of the bond is $1,000 regardless of the remaining time to maturity. This is because the discounted cash flows of the interest payments and the bond’s face value at redemption always equal $1,000 as long as the available interest rate for similar bonds doesn’t change.

SVB invested in 10-year Treasury notes when the interest rates were even lower; let’s say 1.5% as an example. I saw a 3.5% rate quoted around March 2023, so let’s see what happens if there were eight years left to maturity for SVB’s notes. Also assume 3.5% applies to eight-year notes. There are 16 semiannual periods left, the required semiannual rate of return is 1.75%, and the coupon payment is $7.50. The results aren’t pretty. =PV(0.0175,16,–7.5,–1000) returns $861.50, i.e., if the bonds must be sold immediately, each will incur a $148.50 (14.85%) loss on the principal.

Car payments

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.

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.

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.

Now suppose the car’s sticker price is $30,000, and the dealer quotes 75 easy payments of $600. We can use Excel’s Solver function to find the internal rate of return (IRR) that returns a net present value of zero. That is,

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.