### Glossary of Time Value of Money Terms

Michael Sack Elmaleh , CPA, CVA

###### Video: Capital Budgeting

In the previous page we looked at bond pricing. In those cases, we knew with a fair degree of certainty the fixed constant interest payments, the maturity value and the plausible range of interest rates to apply in order to set a price on a given bond. Here we turn to an application where there is far less certainty in future cash outlays and returns and the interest rates to convert future dollars to present dollars are far from obvious. Also, for the first time we will encounter applications where the stream of future cash flows are both positive and negative.

###### Example

A coffee shop owner must replace the shop’s commercial grade coffee maker. The coffee maker could be purchased outright for \$20,000 or leased for a monthly payment of \$386.66 for five years. Assume that the machine will last for precisely five years. In nominal dollars the lease option will require \$3,220 more than the purchase option (\$387*60 = \$23,220-\$20,000). This table shows the non-discounted annual outlays under a lease option.

But of course, the timing of the payments varies. Under the purchase option a full immediate outlay of \$20,000 is required. Under the lease option the payments are spread over five years.

###### Discount Rates are Critical

So, in taking account of time value how do we decide the best option? Well, it turns out the best option depends on the interest or discount rate used to convert the future lease payments to a present value.

This table below shows the difference in present values as a function of different discount rates. As you can see the net present value of the lease option is more expensive than the buy option when we use a discount interest rate of 5% or less but less expensive when we apply a discount/interest rate of greater than 6%. At 6% the options are equivalent. This is no accident.

In this example I assumed that the equipment dealer would want what is in effect an imputed interest return if the coffee shop owner chose to lease the machine rather than buy it outright. So, in calculating the required lease payment I used the mortgage payment formula amortizing the cost of the machine over five years of monthly payments at a 6% annual interest rate or .5% monthly.

But from the standpoint of the coffee shop owner what is the correct interest or discount rate to use in evaluating whether to lease or buy the machine? The answer depends on the options available to the owner. If the owner has no funds to buy the machine outright and cannot borrow the funds to buy the machine at any reasonable interest rate, then leasing is the only option. But what if the owner has the cash on hand to buy the machine?

###### The Opportunity Cost Principal

Well, here we again invoke the concept of opportunity cost to guide us. We simply compare the rate of return that could be earned with the funds on hand with the rate of interest imputed in the lease option. If the rate of return on the available funds is less than the imputed interest rate on the lease then the owner should take the buy option. If the rate of the return is greater than the imputed interest rate on the equipment lease, then the owner should take the lease option.

But how do you figure out the imputed interest rate on the lease if the lessor has not openly stated it? The simplest approach is to use an Excel spread sheet formula like Rate which just requires that you enter the present value of the leased asset, the lease payments and the number of payments. Of course, you have to know the present or market value of the machine in order to apply this formula.

Applying this formula would yield the 6% annual interest rate. Which is of course no surprise because this is precisely the interest rate I used in coming up with the \$387 monthly payment. So, if the coffee shop owner makes less than 6% return on her available funds then she should buy the machine outright. If the owner makes more than 6%, she should lease the machine. Now I should note that I am ignoring income tax effects because interest expense is deductible. But if the interest earned is also taxable at the same rate, then we can ignore these effects.

Of course, there may be another option. It may be possible to borrow the funds to buy the coffee machine at an interest rate lower than the imputed interest rate in the equipment lease option. Again though, it is important to compute the imputed interest rate of the lease payments to make the most rational choice.

###### Evaluating New Products or Service Lines

Now there is another type of capital budgeting problem that business owners and managers encounter that is a bit more complicated than figuring out the imputed interest rate on an equipment lease. Often an established business will investigate starting a new product or service line. In these situations, the new product or service often is not forecast to be immediately profitable. In fact, for the first few years there is likely to be net cash losses followed hopefully by several years of net profit. As an example, suppose the profitability projection for a new product line looks like this:

This is the cash flow forecast in the non-discounted form; in other words, these are non-time valued nominal values. Now how should the business decide whether to go ahead with this new product? We can see that as in the case with the equipment lease that the value attached to the project will depend upon the rate at which we discount the cash flows. You can see the lower the discount rate the more valuable the new product appears. So, what is the correct or rational choice here?

Well as in the case of the equipment lease problem this will depend upon the alternative uses of capital available to the business. In this case the business will have to invest \$16K over two years before it yields any profit at all. So, the choice of discount rate will depend upon how high a rate of return the business could achieve on an alternative use of \$16,000.

One easy rule of thumb is to insist that any project have a discount rate greater than the businesses’ cost to borrow funds. So, if the businesses’ borrowing rate is say 5% then use this rate. In this case using such a discount rate yields a positive net present value so the business would decide to introduce the new product…. unless of course there were alternative investments that would yield higher present values using the same discount rate.

###### Internal Rates of Return

Now there is another time value computation that is often used in these kinds of capital budgeting decisions called the internal rate of return or IRR. This computation shows the rate of return that brings the net present value of an investment precisely to zero. Using a built in Excel function, IRR, shows the  in this example to be 10% which with rounding equals the zero point we computed utilizing the net present value computation. Generally, if the computed IRR is higher than the firms’ rate of return on alternative investments than the firm should go ahead with the investment. I realize this is somewhat counterintuitive. But as you saw in the above table the lower the rate of return available to the firm for alternative investments, the lower the discount rate used and the higher the present value of the investment income stream.

I suggested that using a firm’s borrowing rate is the best measure of a business’s opportunity cost and that this is the rate that should be used for discounting forecast cash flows. Many would argue that the opportunity cost should be measured by what is known as the Weighted Average Cost of Capital or WACC. As the name implies this measure is a weighted average of the cost of debt and cost of equity. The problem with this approach is that it is not very clear how to measure the cost of equity although there are whole industries devoted to this endeavor. Most such endeavors try to construct seemingly reliable indexes that in the end are not reliable at all.