## Time Value of Money

This posting is going to be a little bit different from my typical blog postings, in that I’m going to write about a more general analytic concept that is particularly important to keep in mind when trying to understand projects that are either paid for or receive revenues over a period of time — a concept that is often called the “time value of money“.

This posting is prompted by many comments that I have heard from friends, colleagues, fellow public officials, and other community members in reaction to the controversial decision by the Indiana Finance Authority to award a 35-year contract to design, build, operate, maintain, and finance section 5 of I-69, a 21-mile segment of I-69 from south of Bloomington to south of Martinsville — a so-called Public-Private Partnership, or P3.

Please note that the following discussion does not in any way address the issue of whether building the highway at all is a good investment by the state. All it does is address the issue of paying for a large project like this over time vs. paying for it with current revenues/cash on hand.

## I-69 Example

Without getting too far into the weeds on the specifics, the arrangement is that the State of Indiana will pay a contractor $21.8M per year for 35 years for the design, construction, operation, maintenance, and financing of the 21-mile section of highway. The government estimate of the cost of the design and construction of the highway was around $350M. However, if take the $21.8M per year times 35 years, you get a total of $763M. The claim that has been made, therefore, is that by using this P3 financing vehicle in which payments are made to the contractor over time, that the government is paying *more than double* for the highway than it would if paid through conventional means (i.e., up front, with cash).

However, this conclusion ignores the *time value of money* (TVOM). The basic principle behind TVOM is that a given amount of money today is worth more than it is at some future point. When you think about it, this should be self-evident; just ask yourself: given the choice, would you prefer $100 to be given to you right now or in a year? Once you accept that premise, the next question is how *much* more is that given amount of money worth today than it is at a future point? The answer is: a given amount of money is worth more today than it is at some future point by the amount you could earn by investing (or otherwise using) that given amount of money until that future point.

In order to compare alternative investments (or financing schemes) that are made over various periods of time, we need to make sure we compare apples to apples. One way to do that is to convert all alternatives to **present value (PV)** and then compare. Let’s consider our I-69 example. In that example, the state will be paying the contractor $21.8M per year over 35 years. However, the $21.8M in year 2 is not worth as much as the $21.8M in year 1, and the $21.8M in year 35 is *certainly* not worth as much as it is in year 1. So how do we convert the entire 35-year payout to Present Value, as though it were all being paid out immediately? And more importantly — how do we compare the 35-year payout against a $350M cost if the design and construction were paid out of cash today.

Basically, we *discount* future payments by the amount of money we could earn on the money we save by not having to pay it this year! How much do we discount it by — in other words, how do we compute the present value?

## Quick Mathematical Interlude

I’m going to take a moment to do a little math here; however, if you want to skip to the next section, you won’t lose much of the overall argument. The actual equation is: PV = FV / (1+i)^n, in which *PV* is Present Value, *FV* is Future Value, *i* is the interest rate per period that you could earn on the money, and *n* is the number of periods that you could earn that interest rate. For a quick example, let’s consider the following alternatives:

*Alternative A:*I give you $100 today*Alternative B:*I give you $100 a year from today

First of all, we *know *that alternative A — the $100 I give you today has a present value of $100.To to compare it to alternative B (I give you $100 a year from now), though, we need to compute the present value *today *of the $100 I pay you a year from now. In the above equation, $100 is the *future value* (FV) — the amount that the $100 paid in one year will be worth at the time it is paid. i is the interest rate that we could be earning per year (or per any time period) with the $100. This is where TVOM analysis gets a little squishy, and the results you get can differ a lot depending on your assumptions. How much money can you earn with $100 in a year? Depends a lot on how you invest it! If in a savings account, almost nothing — less than 1%. However many investments earn quite a bit more than a savings account. One number that is considered pretty fair to use is the average municipal bond rate. At the very least, the municipal bond rate can be considered a good proxy for the opportunity costs of the money over a given period of time.

The following Web site provides municipal bond rates for various maturity ranges and credit ratings:

Just for the sake of this example, I’m going to take a national rate for a 10-year bond with AAA credit rating (which Indiana has) — an interest rate of 2.20% per year (of course, this amount may change).

Going back to our equation, we have: FV = $100, i = 0.022 (the 2.2% interest rate), and n=1 (1 year). The present value of that $100 paid out in a year is: PV = 100/(1+0.022)^1 = 100/1.022 = $97.85. In other words, with the assumptions we made, $100 paid to you a year from now is only worth $97.85 *today*.

## I-69 Example, Revisited

OK, so let’s apply the TVOM principle to analyzing the 35-year contract for Section 5 of I-69. For the purposes of illustrating TVOM in comparing the 35-year contract with a conventional financing (i.e. paying for the project out of current tax receipts and cash balance), I am simplifying the situation *dramatically* in the following way: the 35-year $21.8M/year payout to the contractor does not only include the design and construction of the road, but also the operations and maintenance of the road — money that would have had to be spent anyway in all 35 of the outyears regardless of the method of financing the design and construction of the road. If we really want to compare alternatives fairly, we would subtract out the costs of maintaining and operating the road for all 35 years — figures I don’t have close to hand. So this TVOM analysis can really be considered to be a *worst-case *from the perspective of the P3 scenario. In other words, the real cost of the P3 versus conventional financing is much more in the favor of P3 than it is in the following numbers.

I created the following table for all 35 years of payments (all numbers are in millions). The first column, Payment, shows the actual payment made to the contractor each year for the 35 year period of performance. The following 5 columns show how much those 35 years of annual payments are worth *today *(the only fair way to compare the arrangement against a conventional financing arrangement where the whole thing is paid with cash/current taxes), using 5 different interest rate assumptions (1%, 2%, 3%, 4%, and 5%). For a project of this size, given the municipal bond rates for 30 year bonds for AAA credit, the most realistic assumption is probably somewhere around 4% (from the recent past history of municipal bonds, probably a little under 4%).

So with the 4% interest rate assumption, we can see that the present value of 35 years of $21.8M annual payments is **not** $763M (i.e. 35 times $21.8M), but the much lower $406.89M. Obviously that number changes depending on the interest rate assumption. The higher the interest rate, the lower the present value. This should make intuitive sense: the more opportunity I have to make use of the money up front, the less valuable it is for me to have the money in the future compared to the present.

So to get back to our hypothetical-not-so-hypothetical example. We want to compare paying for a $350M highway construction project out of current dollars against the 35-year $21.8M/year P3 arrangement. With our assumption of 4% interest rate, the present value (cost) of the 35-year arrangement is $406.89M, compared to $350M if paid in cash — more, to be sure — but dramatically less than simply adding up the 35 annual payments ($763M). And again, this doesn’t even include the fact that some of the annual payments cover the costs of maintenance and operations of the highway, which would be paid out anyway, regardless of how the design and construction are financed.

## Conclusion

None of this discussion should be taken as an endorsement of the P3 process, or the fact that the P3 financing model basically guarantees that the winning contractor will be a large multinational corporation with deep access to finance. But I do want to make sure that as we move forward with an arrangement that by all accounts will only become more common, discussions of these kinds of arrangements are based on facts. And the fact is that a given sum of money is less valuable in the future than it is today. Most certainly we will be paying more for the privilege of stretching out the payments for the road over 35 years. But we won’t be paying double…not even close.

Nicely done. Would discussing inflation take it too far into the weeds?

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