3.3 Calculating the Relationship of Time and Value

Financial calculation is not often a necessary skill since it is easier to use calculators, spreadsheets, and software. However, understanding the calculations is important in understanding the relationships between time, risk, opportunity cost, and value.

To do the math, you need to know:

1. what the future cash flows will be,
2. when the future cash flows will be, and
3. the rate at which time affects value (e.g., the costs per time period, or the magnitude [the size or amount] of the effect of time on value).

It is usually not difficult to forecast the timing and amounts of future cash flows. Although there may be some uncertainty about them, gauging the rate at which time affects money can require some judgment. That rate, commonly called the discount rate because time discounts value, is the opportunity cost of not having liquidity. Opportunity cost derives from forgone choices or sacrificed alternatives, and sometimes it is not clear what those might have been. It is an important judgment call to make, though, because the rate will directly affect the valuation process.

At times, the alternatives are clear: you could be putting the liquidity in an account earning 3 %, so that’s your opportunity cost of not having it. Or you are paying 6.5% on a loan, which you wouldn’t be paying if you had enough liquidity to avoid having to borrow; that’s your opportunity cost. Sometimes, however, your opportunity cost is not so clear.

Say that today is your twentieth birthday. Your grandparents have promised to give you $1,000 for your twenty-first birthday, one year from today. If you had the money today, what would it be worth? That is, how much would $1,000 worth of liquidity one year from now be worth today?

That depends on the cost of its not being liquid today, or on the opportunity costs and risks created by not having liquidity today. If you had $1,000 today, you could buy things and enjoy them, or you could deposit it in an interest-bearing account. So on your twenty-first birthday, you would have more than $1,000—you would have the $1,000 plus whatever interest it had earned. If your bank pays 4% per year (interest rates are always stated as annual rates) on your account, then you would earn $40 of interest in the next year, or $1,000 × .04. So, on your twenty-first birthday you would have $1,040 as expressed in the equation below:

[latex]$1,000 + (1,000×0.04) = $1,000 × (1 + 0.04) = $1,040[/latex]

Assuming there is little chance that your grandparents will not be able to give this gift, there is negligible risk. Your only cost of not having liquidity now is the opportunity cost of having to delay consumption or not earning the interest you could have earned.

The cost of delayed consumption is largely derived from a subjective valuation of whatever is consumed, or its utility or satisfaction. The more value you place on having something, the more it “costs” you not to have it, and the more time that you are without it affects its value.

Assuming that if you had the money today you would save it, by having to wait until your twenty-first birthday to receive it—and not having it today—you miss out on the $40 it could have earned.

So, what would that nominal $1,000 (that future value that you get one year from now) actually be worth today? The rate at which time affects your value is 4% because that’s what having a choice (to spend or invest it) could earn for you if only you had received the $1,000. That’s your opportunity cost; that’s what it costs you to not have liquidity. Since PV × (1+r)^t = FV, then PV = FV/[(1+r)^t], so PV = 1,000/[(1.04)¹]= 961.5385.

Your gift is worth $961.5385 today (its present value). If your grandparents offered to give you your twenty-first birthday gift on your twentieth birthday, they could give you $961.5385 today, which would be the equivalent value to you of getting $1,000 one year from now.

It is important to understand the relationships between time, risk, opportunity cost, and value. This equation describes that relationship:

The strategy implications of this understanding are simple yet critical. All things being equal, it is more valuable to have liquidity (get paid, or have positive cash flow) sooner rather than later and give up liquidity (pay out, or have negative cash flow) later rather than sooner.

If possible, accelerate incoming cash flows and decelerate outgoing cash flows: get paid sooner, but pay out later. Or, as Popeye’s pal Wimpy used to say, “I’ll give you 50 cents tomorrow for a hamburger today.”

Please see the following TVM tables developed by Jodi Letkiewicz (York University), in order to practice calculating future value and present value. A sample of the first TVM table—Future Value of a Lump Sum ($1 at the end of t periods)—is illustrated below. Remember that FV = PV × (1+r)^t.