4

1,040 + (1,040 × 0.04) =
1,040 × (1 + 0.04) =
[1,000 × (1 + 0.04)] × (1 + 0.04) =
1,000 × (1+0.04)²=
1,081.60

As expected, the present value of the annuity is less if your discount rate—or opportunity cost or next-best choice—is more. The annuity would be worth the same to you as the lump-sum payout if your discount rate were 4.16 per cent.

In other words, if your discount rate is about 4 per cent or less—if you don’t have more lucrative choices than earning 4 per cent with that liquidity—then the annuity is worth more to you than the immediate payout. You can afford to wait for that liquidity and collect it over twenty years because you have no better choice. On the other hand, if your discount rate is higher than 4 per cent, or if you feel that your use of that liquidity would earn you more than 4 per cent, then you have more lucrative things to do with that money and you want it now: the annuity is worth less to you than the payout.

For an annuity, as when relating one cash flow’s present and future value, the greater the rate at which time affects value, the greater the effect on the present value. When opportunity cost or risk is low, waiting for liquidity doesn’t matter as much as when opportunity costs or risks are higher. When opportunity costs are low, you have nothing better to do with your liquidity, but when opportunity costs are higher, you may sacrifice more by having no liquidity. Liquidity is valuable because it allows you to make choices. After all, if there are no more valuable choices to make, you lose little by giving up liquidity. The higher the rate at which time affects value, the more it costs to wait for liquidity, and the more choices pass you by while you wait for liquidity.

When risk is low, it is not really important to have your liquidity firmly in hand any sooner because you’ll have it sooner or later anyhow. But when risk is high, getting liquidity sooner becomes more important because it lessens the chance of not getting it at all. The higher the rate at which time affects value, the more risk there is in waiting for liquidity and the more chance that you won’t get it at all.

As r increases, the PV of the annuity decreases.
As r decreases, the PV of the annuity increases.

You can also look at the relationship of time and cash flow to annuity value in Table 4.3.2 “Lottery Payout Present Values.” Suppose your payout was more (or less) each year, or suppose your payout happened over more (or fewer) years.

As seen in Table 4.3.2, the amount of each payment or cash flow affects the value of the annuity because more cash means more liquidity and greater value.

As CF increases, the PV of the annuity increases.
As CF decreases, the PV of the annuity decreases.

Although time increases the distance from liquidity, with an annuity, it also increases the number of payments because payments occur periodically. The more periods in the annuity, the more cash flows and the more liquidity there is, thus increasing the value of the annuity.

As t increases, the PV of the annuity increases.
As t decreases, the PV of the annuity decreases.

It is common in financial planning to calculate the FV of a series of cash flows. This calculation is useful when saving for a goal where a specific amount will be required at a specific point in the future (e.g., saving for university, a wedding, or retirement).

It turns out that the relationships between time, risk, opportunity cost, and value are predictable going forward as well. Say you decide to take the $500,000 annual lottery payout for twenty years. If you deposit that payout in a bank account earning 4 per cent, how much would you have in twenty years? What if the account earned more interest? Less interest? What if you won more (or less) so the payout was more (or less) each year?

What if you won $15 million and the payout was $500,000 per year for thirty years; how much would you have then? Or if you won $5 million and the payout was only for ten years? Table 4.3.3 shows how future values would change.

Going forward, the rate at which time affects value (r) is the rate at which value grows, or the rate at which your value compounds. It is also called the rate of compounding. The bigger the effect of time on value, the more value you will end up with because more time has affected the value of your money while it was growing as it waited for you. Steven Arnott compares compounding to that of a snowball,

What starts as a small ball begins to collect additional snow as it’s rolled through the field. The snowball grows, and as you continue pushing, it collects snow at a faster and faster rate. The longer you continue, the faster it picks up snow and the larger it becomes. This is the same process your money goes through with compound interest. Your initial savings collect larger and larger amounts of interest, allowing your money to work for you. (2021, p. 10)

An example of how compounding can positively impact your savings, especially if you start saving early in life, is highlighted below:

Hunter begins saving $2,500 annually when he turns twenty-five and continues to save this same amount annually until he turns sixty-five. His savings compound at a rate of 5% annually which results in $301,999.44 (Investment: $100,000, Total Interest Earned: $201,999.44) by the time he retires. Donald begins saving $2,500 annually when he turns forty-two until he turns sixty-five. Donald’s savings also compound at a rate of 5% annually. By the time Donald retires, he has only saved $103,576.19 (Investment: 57,500, Total Interest Earned: $46,076.19). Hunter has saved far more for retirement by starting early.

So, looking forward at the future value of an annuity:

As r increases, the FV of the annuity increases.
As r decreases, the FV of the annuity decreases.

The amount of each payment or cash flow affects the value of the annuity because more cash means more liquidity and greater value. If you were getting more cash each year and depositing it into your account, you’d end up with more value.

As CF increases, the FV of the annuity increases.
As CF decreases, the FV of the annuity decreases.

 

The more time there is, the more time can affect value. As payments occur periodically, the more cash flows there are, and the more liquidity there is. The more periods in the annuity, the more cash flows, and the greater the effect of time, thus increasing the future value of the annuity.

As t increases, the FV of the annuity increases.
As t decreases, the FV of the annuity decreases.

There is also a special kind of annuity called a perpetuity, which is an annuity that goes on forever (i.e., a series of cash flows of equal amounts occurring at regular intervals that never ends). It is hard to imagine a stream of cash flows that never ends, but it is actually not as rare as it sounds. The dividends from a share of corporate stock are a perpetuity, because in theory, a corporation has an infinite life (as a separate legal entity from its shareholders or owners) and because, for many reasons, corporations like to maintain a steady dividend for their shareholders.

The perpetuity represents the maximum value of the annuity, or the value of the annuity with the most cash flows and therefore the most liquidity and therefore the most value.

Life Is a Series of Cash Flows

Once you understand the idea of the time value of money, and its use for valuing a series of cash flows, and the idea of annuities in particular, you won’t believe how you ever got through life without them. These are the fundamental relationships that structure so many financial decisions, most of which involve a series of cash inflows or outflows. Understanding these relationships can be a tool to help you answer some of the most common financial questions about buying and selling liquidity, because loans and investments are so often structured as annuities and certainly take place over time.

Loans are usually designed as annuities, with regular periodic payments that include interest expense and principal repayment. Using these relationships, you can see the effect of a different amount borrowed (PV_annuity), interest rate (r), or term of the loan (t) on the periodic payment (CF).

For example, as seen in Table 4.3.4 “Mortgage Calculations,” if you get a $250,000 (PV), thirty-year (t), 6.5 per cent (r) mortgage, the monthly payment will be $1,577 (CF). If the same mortgage had an interest rate of only 5.5 per cent (r), your monthly payment would decrease to $1,423 (CF). If it were a fifteen-year (t) mortgage, still at 6.5 per cent (r), the monthly payment would be $2,175 (CF). If you can make a larger down payment and borrow less—say $200,000 (PV)—then with a thirty-year (t), 6.5 per cent (r) mortgage your monthly payment would be only $1,262 (CF).

Note that in Table 4.3.4, the mortgage rate is the monthly rate—that is, the annual rate divided by twelve (months in the year), or r / 12, and that t is stated as the number of months, or the number of years × 12 (months in the year). That is because the mortgage requires monthly payments, so all the variables must be expressed in units of months. In general, the periodic unit used is defined by the frequency of the cash flows and must agree for all variables. In this example, because you have monthly cash flows, you must calculate using the monthly discount rate (r) and the number of months (t).

Saving to reach a goal—to provide a down payment on a house, or a child’s education, or retirement income—is often accomplished by a plan of regular deposits to an account for that purpose. The savings plan is an annuity, so these relationships can be used to calculate how much would have to be saved each period to reach the goal (CF), or given how much can be saved each period, how long it will take to reach the goal (t), or how a better investment return (r) would affect the periodic savings, or the time needed (t), or the goal (FV).

For example, if you want to have $1 million (FV) in the bank when you retire, and your bank pays 3 per cent (r) interest per year, and you can save $10,000 per year (CF) toward retirement, can you afford to retire at age sixty-five? You could if you start saving at age eighteen, because with that annual saving at that rate of return, it will take forty-seven years (t) to have $1,000,000 (FV). If you could save $20,000 per year (CF), it would only take thirty-one years (t) to save $1,000,000 (FV). If you are already forty years old, you could do it if you save $27,428 per year (CF) or if you can earn a return of at least 5.34 per cent (r). See Table 4.3.5.

Financial Calculations

Modern tools make it much easier to do the math. Widely available calculators, spreadsheets, and software have been developed to be very user friendly.

Financial calculators have the equations relating the present and future values, cash flows, the discount rate, and time embedded, for single amounts or for a series of cash flows, so that you can calculate any one of those variables if you know all the others.

Personal finance software packages usually come with a planning calculator that performs a similar function. These tools are usually presented as a “mortgage calculator” or a “loan calculator” or a “retirement planner,” and they are set up to answer common planning questions such as, “How much do I have to save every year for retirement?” or, “What will my monthly loan payment be?”

Spreadsheets also have the relevant equations built in, as functions or as macros. There are also stand-alone software applications that may be downloaded to a mobile device, such as a smartphone. They are useful in answering planning questions, but they lack the ability to store and track your situation in the way that a more complete software package can.

The calculations are discussed here not so that you can perform them—you have many tools to choose from that can do that more efficiently—but so that you can understand them, and most importantly, so that you can understand the relationships that they describe.

4.4 USING FINANCIAL STATEMENTS TO EVALUATE FINANCIAL CHOICES

Now that you understand the relationship of time and value, especially looking forward, you can begin to think about how your ideas and plans will look as they happen. More specifically, you can begin to see how your future will look in the mirror of your financial statements. Projected or pro forma financial statements can show the consequences of your choices. To project future financial statements, you need to be able to envision the expected results of all the items on them. This can be difficult, for there can be many variables that may affect your income and expenses or cash flows, and some of them may be unpredictable. Predictions always contain uncertainty, so projections are always, at best, educated guesses. Still, they can be useful in helping you to see how the future may look.

For example, we can glimpse Brittany’s projected cash flow statements and balance sheets for each of her choices, along with their possible outcomes. Brittany can actually project how her financial statements will look after each choice is followed.

When making financial decisions, it is helpful to be able to think in terms of their consequences on the financial statements, which provide an order to our summary of financial results. For example, in previous chapters, Brittany was deciding how to decrease her debt. Her choices were to continue to pay it down gradually as she does now; to get a second job to pay it off faster; or to go to Vegas, hit it big (or lose big), and eliminate her debt altogether (or wind up with even more). Brittany can look at the effects of each choice on her financial statements as seen in Table 4.4.1.

Looking more closely at the actual numbers on each statement gives us a much clearer idea of Brittany’s situation. Beginning with the income statement, income will increase if she works a second job or goes to Vegas and wins, while expenses will increase (travel expense) if she goes to Vegas at all. Assume that her second job would bring in an extra $20,000 income and that she could win or lose $100,000 in Vegas. Any change in gross wages or winnings (losses) would have a tax consequence; if she loses in Vegas, she will still have income taxes on her salary. See Table 4.4.2.

While Vegas yields the largest increase in net income or personal profit if Brittany wins, it creates the largest decrease if she loses; it is clearly the riskiest option. The pro forma cash flow statements (shown in Table 4.4.3) reinforce this observation.

If Brittany has a second job, she will use the extra cash flow, after taxes, to pay down her student loan, leaving her with a bit more free cash flow than she would have had without the second job. If she wins in Vegas, she can pay off both her car loan and her student loan and still have an increased free cash flow. However, if she loses in Vegas, she will have to secure more debt to cover her losses. Assuming she borrows as much as she loses, she will have a small negative net cash flow and no free cash flow, and her other assets will have to make up for this loss of cash value.

So, how will Brittany’s financial condition look in one year? That depends on how she proceeds, but the pro forma balance sheets, as seen in Table 4.4.4, provide a glimpse.