## Valuing a Series of Cash Flows

This section discusses how to value a series of cash flows and offers a few exercises related to mortgage loans that illustrate how annuities pertain to everyday situations.

### Valuing a Series of Cash Flows

#### LEARNING OBJECTIVES

1. Discuss the importance of the idea of the time value of money in financial decisions.
2. Define the present value of a series of cash flows.
3. Define an annuity.
4. Identify the factors you need to know to calculate the value of an annuity.
5. Discuss the relationships of those factors to the annuity's value.
6. Define a perpetuity.

It is quite common in finance to value a series of future cash flows ($CF$), perhaps a series of withdrawals from a retirement account, interest payments from a bond, or deposits for a savings account. The present value ($PV$) of the series of cash flows is equal to the sum of the present value of each cash flow, so valuation is straightforward: find the present value of each cash flow and then add them up.

Often, the series of cash flows is such that each cash flow has the same future value. When there are regular payments at regular intervals and each payment is the same amount, that series of cash flows is an annuity. Most consumer loan repayments are annuities, as are, typically, installment purchases, mortgages, retirement investments, savings plans, and retirement plan payouts. Fixed-rate bond interest payments are an annuity, as are stable stock dividends over long periods of time. You could think of your paycheck as an annuity, as are many living expenses, such as groceries and utilities, for which you pay roughly the same amount regularly.

To calculate the present value of an annuity, you need to know

• the amount of the future cash flows (the same for each),
• the frequency of the cash flows,
• the number of cash flows ($t$),
• the rate at which time affects value ($r$).

Almost any calculator and the many readily available software applications can do the math for you, but it is important for you to understand the relationships between time, risk, opportunity cost, and value.

If you win the lottery, for example, you are typically offered a choice of payouts for your winnings: a lump sum or an annual payment over twenty years.

The lottery agency would prefer that you took the annual payment because it would not have to give up as much liquidity all at once; it could hold on to its liquidity longer. To make the annual payment more attractive for you – it isn't, because you would want to have more liquidity sooner – the lump-sum option is discounted to reflect the present value of the payment annuity. The discount rate, which determines that present value, is chosen at the discretion of the lottery agency.

Say you win $10 million. The lottery agency offers you a choice: take$500,000 per year over 20 years or take a one-time lump-sum payout of $6,700,000. You would choose the alternative with the greatest value. The present value of the lump-sum payout is$6,700,000. The value of the annuity is not simply $10 million, or$500,000 × 20, because those $500,000 payments are received over time and time affects liquidity and thus value. So the question is, What is the annuity worth to you? Your discount rate or opportunity cost will determine the annuity's value to you, as Figure 4.8 "Lottery Present Value with Different Discount Rates" shows. Figure 4.8 Lottery Present Value with Different Discount Rates 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 percent. In other words, if your discount rate is about 4 percent or less – if you don't have more lucrative choices than earning 4 percent 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 percent, or if you feel that your use of that liquidity would earn you more than 4 percent, 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. Suppose your payout was more (or less) each year, or suppose your payout happened over more (or fewer) years (Figure 4.9 "Lottery Payout Present Values"). Figure 4.9 Lottery Payout Present Values As seen in Figure 4.9 "Lottery Payout Present Values", 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 are, 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 college, 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 percent, 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? Figure 4.10 "Lottery Payout Future Values" shows how future values would change.

Figure 4.10 Lottery Payout Future Values

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. 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, 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 so 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.