Chapter 7 Time Value of Money: Multiple Flows

7.1 Multiple Cash Flows

Jamie was up late watching infomercials, and saw that she could purchase the latest wonder product for either $43, or three easy monthly installments (the first payment is immediate) of $15. If she earns 4% APR compounded monthly in her savings account, which option should she pick?

The trick to comparing cash flows usually is to PV them all to the same point in time. When they have been discounted to the same time, then they can be added, subtracted, or compared without worry. Discounting all of the cash flows for an investment to the present, adding inflows and subtracting outflows, is called finding the net present value (NPV)Discounting all of the cash flows for an investment to the present, adding inflows and subtracting outflows.. When deciding among investments, we typically wish to choose the one with the highest (or, in some cases, least negative) NPV.

The NPV of paying $43 immediately is obviously−$43. To find the NPV of the installment plan, we should look at the timeline.

4% APR compounded monthly gives a monthly interest rate of 0.3333%. (More discussion of example)

Many financial calculators have an NPV solving functionality to handle multiple cash flows. Please check the appendix for differences between the models, but the standard process is:

• Step 1: Enter Cash Flow worksheet.
• Step 1a: If entering new data (as opposed to editing what is already entered), clear the worksheet.
• Step 2: Enter CF0 (Net cash flows at time 0).
• Step 3a: Advance worksheet to enter C01 (Cash flow at time 1).
• Step 3b: Advance worksheet to enter F01 (The number of times C01 repeats).
• Repeat steps 3a and 3b until all are entered.
• Step 4: Press NPV and enter periodic interest rate.
• Step 5: Compute NPV.

In our example problem, the initial cash flow (CF0) is −$15. Both of the next two cash flows are −$15, so we can either enter them separately (C01: −15; F01: 1; C02: −15; F02: 1) or tell the calculator to repeat the entry (C01: −15; F01: 2). Both methods will give the proper answer. Here are the precise keystrokes for the latter method:

<CF> <2ND> <CLR WORK> −15 <ENTER> <DOWN ARROW> −15 <ENTER> <DOWN ARROW> 2 <ENTER> <NPV> 0.3333 <ENTER> <DOWN ARROW> <CPT>

Remember that interest rate should be entered as a percentage (0.3333 in this case, not 0.003333).

The corresponding spreadsheet function is:

=NPV(periodic rate, cash flows from period 1 to n) + net of initial cash flows

Note that the spreadsheet function expects cash flows to start in the first period: any initial cash flows need to be netted and added to the result. Like the financial calculator, positive values should be used for cash inflows, and negative values for outflows.

7.2 Perpetuities

As discussed in the previous section, finding the NPV for a fixed set of cash flows will always work. But what if the cash flows were to continue forever? If the payments are constant over time, we can intuit what the value must be for the entire set, instead of figuring out each one separately: if I put $200 into a bank account that promises me 5% per year, and I just withdraw the interest each year without ever touching the principal, how much will I get each year? The first year, the interst will be 5% of $200, or $10. Since I withdraw the interest, my second year balance is also $200, so my interest is again $10. In fact, I (or my heirs) can withdraw $10 a year forever as long as I keep the original $200 in the bank.

Note that this is essentially the same as the bank offering the following: pay $200 today, and the bank will pay the customer $10/year. Therefore, at 5% interest, $10/year forever must be the equivalent of exactly $200 today! This arrangement, with constant periodic payments lasting forever, is called a perpetuityConstant periodic payments lasting forever.. The periodic payment must be the interest rate times the present value, since it is analogous to our bank deposit example above.

PV × r = PMT $\text{PV }=\frac{PMT}{r}$

Note that the interest rate and the payment must be quoted in the same period; if the payments are monthly, than the monthly rate should be used. Also, the first payment is assumed to be at the end of the year; if the first payment was immediate, then the value would be higher by exactly the amount of one payment.

The fact that a stream of payments lasting forever has a specific finite value today is a surprising result for some people. The key is to realize that, because of the time value of money, each successive payment contributes a smaller and smaller amount to the PV.

Returning to the bank deposit example: if I don’t withdraw the full 5% of the $200, instead only withdrawing 4% (that is, 1% less), then I won’t receive as much money this year ($8 vs. $10), but my principal balance will grow by 1% to $202. If I continue to withdraw only 4%, my second year’s withdraw will be $202 × 4% = $8.08, which happens to be exactly 1% higher than my first year’s withdrawal. Withdrawing 1% less causes both my principal to grow by 1% at the end of the year, and my following year’s payment to grow by the same 1%.

If I consistently withdraw less than the interest rate (the difference, interest rate minus withdrawal rate, being the growth rate, g), I can model the relationship using the following modified perpetuity equation:

PV now × (interest rate – growth rate) = payment at end of year $P{V}_n\times (r–g)=PM{T}_{n+1}$ $P{V}_n=\frac{PM{T}_{n+1}}{(r-g)}\{\text{for }g<r\}$

Note that we are assuming that some money is withdrawn in each year, meaning that g must be less than r. And, when g equals zero, this equation reduces to our earlier equation.

Since we know that our growth rate, g, is constant, given any payment we can find the following year’s payment by increasing it by g.

$PM{T}_{n+1}=PM{T}_n+(g\times PM{T}_n)=PM{T}_n(1+g)$ $P{V}_n=\frac{PM{T}_{n+1}}{(r-g)}=\frac{PM{T}_n(1+g)}{(r-g)}\{\text{for }g<r\}$

Here PM{T}_n is the payment just paid, so technically its value isn’t included in the PV; we are only including it since we can use it to find the payment due next year.

7.3 Annuities

Since, in practice, very few financial arrangements include “forever” as a reasonable timeframe, we should consider the value of cash flows over shorter periods. Any set of constant periodic payments lasting for a fixed amount of time is called an annuityConstant periodic payments over a fixed amount of time.. Paying down a credit card, mortgages, lottery payouts, and more can all be modeled as annuities. All annutities can be valued using the “long way”, by calculating the NPV of the cash flows, but we can also value them using mathematical intuition.

If we want to figure out the value of $10/year for 15 years, and the current interest rate is 5%, then we can use our perpetuity formula to determine the value of the annuity in the following way:

1. Determine the value of an equivalent perpetuity (in this case, $10/year at 5% gives a PV for the perpetuity of $200).
2. Determine the PV of the value of the perpetuity at the end of the annuity period (in this case, the PV of $200 in 15 years is $96.20).
3. Subtract #2 from #1 to get the value of the annuity ($200−$96.20 = $103.80).

$\text{PV }=\frac{PMT}{r}-\frac{\frac{PMT}{r}}{{(1+r)}^n}$ $\text{PV }=\frac{PMT}{r}\times (1-\frac{1}{{(1 + r)}^n})$

Sometimes an annuity will also include an additional one-time cash flow at the conclusion of the annuity. In this case, it is a simple matter of adding the PV of this cash flow to your calculated value to get the total value of the annuity. Continuing our example, if the annuity delivered $100 at its conclusion, then the PV would increase by the PV of $100 in 15 years at 5%, or $48.10, making the total value $103.80 + $48.10 = $151.90.

Financial calculators and spreadsheets can also easily solve annuities. On the calculator, the PMT key accepts the input for the periodic payment, and any four inputs can be used to calculate the fifth.

Using a financial calculator:

<CLR TVM> 15 <N> 5 <I> 10 <PMT> 100 <FV> <CPT> <PV> should show the proper solution of −151.90.

Spreadsheet:

=PV(rate, nper, pmt, fv) =PV(5%, 15, 10, 100) −151.90

Also, PMT can be solved for given the other four inputs. =PMT(rate, nper, pv, fv) is the corresponding spreadsheet function. It is critical that we always input our rate, number of periods, and payment based upon the same time period. For example, if payments are monthly, then rate should be a monthly rate, and number of periods will be the number of months.

With most annuities, each payment is assumed to occur at the end of each period. When payment in each period occurs at the beginning instead of the end, the annuity is called an annuity dueAn annuity with payments due at the beginning of each period instead of at the end..

While technically each payment is occurring at the beginning of its period instead of the end, in effect, the only change is that the very last payment has been moved to the very beginning! Thus, the value of an annuity due is higher by the difference between the PV of a payment at the end and the PV of a payment now (which, by definition, is equal to the total payment).

$P{V}_{due}=P{V}_{ordinary}+(\text{PMT}-\frac{PMT}{{(1+r)}^n})$

If our example were an annuity due, its value would increase by $10 minus the PV of $10 in 15 years at 5%, or ($10 – $4.81) = $5.19.

Financial calculators can be set to solve for annuity due (see appendix). Most spreadsheet functions can be set to automatically calculate an annuity due by taking a fifth argument, “type”. When type is omitted, it is assumed to be calculating an ordinary annuity, but by using an input of “1” instead, an annuity due is calculated:

=PV(rate, nper, pmt, fv, type) =PV(5%, 15, 10, 100, 1) −157.09

7.4 Loan Amortization

Fred owes $1,500 on his credit card! When his monthly bill arrives, Fred is relieved to see that the minimum payment is only $30. If the advertised rate is 20% APR (assume monthly compounding), how many months will it take Fred to pay off his debt?

If Fred makes his payments every month, the debt will finally be fully paid after 109 months, or over 9 years! We can verify this quickly using a financial calculator or spreadsheet.

=NPER(20%/12,30,−1500) =108.4

During this time, the total amount of money paid by Fred will be about 109 * $30 = $3,270, or over twice the original amount owed. This means that Fred paid more in interest ($3,270−$1,500 = $1,770 in interest) than he originally borrowed ($1,500). On the plus side, Fred was able to slowly pay down the balance owed, so that he never had to pay more than $30 in any month, but the total principal was paid by the end of the loan. This process of spreading out principal payments of a loan over time is called amortizationThe process of spreading out the principal payments of a loan over time. (from the Latin root “mort-”, meaning death, as the loan balance slowly “dies” as the principal is paid down). Mortgages are a common example of this type of loan (also coming from the same Latin root), as the principal is paid down over 15, 30 or more years. A loan that pays only the interest payments, with no principal payments, is called an interest only loan. If the periodic payments are so low they don’t even cover the interest on the initial principal, then the loan will be a negative amortization loan, and the extra interest will be added to the principal due over time.

As each month passes, the outstanding principal balance decreases, so the interest due also decreases. Thus, over time, the amount paid toward principal grows from a small portion of the monthly payment to a larger portion. This is particularly important when evaluating mortgages, as often the interest portion is tax deductible, but the principal portion is not. If we were to continue the chart for the full 109 months, we would see that the last month would only require a partial payment (that is, less than $30) to pay off the remaining principal.

Of course, not all loans have constant payment schedules. Some loans have “teaser” periods, with low interest rates for a certain period of time. These might seem like fantastic deals, until we realize that the loan is actually negatively amortizing over the period, causing higher rates (or a longer payoff time) once the period is over.

7.5 End-of-Chapter Assessment