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# Chapter 7 Time Value of Money: Multiple Flows

## The Cash Comes In, The Cash Goes Out…

PLEASE NOTE: This book is currently in draft form; material is not final.

## 7.2 Perpetuities

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Define a perpetuity.
2. Calculate the PV of a perpetuity.
3. Explain how the PV of a perpetuity is derived.

## 7.3 Annuities

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. For an annuity, calculate the fifth variable given four of: PV, FV, rate, PMT, and n.
2. Calculate the PV of an annuity due.

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.

Figure 7.6 Annuity Timeline (General)

Figure 7.7 Annuity Example Timeline

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).

Equation 7.4 PV of an Annuity

PV of annuity = PV of equivalent perpetuity – PV of perpetuity at end of annuity

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.

### Watch the Direction of Cash Flows

As we get more complex with or calculations, it becomes all the more important to keep track of whether our cash flows are inflows or outflows. For example, if a one-time payment in this example were instead a cash outflow, then our value would drop by $48.10 to a total value of$55.70.

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.

=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.

### Why Is the Result Negative?

Calculators and spreadsheets always assume that a financial transaction is occurring, so they figure out the “fair value” based on the inputs. Since the NPV of the annuity’s inflows is $151.90, the calculated answer is that you should be willing to pay$151.90 (an outflow) to receive the scheduled payments in a completely “fair value” transaction.

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..

Figure 7.8 Annuity Due Timeline

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).

Equation 7.5 PV of an Annuity Due

PV of annuity due = PV of annuity + (payment – PV of payment at end)

## 7.4 Loan Amortization

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Explain loan amortization.
2. Create an amortization schedule.

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. Figure 7.9 Amortization Schedule for Credit Card Payments 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.

Figure 7.10 Principal and Interest Portion of Payments Over Time

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.

### Key Takeaways

• Payments can be broken into principal and interest components.
• As principal is paid down, interest decreases. Future payments will, over time, accelerate in their rate of paying down principal.
• Negative amortization can actually increase principal balances over time.

### Exercises

1. If Fred could afford to pay $5 more than the minimum payment, how many months would it take to pay off the balance? 2. A 30 year fixed rate mortgage has monthly payments (and no payment due at the end). If the interest rate is 6% APR and the amount borrowed is$200,000, what is the monthly payment due? Create an amortization schedule for this mortgage.