# Chapter 2: financial computations for financial planning.

Financial Computations: The Basics

Almost all aspects of personal financial planning involve mathematical calculations. The good news is that the math is not extremely complex. Furthermore, sophisticated calculators and software programs are available to do most of the work. The most important caveat with financial planning calculations is that outputs depend entirely upon the quality of inputs.

The formulas needed for the majority of the calculations are relatively easy to memorize. Applying the correct formula and related inputs within a calculation is usually what causes the most difficulty. More important than the formulas themselves is a fundamental understanding of the logic, or reasoning, required to identify the correct mathematical formula for a particular purpose.

Once the logic is correct, there may be multiple ways to derive the correct solution to a financial planning question. For example, one financial planner may choose to adjust a future value calculation for taxes at the end of period, while another will adjust a savings need at the beginning to account for taxes. Both planners should arrive at the same answer. This can sometimes be confusing to students versed in hard sciences, where one method is preferred and is often the only way to calculate an outcome.

What is important, then, is for planners to choose the method that they are comfortable with and practice that method until the process becomes second nature. Then, it is equally important to ensure accuracy--both in the input of the data and the supporting assumptions on which the calculation is based.

The purpose of this chapter is to briefly review some of the most important calculations made in personal financial planning. After reading this chapter, readers should understand rates of return and time value of money formulas and how to set up equations for later use in a spreadsheet application or financial calculator. It is assumed that much of the content of this chapter is a review. However, a thorough grounding in personal financial math is fundamental to financial planning.

Historical Rate of Return Calculations

One of the most elementary of all personal finance calculations is determining the total return earned on an investment. Financial planners use this calculation to determine how much profit has been earned on an investment during a certain period of time. For example, an investor deposits \$1,000 into an account, the account has a stated rate of return of 6%, and the investor leaves that money in the account for one year. What will be the account balance at the end of the investment period? How much of the account balance is principal and how much is interest? How much will the account be worth in real terms if inflation averaged 4% over the same time period? How much money will be available after paying taxes on the earnings? All of these questions are answered by using various rate of return calculations.

In order to calculate return on an investment, all sources of income must be identified. The two most common sources of return are income and capital gain. In general, income is derived from periodic payments received by the investor, either in cash or as deposits to the account during the investment period. Examples include stock dividends, bond interest payments, mutual fund dividends, or savings account interest payments. Capital gains (or losses), on the other hand, are generally received or realized when the investment period ends because the investment is sold, liquidates, or matures. If the security is sold for more (or less) than the purchase price, the investor will realize a capital gain (or loss).

It is these two sources of cash inflow that serve as the basis for the nominal return on investment. A nominal value, in its simplest form, means "unadjusted." It can be thought of as the "gross" return. The most common adjustment that is made to a nominal value is an inflation adjustment; however, nominal values can be adjusted for many other factors. Later in the chapter, nominal values will be the basis of adjustments for inflation (real returns), taxes (after-tax returns), compounding frequencies (effective returns), and others. It is as important to understand the interrelationship between the nominal and adjusted rate when calculating total returns or future values, as it is to be able to accurately calculate the nominal rate of return.

Holding Period-Based Returns

Simple Holding Period Return

A simple method of calculating nominal returns involves determining an investor's holding period return (HPR). The formula considers a beginning and ending value for the investment, income earned during the holding period, and transaction costs. The income received could be dividends from an equity, or stock, investment or the income could be coupon "interest" payments from a debt, or bond, investment.

HPR = Income + (Ending Value - Beginning Value) - Transaction Costs/ Beginning Value

Example: Bob purchased one share of stock for \$45 per share, received two dividend payments of \$1 each that were not reinvested, and paid a transaction fee of \$5. Bob then sells the stock for \$55. What was Bob's holding period return?

HPR = [SIGMA] Div + (End - Beg) - Costs/Beg

HPR = (1 + 1) + (55 - 45) - 5/45 = 2 + 10 - 5/45 = 15.6%

Another common modification to nominal rates of return is an adjustment for taxes paid at the state and federal level. However, calculating tax-adjusted returns can be complex. First, a planner needs to know what type of income was generated as a result of the return. Second, the planner must know what tax rate applies to each type of return. For example, interest income, dividends (qualified or non-qualified), and capital gains may be taxed at different rates, and the rates may also vary depending on the holding period for the investment (e.g., short-term or long-term capital gains). Different investments may be taxed at different rates.

To accurately calculate the tax-adjusted rate of return, the HPR must be offset by any tax liability generated as a result of owning the investment. The following adjustment can be used for historical return calculations.

(Inc - Inc Tax) + ((End - Beg) - (Costs - Tax Impact of Costs) - Capital Gains Tax)/ Beginning Value

Obviously, this equation can only be used for historical calculations because the actual amount of tax liability must be determined prior to calculating the return. However, to estimate the after-tax rate of return for projection purposes, where taxation occurs annually, a financial advisor could also use the following equation.

[r.sub.t] = R x (1 - t)

Where:

R = Nominal return

t = Investor's marginal tax bracket

Example: Expanding on the previous example, assume Bob owned the stock in a non tax-qualified account and dividends and capital gains were taxed at a short-term tax rate of 15%. His after-tax rate of return would be 13.26%, as shown below.

[r.sub.t] = 15.6% x (1 - 0.15) = 13.26%

Dollar-Weighted Return

The problem with the holding period return is that the formula does not account for the timing of subsequent cash flows into or out of the account. A very basic formula that corrects for the timing of subsequent cash flows is the dollar-weighted return formula, as shown below:

Dollar Weighted Rate of Return = Inc + [End - (Beg + D - W)] - Costs/ Beg + [D.sub.t] - [W.sub.t]

Where:

Inc = Income

End = Ending Value

Beg = Beginning Value

D = Deposit Amount

W = Withdrawal Amount

Costs = Transaction Costs

[D.sub.t] = D x (1 - timing of deposit)

[W.sub.t] = W x (1 - timing of withdrawal)

Example: Altering the example from above, assume that one year after Bob made his initial one share purchase he bought an additional two shares of stock at \$48 (\$48 x 2 = \$96). Further, assume that he sold one share at \$45 at the end of the second year, and liquidated the account after three years at \$50 per share (\$50 x 2 = \$100). He still receives semiannual dividends of \$1 per share (\$12 total) and the total transactions cost is still \$5. What is Bob's dollar-weighted rate of return?

Dollar Weighted HPR = \$12 + [\$100 - (\$45 + \$96 - \$45)] - \$5/ \$45 + [\$96 (1 - 12/36)] - [\$45 (1 - 24/36)]

= \$12 + \$4 - \$5/ \$45 + \$64 - \$15 = 11.7%

Time-Weighted Return

An alternative to dollar weighted returns is a time-weighted return. This method of calculation ignores the actual value of the investment during each time period.

Example. Once again, use Bob's situation, but this time solve for the time-weighted return. From the previous example, it is known that the stock was initially purchased for \$45 per share and was valued as follows:

End-of-year 1 = \$48

End-of-year 2 = \$45

End-of-year 3 = \$50

Using the holding-period return formula results in the following annual rates of return.

Year 1 = \$2 + (\$48 - \$45)/\$45 = 11.1%

Year 2 = \$2 + (\$45 - \$48)/\$48 = -2.1%

Year 3 = \$2 + (\$50 - \$45)/\$45 = 15.6%

Average = 11.1% - 2.1% + 15.6%/3 = 8.2%

Thus, the average of the three annual returns results in a time-weighted return of 8.2%. Note that the dividend was the same for all three periods because the time-weighted return ignores the number of shares (dollar amount) owned in each period.

So, which measure is superior? It depends on asset control. Investors who control their own investments (both the timing and amount of purchases) would want to use the dollar-weighted average. In this case, the more money that is invested when a stock is performing well, the more money the investor will have in the end. However, a person who does not control the timing and amount of the investment, such as a portfolio manager, would want to measure performance using the time-weighted measure. It would be inappropriate to measure a manager's success or failure based on a measure over which the manager does not have complete control.

Average Annual Returns

Arithmetic Mean

Although holding period-based returns are very simple to use and calculate, most returns are quoted on an average annual basis. Annualized returns are more intuitive for investors to understand and are often required by regulators. There are a number of different methods a financial planner can use to measure average rates of return. The simplest method is to calculate the arithmetic mean.

Average Return (AR) = [r.sub.1] + [r.sub.2] + [r.sub.3] + ... + [r.sub.n]/n

Where:

r = Return for Period

n = Number of Periods

Example. Consider a client who makes 8%, 2%, and -5% in each of three years, respectively. The average of these returns is 1.67%

AR = 8% + 2% - 5%/3 = 1.67%

The arithmetic mean can be used to generate a quick estimate of average returns, particularly when the sample size is small. However, the variability (measured by standard deviation) of returns may make this method less effective, or perhaps even inappropriate, for measuring long-term periods or for forecasting.

Geometric Mean

Consider a \$100 stock that rises 50% in one year and falls 50% the next. The arithmetic mean return is zero. However, when the return is tracked, it is apparent that in the first year the final price was \$150. Then the 50% drop resulted in a final value of \$75. The actual return is less than zero. For this reason, the geometric mean is recommended for most long-term averages and forecasting. The geometric mean is also referred to as the annualized return. The formula for calculating a geometric average is as follows:

Geometric Mean Return = [[(1 + [r.sub.1]) x (1 + [r.sub.2]) x (1 + [r.sub.3]) x ... x (1 + [r.sub.n])].sup.1/n] - 1

Where:

r = Return for Period

n = Number of Periods

Example. Tonya purchased an investment seven years ago. According to the annual report, the investment earned the rates of return shown in the table below. She wants to know what return she could project for the next seven years by calculating the annualized return.
```Year     Rate of Return

1            9%
2            8%
3            2%
4           12%
5            4%
6            2%
7            4%
```

Geometric Mean Return = 5.80%

[[(1 + 0.09) x (1 + 0.08) x (1 + 0.02) x (1 + 0.12) x (1 + 0.04) x (1 + 0.02) x (1 + 0.04)].sup.1/7] - 1

Consider the hypothetical values and returns presented in the table above. The mathematical average for the returns is 5.86% [(9% + 8% + 2% + 12% + 4% + 2% + 4%) / 7]. However, using this average would actually overstate the client's expected return if this average was projected into the future. In 7 years, \$1,000,000 would compound annually at 5.86% to \$1,489,784. This is over \$6,000 greater than the \$1,483,663 for compounding of \$1,000,000 using the actual rates in the table. Using 5.80% compounded annually, the \$1,000,000 would compound to \$1,483,883. This is pretty close to \$1,483,663; the \$220 difference between this and actual compounding is due to the fact that the geometric rate was rounded to 5.80%.

Using a geometric average will result in a more precise measure of the client's real return when projecting account values to the future. The arithmetic mean can be useful in certain circumstances, or for a quick estimate, but for actual reporting and forecasting, the geometric mean is generally preferred.

The rule is that, unless the variation of returns is zero, then the geometric return will be less than the arithmetic return. How much less depends on the variation (standard deviation) of returns. Standard deviation is the average difference between an individual outcome (in this case investment return) and the average, or expected, outcome for the applicable group. Standard deviation will be discussed in more depth later.

Arithmetic vs. Geometric: Margin for Error

When comparing the arithmetic mean and the geometric mean, it is interesting to note that as the standard deviation of the returns approaches zero, the difference between the arithmetic mean and the geometric mean also approaches zero. This is because the arithmetic mean and the geometric mean are equal if the return is fixed, meaning that the exact same outcome (return) occurs in each period, resulting in a standard deviation of zero. However, the inverse is also true. As the standard deviation of outcomes increases, the difference between the arithmetic mean and the geometric mean also increases.

Consider the S&P 500 total returns below. The difference in the two means is quite large, 1.23% (13.03%-11.80%), because the standard deviation is nearly 17%. As this example illustrates, the relationship between the two statistics becomes quite apparent and very important.

Example: A client has an account balance of \$50,000. The client assumes that since the average annual return of the stock market has been 13% over the last 20 years, the next 20 years should result in the same return. Even if history were to repeat itself, the client still should not use the 13% average annual return, but should use the 11.8% geometric average return. By using the incorrect average return for projecting the value of the account in 20 years, the client would overstate the account balance by more than \$135,000. Using the incorrect 13% return results in an account balance of \$576,154; using the correct 11.5% return results in an account balance of \$441,029.

Internal Rate of Return and Net Present Value

Internal rate of return (IRR) and net present value (NPV) are more sophisticated methods that determine whether or not an investment is expected to meet a client's required rate of return. IRR measures the discount rate at which the present value of cash inflows (returns or benefits) equals the present value of cash outflows (investments or costs). IRR takes into account the time value of money by considering the amount and timing of cash inflows and outflows over the life of an investment. Calculating the IRR is done through a trial-and-error process. IRR can be calculated by doing multiple iterations of net present value with different interest rates until NPV is equal to zero. The trial-and-error calculation can also be automated by using the IRR function in a spreadsheet program or programmable calculator.

The IRR and NPV concepts are related but they are not equivalent. Net present value (NPV) is defined as the present value of a stream of earnings less the present value of contributions. The formula for calculating NPV is:

NPV = PV of Cash Inflows - PV of Cash Outflows

Where:

NPV = Net Present Value

PV = Present Value

NPV compares benefits and cost. Therefore, the rule for NPV is that if the present value of benefits exceeds the present value of costs (NPV > 0), then the investment should be accepted; if the present value of benefits does not exceed the present value of costs (NPV < 0), then the investment should be rejected.

A similar rule applies to IRR, given that NPV and IRR are related. The rule is that if the IRR for an investment exceeds the required rate of return, then accept the investment; if IRR does not exceed the required rate of return, reject the investment.

Again, the internal rate of return (IRR) measures the present value of a series of cash flows in relation to an initial (and possibly subsequent) investment amount. The formula for calculating internal rate of return is:

PV = [PMT.sub.1]/[(1 + r).sup.1] + [PMT.sub.2]/[(1 + r).sup.2] + ... + [PMT.sub.n]/ [(1 + r).sup.n] + [FV.sub.n]/ [(1 + r).sup.n]

Where:

PV = Present Value

FV = Future Value

PMT = Periodic payment

r = Internal Rate of Return

However, as mentioned above, it would take multiple iterations--cycling through the equation using different discount rates--to determine the rate that would result in the NPV being equal to zero. Therefore, using the formula for calculating internal rate of return can be tiresome. Fortunately, internal rates of return that do not involve too many cash flows can be determined using a financial calculator relatively easily..

Example: Assume that John invests \$1,000 today into an investment that will pay him \$100 per year for 10 years. At the end of the investment period, John receives \$2,000. What rate of return did he make? Entering these data into a Texas Instruments (TI) BA II Plus calculator, as shown below, results in an internal rate of return equal to 14.94%.

10 [N]

1000 [+/-] [PV] (Initial investment is a negative cash flow.)

2000 [FV]

100 [PMT

[CPT] [I/Y] (result = 14.94)

While the method for calculating internal rate of return looks innocuous enough, application can become quite tedious, especially when the periodic payments fluctuate every period. Luckily, a financial calculator or software spreadsheet can also be used if the earnings, or periodic cash flows, are not fixed. By using the built-in cash flow tables or registers, individual cash flows may be entered separately.

Example: Assume that Shameka is considering an investment that requires a \$5,000 payment today. The required rate of return, based on her next best investment choice, is 7% in a mutual fund. If she anticipates receiving the following payments, should she make the investment?
```Year 1:   \$1,500
Year 2:   \$1,000
Year 3:   \$500
Year 4:   \$250
Year 5:   \$5,000
```

A quick way to answer the question is to compute the net present value using a financial calculator. (The same procedure can be used to calculate an internal rate of return.) The key strokes to follow for the TI BA II Plus calculator are shown below. Note that the cash flow register should always be cleared prior to beginning a new problem. To clear the register, enter the cash flow tables and then successively press the 2nd key and then the CLR WORK key.
```CF                             (enters the cash flow table)
[2ND] [CLR WORK]               (clears cash flow worksheet memory)
5000 [+/-] [ENTER]
[[down arrow]]               (CF0) (Initial investment is a negative
cash flow.)
1500 [ENTER] [[down arrow]]    (CO1)
1 [ENTER] [[down arrow]]       (FO1)
1000 [ENTER] [[down arrow]]
[[down arrow]]               (CO2)
500 [ENTER] [[down arrow]]
[[down arrow]]               (CO3)
250 [ENTER] [[down arrow]]
[[down arrow]]               (CO4)
5000 [ENTER] [[down arrow]]
[[down arrow]]               (CO5)
```

Now that the cash flows are entered into the cash flow table. IRR is computed by pressing the IRR key and then pressing CPT. In this case, the IRR is 15.12%. To solve for NPV, however, an estimated rate of return must be entered. After the NPV key is pressed, the calculator will ask for the required rate of return--in this case, 7%. Once the return is entered, pressing the down arrow key again will display the NPV, which will be computed when the CPT key is pressed. For this example, the NPV is \$1,439.11.
```[IRR] [CPT]                            (IRR = 15.12)
[NPV] 7 [ENTER] [[down arrow]] [CPT]   (NPV = 1,439.11)
```

Weighted Average Return

Sometimes it is necessary to calculate a weighted average return of different assets or values. A weighted average is calculated by multiplying an investment return by the investment's proportion, stated as a percentage, of the total portfolio and then summing the total for all of the investments. For example, the following table provides rate of return (RoR) data from seven mutual funds. The amount invested in each fund is also provided. These data are used to determine that the portfolio had a weighted average RoR of 5.64%.

Types and Forms of Interest Rates

Real vs. Nominal Rates

Nearly all interest rates quoted in newspapers, magazines, and annual reports are nominal rates, meaning that the rate has not been adjusted to account for inflation. Nominal rates are used when the actual dollar amount of an investment is important. By contrast, the real rate of return is the return "left-over" after adjusting for inflation. This provides a more accurate representation of purchasing power, which is an important consideration for long term investments. A client may be interested in having a portfolio value of \$1 million when she turns 65, but needs to recognize that the purchasing power of that portfolio will be far less than what \$1 million provides today. This is why adjusting returns to account for inflation is of paramount importance.

There are two equations used to account for the effects of inflation: one if the nominal rate of return is continuously compounded and one if the nominal rate is not continuously compounded. If the nominal rate of return is continuously compounded it is appropriate to subtract the inflation rate from the nominal rate to determine an approximations of the real rate.

r = R - i

Where:

Real Rate = r

Nominal Rate = R

Inflation Rate = i

Example. The nominal annual rate of return on Mary's portfolio is 9% and the annual inflation rate is 4%. In this case, the nominal rate of return is continuously compounded. What is her real rate of return?

r = 9% - 4% = 5%

However, if the nominal rate is not continuously compounded, a different equation should be used to take into account the effects of this compounding difference. The result of this calculation is sometimes referred to as a serial rate.

= (1 + R )/(1 + i) - 1

Where:

Real Rate = r

Nominal Rate = R

Inflation Rate = i

Example. The nominal annual rate of return on Mary's portfolio is 9% and the annual inflation rate is 4%. In this case, the nominal rate of return is not continuously compounded. The real rate of return is no longer 5%, but slightly less than 5%.

r = (1.09)/(1.04) - 1 = 4.81%

This calculation should be used whenever a client will face inflation risk while making an investment. These are particularly important formulas whenever determining the present value of an inflation-adjusted dollar amount--as would be the case anytime a planner or client is calculating future values, but stating the results in today's dollars--in other words, the inflation-adjusted dollar values.

Tax-exempt vs. Taxable Rates

In addition to the difference between nominal and real rates of return is the difference between taxable and tax-exempt rates of return. The effect of taxes must always be considered, because a client cannot keep what the government gets. In the earlier discussion of holding period return, the concept of "tax impact" was briefly introduced. For the most part, exact after-tax returns can't be determined until the end of the investing period. However, applying a client's marginal tax brackets to a return calculation can at least offer a reasonable estimate of the future tax impact on a projected return.

Example: A client is in the 25% marginal tax bracket and the expected return on an investment is 7%. By applying the formula shown below, a financial advisor can determine how much of the return will actually benefit the client. In other words, how many of the dollars earned would be kept?

r' = R x (1 - t)

Where:

After-tax Rate = r'

Nominal Rate = R

Fed Tax Rate = t

r' = 0.07 x (1 - 0.25) = 5.25%

In this case the investor's after tax return would be 5.25%. Therefore, it can be estimated that if that client invested \$1,000 in a taxable account and the account returned 7% per year, or \$70, then the client would keep \$52.50 of those dollars and would pay \$17.50 in taxes. This formula can also be expanded to include the state tax impact on nominal returns.

In many states, state income taxes are levied against gains that remain after federal tax liability has been applied; in such states, the state tax rate cannot be simply added to the federal tax rate in the above formula. The state tax rate is, in essence, reduced by the client's federal tax bracket. This is accomplished by using the same formula as above, except that the nominal rate in the above formula is replaced by the state tax rate.

s' = s x (1 - t)

Where:

s' = Applicable State Tax Rate

s = State Tax Rate

t = Fed Tax Rate

Now the combined after tax rate can be determined by combining both of the above equations.

r' = R x [1 - (t + (s x (1 - t))] or r' = R x [1 - (t + s')]

This may seem like a lot of manipulation for some fairly simple projections; however, where this may be useful is in determining taxable equivalent yields. A taxable equivalent yield is the investor's implied required rate of return on a taxable investment with identical risk/reward characteristics. In other words, if an investor could earn 6% on a tax-free investment, the taxable equivalent yield would need to be higher to compensate the investor equally on an after-tax basis. The formula to determine exactly how much higher is simply a variation on the above.

The formula below is used to determine the taxable equivalent yield (TEY) on a taxable investment that can then be compared with a federal tax-free investment (e.g., an out-of-state municipal bond) using the federal marginal tax bracket (FMTB)

TEY = Bond Yield/(1 - FMTB)

However, this formula only accounts for federal taxes and, it only applies to certain situations including: (1) the purchase of out-of-state municipal bonds; (2) the comparison of in-state municipal bonds to Treasury bonds, because neither bond is subject to state taxes; or (3) the client's state of residence does not levy a state income tax. But outside of these situations, the impact of both federal and state taxes should be considered. To include the impact of state taxes, then the formula would need to be slightly altered to include the state marginal tax bracket (SMTB).

TEY = Bond Yield/1 - [FMTB + SMTB x (1 - FMTB)]

Example: A client desires a fixed income investment and has narrowed the list of choices to two bonds: (1) an in-state municipal bond with an annual coupon rate of 6%, and (2) a corporate bond with an annual coupon rate of 8%. The investor is in the 25% federal tax bracket and the 5% state tax bracket. Which bond would be the better bond to purchase?

On the surface, it would seem that the corporate bond that pays \$80 per \$1,000 invested would be the better choice. However, the client would have to pay taxes on the \$80, leaving the client with less than \$80 after taxes. The other choice is the municipal bond that only pays \$60, but the client would, under most circumstances, not owe any federal or state taxes on that amount.

By using the taxable equivalent yield formula, the client can determine the required yield on a taxable investment that would provide equal compensation. In other words, this is the yield that is required to "net" the client the same amount of "spendable" dollars after paying the tax liability on the earnings.

TEY = 0.03/1 - [0.25 + 0.05 x (1 - 0.25)]

TEY = 0.06/0.7125 = 8.421%

Completing the above formula, the advisor can determine that a taxable return would need to be 8.421% in order for the client to maintain the same "net" return. Therefore the client would be better-off purchasing the in-state municipal bond.

Risk and Returns

Inflation risk, default risk, interest rate risk, reinvestment risk, political risk, exchange rate risk, and others must be analyzed and either removed, mitigated or compensated for. In order to be compensated for these risks, an investor needs to receive a rate of return high enough to adjust for the risk--the investor's required rate of return. This rate is greater than the risk-free rate that simply compensate the investor for delaying consumption.

Time Value of Money

It has been said that the financial plan is the foundation of a client's financial future. However, if that is the case, then the time value of money is the cornerstone of that foundation. The concept of the time value of money (TVM) has two primary purposes: (1) to allow money to be moved around in time while accurately reflecting its true value or purchasing power; and (2) to solve for complex spending and savings scenarios, such as college or retirement funding. The TVM formulas can answer such questions as: What will be the value of a client's savings in ten years? How much does a client need to save today to fund something in the future? What will be the impact of inflation on a client's standard of living?

The underlying assumption behind all time value of money calculations is that generally it is more beneficial to receive a dollar today rather than later. Someone who waits to receive a dollar, rather than taking it today, must be compensated for waiting. That person would be willing to spend less than a dollar today, in order to receive a dollar in the future.

Effective and Nominal Rates

Interest rate, or the investor's required rate of return, is the variable that compounds or discounts the cash flows. This is the variable that keeps the values equal across time. When using the TVM formulas, it is critical that the interest rate used is an effective rate--one that considers the effects of compounding when compounding occurs more frequently than once per year. In other words, this is the actual accrued annual interest rate.

If the rate is not an effective rate, then it is a nominal rate, often referred to as an annual percentage rate (APR). The nominal rate may be used in a financial calculator because the calculator can automatically adjust for the compounding period (payment frequency). It generally may not be used in the mathematical formulas unless adjustments are made to the formulas. The following formula is used to adjust a nominal rate to an effective rate.

EAR = [(1 + APR/m).sup.m] - 1

Where:

EAR = Effective Annual Rate

APR = Annual Percentage Rate

m = Number of compounding periods per year

The effective annual rate (EAR) is the rate that, under annual compounding, would have produced the same future value at the end of one year as was produced by more frequent compounding. If the compounding occurs on an annual basis, the effective annual rate and the nominal rate are the same. If compounding occurs more frequently, the effective annual rate is greater than the nominal rate.

In addition to effective annual rates, there are also effective periodic rates (EPR). The periodic rate (EPR), [i.sub.PER], is the rate charged by a lender or paid by a borrower each period. It can be a semiannual, quarterly, monthly, or daily rate; in fact it can be any time interval including, for example, every ten years.

Example: What is the effective annual rate in a bank account that has an APR of 6%, but the interest is compounded quarterly?

Effective Annual Rate (EAR) = [(1 + APR/m).sup.m] - 1

= [(1 + .006/4).sup.4] - 1

= [(1.015).sup.4]

= 1.0614 - 1

= 6.14%

Example: What is the effective monthly interest rate on a mortgage with a stated APR of 8%? This would be the rate necessary to determine the monthly interest charges on a simple-interest mortgage with a monthly payment.

Effective periodic rate (EPR) = APR/m

= 0.08/12

= 0.667%

The following illustrate the common variations of the equations used to adjust between nominal and effective interest rate.

* If the EAR is known and the EPR is desired, then: EPR = [(1 + EAR).sup.1/m] - 1

* If the EAR is known and the APR is desired, then: APR = [[(1 + EAR).sup.1/m] - 1] x m

* If the EPR is known and the EAR is desired, then: EAR = [(1 + EPR).sup.m] - 1

* If the EPR is known and the APR is desired, then: APR = EPR x m

* If the APR is known and the EPR is desired, then: EPR = APR/m

* If the APR is known and the EAR is desired, then: EAR = [[(1 + (APR/m)].sup.m] - 1

One of the most important concepts to understand when manipulating the rates of return in time value of money equations is that the compounding frequency of the effective rate must match the payment frequency. In other words, the rate used must match the compounding frequency to how often cash flows, or payments, are occurring. For example, if the payment is occurring monthly, then the effective rate needs to be adjusted so that is also compounded monthly. However, this adjustment does not affect the actual interest earned; it simply adjusts how the rate is interpreted within the equation.

Payments are the periodic inflows or outflows that occur during the valuation period, and are only used with annuity equations. In order for a cash flow to be considered a "payment," it must happen more than once and must occur on a fixed periodic basis (e.g., monthly, semiannual, annual, etc.). Since the payment and the interest rate must "match" in frequency and the payment frequency is independent, the interest rate must be adjusted to match the payment frequency.

Example: A client is planning to save for retirement by contributing to the employer-sponsored 401(k) plan on a semi-monthly (twice per month) basis. However, the account reports and compounds at a monthly rate of 0.721%. In order for the TVM equation to be accurate, the rate needs to reflect the semi-monthly compounding period.

To make this conversion, two of the effective rate equations must be used. First convert the monthly effective rate into an EAR.

EAR = [(1 + EPR).sup.m] - 1

EAR = [(1 + 0.00721).sup.12] - 1 (M = 12 because of monthly compounding)

EAR = 9.0%

Now the EAR must be converted back into a semi-monthly effective rate.

EPR = [(1 + EAR).sup.1/m] - 1

EPR = [(1 + 0.09).sup.1/24] - 1 (M = 24 because of semi-monthly compounding)

EPR = 0.360%

This new EPR can be used in the formula to determine what the required savings amount is, or the future value of the account, or whatever else the advisor may be calculating.

Basic Time Value of Money Calculations

Future Value of a Single Sum

One of the most common questions a client will ask is how much a sum invested today will be worth in the future. To answer this question, a financial planner should use a basic future value formula. The future value of a single sum formula is as follows.

FV = PV x [(1 + i).sup.n]

Example. Assume that Jack made a \$10,000 deposit into a three-year certificate of deposit that paid an annual compounding interest rate of 5%. How much will Jack accumulate when the certificate of deposit matures? To answer this question, simply insert the facts into the formula as follows.

[FV.sub.3] = \$10,000 x [(1 + 0.05).sup.3] = \$11,576.25

Financial Computation Tip: Rule of 72

A quick estimate for calculating the future value of a single sum is the Rule of 72. The rule of 72 can be used whenever a client wishes to know approximately how long it will take to double an investment. To answer a "doubling" question, 72 is divided by the interest or discount rate earned. So, for example, it will take approximately eight years to double an investment that earns 9% annually (72/9 = 8).

A planner can also use this shorthand tool to determine the rate of return needed to double a client's initial investment. For example, if a client has six years to meet a goal that requires the doubling of an asset's value, what rate of return must be achieved? In this case, the client must earn 12% on average over the six-year period (72/6 = 12).

However, caution must be used, as this is only an estimate. With small interest rates, this estimate works just fine. However, when larger interest rates are used, this method becomes a less accurate estimate. This should be fairly obvious, because an interest rate of 72% would not result in a one year doubling of value.

Present Value of a Single Sum

Present value is used to determine the value of a future asset in today's dollars. The present value of a single sum formula is as follows.

PV = FV/[(1 + i).sup.n]

Example. Assume that Melissa needs to accumulate \$50,000 in 10 years. She is confident that she can earn an annual effective rate of 8% per year in her portfolio. How much does Melissa need to deposit into the account today to achieve her goal? The present value of a single sum formula can be used to solve this question. The answer to Melissa's question is easily solved by entering the known variables into the formula as follows.

PV = \$50,000/[(1 + 0.08).sup.10] = \$23,159.67

Future Value of an Annuity

The future value and present value of a single sum calculations assume that a client is dealing with a single lump sum. Sometimes a client needs to know the future or present value of a series of equal payments rather than a lump sum. A series of equal payments or receipts is known as an annuity. The formula for the future value of an ordinary annuity is as follows.

[FVA.sub.n] = PMT/i = [[(1 + i).sup.n] - 1]

Example. Thomas would like to know how much he will accumulate if he invests \$5,000 per year into a Roth IRA for 20 years earning 9% interest? Answering Thomas' question is easy using the future value of an ordinary annuity formula.

[FVA.sub.20] = \$5,000/0.09[[(1.09).sup.20] - 1] = \$255,800.60

Present Value of an Annuity

The present value of an annuity is used to determine how much a client needs today, in a lump sum, to equal a given number of fixed payments over time in the future. This equation can also be used to determine the amount of a periodic payment given a certain present value and number of desired payments. College funding scenarios often involve the use of a present value of annuity calculation. The present value of an ordinary annuity formula is as follows.

PVA = PMT/i [1 1/[(1 + i).sup.n]]

Example. Consider Erik and Erin who want to make an immediate lump sum deposit into an account earning an effective annual rate of 5% in order to fund four years of college expenses starting at the end of this year. Current college costs are assumed to be \$15,000 per year and are not expected to change. How much do they need in the account today?

To gauge the true amount needed, the present value of an annuity formula should be used as follows.

PVA = \$15,000/0.05 [1 - 1/[(1.05).sup.4]] = \$53,189.26

In other words, Erik and Erin need to deposit \$53,189.26 into an account today, earning 5% per year, to fund college costs of \$15,000 per year for four years.

Present Value of a Delayed Annuity

It may be necessary to calculate the present value of an annuity at some point in the future. Again consider Erik and Erin planning for future education expenses, perhaps for a younger child. In the previous example, it was assumed that the first tuition payment occurred at the end of the first year. But what would happen if that tuition payment was not due for 10 years?

Again the clients would need to know how much they need to save by the start of college, but they now have 10 years for an account to accumulate the required amount prior to the first tuition payment. The formula to solve for this type of equation is a combination of two of the previous formulas--the present value of an annuity and the present value of a single sum. The two equations are:

[PVA.sub.n] = PMT/i [1 - 1/[(1 + i).sup.n]] and [PV.sub.n] = FV/[(1 + i).sup.n]

The solution to this situation begins the same way as the previous example, by solving for the present value of the annuity for the four years of college. However, in the first example this answer was as of today ([T.sub.0]), now it is as of ten years from today ([T.sub.10]). (2) Therefore the value would need to be discounted a second time--back to today. This is referred to as double discounting, because the first equation discounted the four payments to the beginning of the payment period and the second equation discounted that result back to today. Functionally, the first answer, which was solved for as a present value, becomes a future value when solving for the second answer. These equations may be combined into one equation that handles both steps. The equation on the left is then substituted into the numerator of the equation on the right, as shown below:

[PVA.sub.n] = PMT/i [1 - 1/[(1 + i).sup.n]] [right arrow] [PV.sub.o] = [PVA.sub.n]/[(1 + i).sup.d]

Where:

n = number of payments

d = number of periods delayed

Completely combining the two equations results in the following:

[PV.sub.o] = PMT/i [1 - 1/[(1 + i).sup.n]]/[(1 + i).sup.d]

Example. To return once again to Eric and Erin, the advisor would solve for the present value of the annuity at year 10, which is \$53,189.26 and then discount that result back to today. The final need can then be determined to be \$32,653.59, as shown in the following equation.

[PV.sub.d] = \$15,000/0.05 [1 1/[(1.05).sup.4]]/[(1.05).sup.10]

[PV.sub.d] = \$53,189.26/1.62889 = \$32,653.59

In essence, what is happening with the second discounting (during periods 1-10) is that the equation is backing-out the interest earned between today ([T.sub.0]) and the beginning of the year in which the first payment occurs ([T.sub.10]).

This type of combined equation can be used any time a stream of payments begins at a point in time beyond the end of the first year. In such a case, it is appropriate to discount the cash flow twice--once to bring the stream back to the beginning of the year in which the first payment occurs, [T.sub.x,], and once again to bring that result back to the present time, [T.sub.0].

Future Value of a Geometrically-varying "Growing" Annuity

An assumption about the payment must be made whenever using an annuity equation. Sometimes, the assumption is that the client's payment will change from year to year by a constant percentage. (Note: The rate of change must always be at a constant percentage amount and not a constant dollar amount.) This could be due to a number of factors (e.g., changes in salary, changes in disposable income, etc.), but the most common reason for a changing payment is inflation.

If a client wants the payment, either in or out of an annuity, to keep pace with inflation, than a grow rate must be applied. The growth rate for this equation may be positive or negative, but it cannot be equal to the rate of return. If the growth rate is zero, then the fixed annuity equation should be used. Growing annuities apply to either present value or future value situations. The equation for the future value of a growing annuity is as follows.

[FVGA.sub.n] = [PMT.sub.1]/(i - g) [[(1 + i).sup.n] - [(1 + g).sup.n]]

Where:

i = Interest Rate (i [not equal to] 0)

g = Growth Rate of Payment (i [not equal to] g)

Consider a client, for instance, who participates in a 401(k) or 403(b) plan. Usually, defined contribution plan salary reductions are based on a percentage of the employee's annual income. If the income increases by, say, 3% annually, it follows that the retirement plan contribution will also increase. A basic future value of annuity formula cannot account for this assumption. A geometrically varying annuity formula can be used to determine a future value whenever a payment is expected to increase at a fixed geometric rate.

Example: Assume that Thomas will make annual payments (3) into a 401(k) for 20 years earning an effective annual rate of 9%. He will begin with a \$3,000 contribution, which is 5% of his income. But every year thereafter, he will increase his deposit by 3% to reflect the expected increase in his salary. Using this assumption, how much will Thomas accumulate at the end of 20 years? To solve this problem use the geometrically varying ordinary annuity formula:

[FVGA.sub.n] = \$3,000/(0.09 - 1.03) [[(1.09).sup.20] - [(1.03).sup.20]]

[FVGA.sub.n] = \$11,394.90/0.06 = \$189,914.98

In this case, the fact that Thomas' initial deposit into the 401(k) grows by 3% annually means that he will have approximately \$189,915 in the account at the end of 20 years. When using growing annuities it is often beneficial, or at least interesting, to compare the results of a growing annuity and a fixed annuity, keeping all of the variables--other than the growth rate--fixed.

Solving the previous example, but without growing the payment by 3% per year, can be done using the same formula and entering a zero for the growth rate or it can be solved for using the fixed annuity equation as shown below.

[FVGA.sub.n] = \$3,000/(0.09 - 0.00) [[(1.09).sup.20] - [(1.00).sup.20]]

[FVGA.sub.n] = \$3,000/0.09 [[(1.09).sup.20] - 1] Note: This is the fixed annuity equation

[FVGA.sub.n] = \$13,813.23/0.09 = \$153,480.36

By removing the growth rate, the value in Thomas' account at the end of 20 years is \$36,435 (\$189,915-\$153,480) less if he chooses not to grow his annual payment. Therefore, growing payments over a long-term planning horizon can make a significant impact on meeting a client's financial goals.

It is also important to note that when using a growing annuity equation, the payment used is the first payment. Similarly, when using the equation to solve for "payment," the equation solves for the first payment (PMT1). All other payments must be solved for using PMT1 and then increasing payments by the growth rate.

Present Value of a Geometrically-varying "Growing" Annuity

An equation that is very similar in form and function to the present value of a fixed annuity is the present value of a growing annuity. Just like in the future value of a growing annuity above, a growth rate must be included. For instance, a client may also desire to know the amount that must be saved, or available, at the beginning of a withdrawal period if the desired withdrawal amount changes over time. This is a highly likely scenario because of the effects of inflation on purchasing power. In this situation, the client wants to know the present value of a growing annuity.

[PVGA.sub.n] = [PMT.sub.1]/(i - g) [1 - [(1 + g).sup.n]/[(1 + i).sup.n]]

Example: Middle-aged clients, Joshua and Rebecca Rosenbaum, want to know how much they need to have saved by retirement in order to withdraw \$80,000 in the first year and to have that amount increase by 4% per year. They assume that they can earn an EAR of 8.5% throughout their 25 years of retirement.

[PVGA.sub.n] = \$80,000/(0.085 - 0.04) [1 - [(1.04).sup.25]/[(1.085).sup.25]]

[PVGA.sub.n] = \$1,777,777.78 [0.65319] = \$1,161,228.92

In this case the Rosenbaums need to save \$1,611,228.92 by the time they retire in order to meet their withdrawal goal. However, similar to the previous comparison done with the future value of a growing annuity, they only need to save \$818,735.26 if they choose to not increase their annual withdrawal by 4%.

[PVGA.sub.n] = \$80,000/(0.085 - 0.00) [1 - [(1.00).sup.25]/[(1.085).sup.25]]

[PVGA.sub.n] = \$80,000/0.085 [1 - 1/[(1.085).sup.25]] Note: This is the fixed annuity equation

[PVGA.sub.n] = \$69,592.50/0.085 = \$818,735.26

Serial Interest Rates

Another method for solving growing annuities is to calculate the serial rate. The serial rate is calculated using the same equation as is used to determine the real rate.

Serial Rate = [(1 + i)/ (i + g)] - 1

Where:

i = Interest rate

g = Growth rate of payment (Could be the inflation rate)

Example. Consider the first step to the previous retirement savings example, where Joshua and Rebecca are determining how much they need to save by retirement to fund a growing stream of retirement payments. Using the identical assumptions from above, the problem may also be solved using the fixed annuity equation by substituting the serial rate for i. First solve for the serial rate.

Serial Rate = [(1.085)/(1.04)] - 1 = 4.327%

The serial rate should then be inserted into the adjusted present value of a fixed annuity formula as follows:

[PVA.sub.n] = PMT/(1 + g)/i [1 - 1/[(1 + i).sup.n]]

[PVA.sub.n] = \$80,000/1.04/0.04327 [1 - 1/[(1.04327).sup.25]]

[PVA.sub.n] = \$1,777,746.17[0.65320] = \$1,161,219.63

The difference from the earlier calculation is due to rounding. If the serial rate (0.04327) were not rounded, [PVA.sub.n] would equal \$1,161,228.91.

Present Value of a Growing Perpetuity

While not very common, there are certain instances when a payment will either be made or received forever. This equation could be used in several retirement planning scenarios when a client wants to be very conservative, because the equation assumes that the principal of the investment is never liquidated--it continues in "perpetuity."

Example: A client wants to make sure that he leaves a legacy for is wife and children, but he also wants to ensure a comfortable retirement. He requires an annual rate of return of 7.0%. He wants an initial annual payment of \$100,000 that will keep pace with expected inflation at 3.5%. How much does the client need to have saved at the beginning of the withdrawal period?

PVP = PMT/(i - g)

PVP = \$100,000/(0.07 - 0.035)

PVP = \$10,000/0.035 = \$2,857,142.86

Therefore, in order for the client to ensure that he will not out live his money, nor lose purchasing power to inflation, the client needs to have nearly \$2.9 million saved.

An interesting thing about perpetuities is that their present values are not significantly larger than long annuities. See the following table for a summary of present values for different terms.

Mixed-Stream Calculations

The net present value (NPV) method of calculation (see the NPV and IRR section, above) is used when a series of payments, either made by the client or received by the client, fluctuate on a per period basis. The traditional present value of annuity calculation assumes that payments remain fixed, or if they do change, the rate of growth is fixed. A simple example can be used to illustrate the net present value method. The formula is as follows.

NPV = [PMT.sub.1]/[(1 + r).sup.1] + [PMT.sub.2]/[(1 + r).sup.2] + ... + [PMT.sub.n]/[(1 + r).sup.n] + [FV.sub.n]/ [(1 + r).sup.n] - PV

Example. Assume that a client is considering an investment for \$5,000. Currently, the client is making 7% on the \$5,000 in a conservative mutual fund. The client anticipates that the investment will make payments back to him as shown below:
```Year 1:   \$1,500
Year 2:   \$1,000
Year 3:   \$500
Year 4:   \$250
Year 5:   \$5,000
```

Should the client make the investment? A quick way to answer the question might be to compute net present value using the cash flow tables in a financial calculator. However, the formula may also be used to find a solution.

NPV = \$1,500/[(1.07).sup.1] + \$1,000/[(1.07).sup.2] + \$500/[(1.07).sup.3] + \$250/[(1.07).sup.4] + \$5,000/[(1.07).sup.5] - \$5 000

NPV = \$1,401.87 + \$873.44 + \$408.15 + \$190.72 + \$3,564.93 - \$5,000

NPV = \$6,439.11 - \$5,000 = \$1,439.11

When using the NPV method, it is critical to understand what the answer is saying. When the NPV is positive, it means that the internal rate of return (IRR) is greater than the required rate and therefore the investment should be accepted. IF the NPV is negative, it means that the IRR is less than the required rate and therefore the investment should be rejected. If the IRR equals the required rate, then the investment should be accepted because the required rate is achieved.

A third method for this situation is to create a spreadsheet to solve for net present value, as an alternative to using either a financial calculator or the formulas. The table below illustrates how the answer can be computed in spreadsheet format.

According to the net present value calculations shown in both the formula-based calculation and and the spreadsheet calculation, the client should accept this investment alternative because the NPV is positive.

Loans and Amortization Schedules

One of the most common uses for time value of money formulas is when a client or planner is working with an amortized loan. An amortized loan is a liability where the payment is normally fixed and that payment is set to repay the obligation over a fixed period of time. Every payment for the loan is composed of principal and interest and, as the loan is paid back, the percentage of each payment that is attributable to interest declines. This happens because a simple interest loan (which represents the majority of loan structures to individuals) recalculates the interest expense each period based on the remaining outstanding balance. Therefore as the outstanding balance declines, so does the periodic interest expense.

A prevalent method of presenting this information to a client is with an amortization table. An amortization schedule can be created for any loan that has a fixed rate of interest and a known number of payment periods. While an outstanding loan balance can be calculated using a financial calculator or present value of an annuity formula, it is sometimes more helpful to create an amortization schedule using a computer spreadsheet. A spreadsheet can more easily show different loan payoff amounts at different times. A spreadsheet can also be used to illustrate how a change in interest rates or duration can change a client's cash flow situation. The table below provides an example of how an amortization schedule can be created in a computer spreadsheet.

However, another method may be more useful and concise when trying to determine the outstanding balance on a loan. This method uses the previously discussed present value of an annuity (PVA) formula. It stands to reason, since the original payment for the loan could be determined with the PVA formula, that the same formula could be applied with that payment to determine the outstanding balance. The key is to change the n (number) in the equation to reflect the number of payments remaining.

Example: Recall the example loan from the above table. The table illustrates how the loan is amortized, but in this example it does not show the outstanding balance at all points in time. If the client wanted to pay off this loan after the 120th payment (in conjunction with the 121st payment), the client would need to know the outstanding balance at that time. To provide the client with the needed information, the planner would simply solve for the present value of an annuity with the PMT equaling \$665.30, the periodic interest rate equaling 0.583%, and the number of payments remaining equal to 240 (360 total payments--120 payments previously paid).

[PVA.sub.n] = PMT/i [1 - 1/[(1 + i).sup.n]]

[PVA.sub.240] = \$665.30/0.00583 [1 - 1/[1.00583.sup.240]]

[PVA.sub.240] = \$114,116.64[0.75220] = \$85,838.54

While it hardly seems to make sense that over 85% of the loan is still outstanding, this is the common occurrence as the progression of a geometric function is not linear over time. In other words, the loan balance does not decline in a straight line.

Another useful application based on the time value of money formulas is to determine how much money has already been paid on a loan. This is necessary information when determining the total amount of interest paid, as the difference between the total of payments and the change in the outstanding balance is the total interest paid to date. The following formula is used to determine the total amount paid to date on a loan.

Total Payments = PMT Amount x number of PMT

Example: Revising the previous example, it has already been determined that the client has reduced the outstanding balance on the loan by \$14,161.46 (\$100,000-\$85,838.54); however, this does not show how much the client has paid in-total. Nor does it tell how much interest the client paid over the first ten years of the loan. To determine the total paid the monthly payment is multiplied by the number of payments made as shown below.

Total Payments = \$665.30 x 120 = \$79,836.00

To determine the total amount of interest paid, the change in outstanding balance is subtracted from the total of payments. In this example, the total interest paid to date is determined to be \$65,674.54.

Total Interest = Total of PMT - Change in Balance

Total Interest = \$79,836.00--\$14,161.46 = \$65,674.54

Leimberg, Tools & Techniques of Financial Planning, 8th Edition (Cincinnati, OH: National Underwriter Company, 2007).

Leimberg, Tools & Techniques of Investment Planning, 2nd Edition (Cincinnati, OH: National Underwriter Company, 2006).

Quantitative / Analytical Mini-Case Problems

1. Effective Rate Conversion Problems

a. A client was earning an annual effective rate of 7% (EAR). The client was contributing to the account on a monthly basis. What would be the appropriate effective periodic rate (EPR)?

b. A client wants to compute the estimated future value of a bank account. The account pays a semiannual rate of 4% (EPR). The client is investing in the account on a bi-weekly basis. What is the appropriate effective periodic rate (EPR) for the client to be able to solve for an estimated future value?

2. Future Value Problems

a. Ted has \$1,000. He can earn an annual effective rate of 5%. How much will he have in ten years?

b. Lanisha currently earns \$45,000 per year. She expects her salary to increase by 3.5% per year. She plans to work for another 20 years. How much will she earn in her final year of work?

3. Present Value Problems

a. Tammy has determined that she will need \$3 million when she retires in 45 years. The current interest rate is 7% (EAR). How much does she need today in order to fully fund this goal at that rate?

b. David and Iantha expect their 10-year-old daughter to get married some day. They estimate that her wedding would cost \$30,000 today, costs will increase by 3% per year, and she will marry in 15 years. David and Iantha earn a pre-tax annual rate of 5%. They are in the 20% marginal tax bracket. What amount do they need to set aside today for their daughter's wedding?

4. Future Value of Annuity Problems

a. Sue wants to save \$1,000,000 in 20 years. She estimates she can earn 8.5% on savings. She intends to make deposits at the beginning of every year in a series of equal payments starting today. How much does she need to save each year in order to reach her goal?

b. Merita has been offered a choice of either receiving \$100,000 in ten years or receiving \$7,000 per year (at the end of each year) for the next ten years. Which is the better option if her required annual rate of return is 7%?

5. Present Value of Annuity Problems

a. Mike needs to receive \$40,000 per year (at the beginning of each year) from his investments for the next 30 years. His opportunity cost of money is 5%. How much does he need in an account today to generate this level of income if he fully depletes the account?

b. Roshanna has just won the lottery. Her required rate of return is 6%. Should she take the annual annuity payment of \$125,000 for the next 20 years or an immediate lump-sum payout of \$1.5 million?

6. Future Value of Growing Annuity Problems

a. Martha wants to start saving for college. She estimates that she will need \$50,000 when she starts college 4 years from now. She plans to save \$8,000 this year, and increase deposits by 5% annually (payments at the end of each year). She can earn 7% on her savings. Will she meet her savings goal of \$50,000 for college 4 years from now?

b. Todd is saving \$3,000 annually into an account (payments at the end of each year). He plans to increase the annual level of savings by 5% each year. He can earn 9% annually. How much will he have in the account at the end of 20 years?

7. Present Value of Growing Annuity Problems

a. Mike wants to receive \$40,000 per year pre-tax from his investments for the next 30 years. He is concerned about inflation which he expects to average 3%. He can earn a pre-tax rate of 6% on his money. How much does he need in an account to generate \$40,000 in annual inflation adjusted income (payments at the end of each year)?

b. A client is doing some investment/retirement planning. She is attempting to determine how much of her estate she needs to set aside today to fully fund her retirement. She desires annual beginning-of-period withdrawals for the next 25 years. She would like inflation adjusted withdrawals that start at \$50,000 per year. Assume an annual post-tax rate of return of 7.5% and that inflation will increase at an annual effective rate of 3.5%. What amount does she need to dedicate to this goal?

8. Present Value of Delayed Annuity Problems

a. Jonas desires fixed annual income of \$85,000 beginning in 20 years from now and lasting for 20 years. He plans to deplete the account. His annual required return is 9.5%. How much does he need to invest today to achieve his goal?

b. An investor has two options: (1) \$50,000 received at the end of this year, or (2) \$10,000 received each year for ten years but beginning ten years from now. Assume the rate of return is 7%. Which option has the higher present value?

9. Present Value of Perpetuity Problems

a. An investment would provide end-of-year annual income of \$2,000. The client's required rate of return is 12.5%. What price should the client be willing to pay for the investment ?

b. Jose Marie and his wife, Ayna, want to ensure that they do not outlive their money. They want to make make end-of-year withdrawals that start at \$80,000 after-tax and increase by 4% per year. They can earn an annual effective rate of 7.2% after-tax. How much do they need to have invested to make withdrawals that last forever?

10. Loan Amortization Problems

a. Willy and Ursha have just purchased a new car for \$38,000. They financed \$35,000 for five years at 6.75% (APR). What is the monthly payment and total interest expense?

b. Assume an initial loan amount of \$150,000, an APR of 6.5%, fixed monthly payments, and an amortization period of 20 years. What is the outstanding balance on the loan after 60 payments have been paid?

11. Kristin and Dan Peterson wonder how much they will need to save to fully fund four years of college for their daughter, Samantha. Samantha is 12 years old today. She plans to attend college at age 18. They know that their first choice college currently costs \$13,000 per year. College costs are increasing 5% annually. They feel that it is possible to earn an effective annual rate of 8.3% (8% compounded monthly), both before and during college.

a. How much will Samantha's first year of college cost?

b. How much do the Peterson's need to fully fund college costs when Samantha begins college?

c. How much do Kristin and Dan need to deposit in an account today to fully fund four years of expenses?

d. How much will they need to save on a monthly basis starting at the end of this month to fully fund four years of expenses?

12. Tony has been eyeing a new car. Last weekend, he went to the dealership and noted that his dream car would cost \$23,000 if purchased today. Tony currently has \$9,000 saved. He does not want to go into debt to buy the car, so he has decided to save towards the purchase in three years. Tony estimates that inflation will average 4.5% per year. He earns 7% (EAR) on his savings.

a. How much will the car cost in exactly three years?

b. How much must he save per year (at end of each year) to purchase the car in three years?

13. John is an avid stamp collector. He has been noting the rapid rise in prices for stamps printed in the 19th Century. Some of the best stamps from that time period have been increasing in value by 12% per year. John has the rare opportunity to purchase several impressive stamps from a reputable dealer.

The dealer has offered to sell the stamps to John for \$65,000 today. As an alternative, John can put down \$10,000 towards the purchase today and buy the stamps outright for an additional \$89,500 in four years. Assume an annual discount rate of 12%. Assuming John has the cash for either deal, which should he take?

Comprehensive Bedo Case--Analysis Questions

There are no Bedo case questions for this chapter. The quantitative/analytical mini-case problems above can serve as a good review. Time value of money concepts will be applied to the Bedo case in many other chapters to quantify needs and funding.

Chapter Endnotes

(1.) Assuming the frequency for all payments is 1, then the "frequency" step may be skipped by pressing the down arrow twice.

(2.) [T.sub.0] is always the present time, "today." Future points in time are then counted to, with the first period ([T.sub.1]) being one period in the future. For example, 10 periods in the future would be designated as [T.sub.10]. The period could be years, months, or any other definable period.

(3.) Annual payments are assumed since the growth rate is annual. Semi-monthly or bi-weekly payments would be expected. However, the growing annuity equation assumes that the payment change frequency due to applying the growth rate is the same as the payment frequency.
```S&P 500 Total Returns (1987-2006)

Year                 Annual Return          Year          Annual Return

1987                     5.25%              1997              33.36%
1988                    16.61%              1998              28.58%
1989                    31.69%              1999              21.04%
1990                    -3.10%              2000              -9.10%
1991                    30.46%              2001             -11.89%
1992                     7.62%              2002             -22.10%
1993                    10.08%              2003              28.68%
1994                     1.32%              2004              10.88%
1995                    37.58%              2005               4.91%
1996                    22.96%              2006              15.80%

Arithmetic Average      13.03%       Geometric Average        11.80%
Variance                 2.76%       Standard Deviation       16.62%

Source: Based on data obtained from Global Financial Data,
Los Angeles, CA, www.globalfindata.com.

Hypothetical Portfolio Values and Returns

Percent of
Value       Portfolio
Fund          Objective                     (A)           (B)
Calculation                                             (A / E)

Fund A        Moderately Aggressive       \$75,000        15.79%
Fund B        Moderately Conservative    \$100,000        21.05%
Fund C        CD                          \$94,000        19.79%
Fund D        Aggressive                  \$14,000         2.95%
Fund E        Conservative                \$35,000         7.37%
Fund F        CD                          \$45,000         9.47%
Fund G        Conservative                \$112,000       23.58%
Total                                   \$475,000 (E)    100.00%

Weighted
Return
Fund          Objective                   RoR       (D)
Calculation                               (C)     (B x C)

Fund A        Moderately Aggressive      8.00%     1.26%
Fund B        Moderately Conservative    9.00%     1.89%
Fund C        CD                         3.00%     0.59%
Fund D        Aggressive                10.00%     0.29%
Fund E        Conservative               3.00%     0.22%
Fund F        CD                         2.00%     0.19%
Fund G        Conservative               5.00%     1.18%
Total                                              5.64%

Comparison of Present Values by Term Length

Periodic Payment   Discount Rate   Periods    Present Value

\$1,000                  7%            20         \$10,594
\$1,000                  7%            35         \$12,948
\$1,000                  7%            50         \$13,801
\$1,000                  7%            65         \$14,110
\$1,000                  7%         Infinite      \$14,286

Net Present Value Calculations

Cash Inflow (Cash Outflow)   PV of Cash Flow at
Year         For Investment          7% Discount Rate

0               (\$5,000)               (\$5,000.00)
1                \$1,500                 \$1,401.87
2                \$1,000                   \$873.44
3                  \$500                   \$408.15
4                  \$250                   \$190.72
5                \$5,000                 \$3,564.93
Net Present Value                       \$1,439.11

Inputs    Interest Rate
Present Value
Term

A         B             C
Month     Beginning     Total Monthly
Balance       Payment
(PV of Annuity)

1         \$100,000.00   \$665.30
2         \$99,918.03    \$665.30
359       \$1,319.05     \$665.30
360       \$661.44       \$665.30

Inputs    7% annual [AFR] (0.583% monthly EFR)
\$100,000
30 years (360 months)

A         D                  E                   F
Month     Monthly Interest   Monthly Principal   Ending
Payment            Payment             Balance
(B x EPR)          (C - D)             (B - E)

1         \$583.33            \$81.97              \$99,918.03
2         \$582.85            \$82.45              \$99,835.58
359       \$7.69              \$657.61             \$661.44
360       \$3.86              \$661.44             \$0.00
```
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