# Comparing investments

A work in progress. Comments / corrections are welcome. Please post in New in Wiki - Comparing Investments.

This article shows how to compare the past performance of two investments. Solutions are provided to allow the reader to duplicate the results with a spreadsheet.

There are many online financial calculators that do exactly what this article describes. However, the reader is placing full confidence that the math behind the online calculator (1) is accurate and (2) uses assumptions expected by the reader. There should be no surprises, especially when finances are involved. Having your own spreadsheet clearly defines the situation. What-if scenarios are easily done, especially since the data can be manipulated to see trends with a simple chart.

Caveat: Past performance is no guarantee of future performance. Interpretation of results is the responsibility of the reader. If there are any questions, please don't hesitate to ask in the forum for guidance.

Microsoft Excel was used for this article. However, Open Office Calc will work equally well and is a free application that is supported on several operating systems. Open Office financial functions are very similar to Microsoft's functions.

## Present Value

In order to evaluate a financial investment, prices are referenced as if the money is invested today. The amount of money that must be invested today is called present value. Using the same point in time allows comparison between different financial instruments, e.g. an "apples-to-apples" comparison.

The equation for Present Value can be expressed as:

PV = FV * 1 / (1 + i)N or PV = FV * (1 + i)-N
where
PV is the present value (amount of money today)
FV is the future value (amount of money at some point in the future)
i is the interest rate
N is the number of periods

The graph below shows the general idea. The slope of the line depends on the interest rate (i) and the number of payments (N).

### Fundamental Property #1

The first important property of present value is that the higher the interest rate (or discount rate), the lower the present price.

The higher the interest rate today, the less that has to be invested to achieve the same value in the future. ### Fundamental Property #2

The second important property of present value is that for a given interest rate (or discount rate), the farther into the future the investment will be received, the lower the present value.

Longer term investments have more time for the interest to accumulate, resulting in fewer dollars that need to be invested. ## Cash Flow Diagrams

(in process)--LadyGeek 02:52, 15 March 2010 (UTC)

## Financial Functions

### Sign Convention

Financial functions use a standard sign convention to indicate the direction of cash flow.

• A positive sign is used for money that flows into our hand.
• A negative sign is used for money that flows out of our hand.

Whether a cash flow is an inflow (+) or outflow (-) depends on your perspective. For example, money paid on a loan (outflow) is negative.

Note:

• One wrong sign will change the computed results and give an incorrect answer.
• If you consistently change all signs in the problem, you switch roles (for example, switching from being the borrower to being the lender)
• In all problems, both signs must be used. Sometimes the other sign is in the answer.

Double-check signs on examples. Example 1 needs a sign swap. --LadyGeek 02:40, 15 March 2010 (UTC)

### Financial variables

There are six financial variables.

1. PV: The first unequal money in or out of the cash flow diagram is the Present Value.
2. FV: The last unequal money in or out of the cash flow diagram is the Future Value.
3. PMT: A fixed amount received in each period is the Payment.
4. Type: Money received at the “end of the period” is Excel Type = 0. Money received at the “beginning of the period” is Excel Type = 1.
5. N: Number of periods.
6. I: Interest rate per period.

N and I must be expressed in the same terms. If N is years, then I must be expressed in years. If N is months, then I must be expressed in months.

## How to Compare Investments

1. Sum the individual cash flows (every payment and any final value) in today's price for each investment.
2. Compare the prices.

### Comparing Interest Rates #### Example 1

An investing opportunity offers to pay \$500 per year for the next 20 years. The purchase price of this opportunity is \$5,300.

If the investor wants a 5.5% interest rate, is this a good investment?

The present value (price today of the investment 20 years later) is \$5,975.19.

Since the present value is higher than the purchase price of \$5,300, you will be getting more income than desired. This is a good investment.

With Excel:

 Present Value Excel Formula \$5,975.19 =PV(5.5%,20,-500)

#### Example 2

An investing opportunity offers to pay \$100 per year for the next 5 years and an additional \$1,000 at the end of 5 years. The purchase price is \$1,243.83.

If the investor wants a 6.25% interest rate, is this a good investment?

The present value (price today of the investment 5 years later) is \$1,156.90.

Since the present value is lower than the purchase price of \$1,243.83, you will be paying more than what this investment is worth. Investment is not recommended.

With Excel:

 Present Value Excel Formula Cell \$418.39 =PV(6.25%, 5,-100) B8 \$738.51 =PV(6.25%,5,0,-1000) B9 \$1,156.90 =SUM(B8:B9) B10

A single payment at maturity is the Future Value (FV) parameter in the PV() formula. There are no payments, which is set to zero as shown above.

#### Example 3

Would you like to receive \$15,000 today or \$18,000 in four years?

The answer depends on the interest rate. If you can find an investment with good interest rate, it will be better to take the \$15,000 today. How do you find the interest rate to invest \$15,000 today?

Use a starting guess of 4%. The present value of \$18,000 received four years from now is \$15,386.48. Since the present value is higher than \$15,000 (receiving \$386.48 more) it is still better to receive \$18,000 in four years.

With Excel:

 Present Value Excel Formula \$15,386.48 =PV(4%,4,0,-18000)

Now, let Excel remove the guess work and find where it would be more beneficial to receive the \$15,000 now.

Use the Goal Seek function (see Excel help for details) to set the value of Cell F13 to zero by adjusting Cell C13.

With Excel (displayed):

 Rate PV Purchase Price Difference 4.0% \$15,386.48 \$15,000.00 \$386.48 4.7% \$15,000.00 \$15,000.00 \$0.00

 Rate(Column C) PV(Column D) Purchase Price(Column E) Difference(Column F) Row 12 0.04 =PV(C12,4,0,-18000) 15000 =D12-E12 Row 13 0.0466 =PV(C13,4,0,-18000) 15000 =D13-E13

If you can get more than 4.7% for your investment now, take the \$15,000.

### Comparing Interest Rates and Maturity Dates #### Example 4

Which is better: \$140,000 paid over the next 36 months or \$160,000 paid over the next 60 months? Clearly \$160,000 is better than \$140,000 but what is the interest rate on the deferred funds to get the additional \$20,000?

Phrased differently, what is the interest rate for the first investment (36 months) to match the 60 month investment (\$160,000)?

The interest rate where it would be worthwhile to take the \$140,000 in 36 months is 14.90%.

How:

Find the interest rate where the present value of both investments are equal. In other words, make the present value of the two investments in the figure above equal height by adjusting the interest rate (ignore the FV height difference, the graph is easier to read this way). Adjust the interest rate for both investments simultaneously to find where the arrows align.

Let Excel do the work. Use the Goal Seek function (see Excel help for details) to set the value of Cell P11 to zero by adjusting Cell K11.

Displayed values:

 EffectiveAnnual Yield NominalInterest Rate MonthlyInterest Rate 36 Month Investment 60 Month Investment Difference 15.96% 14.90% 1.24% \$112,346.88 \$112,346.88 (\$0.00)

Cell Formulas:

 EffectiveAnnual Yield(Column K) NominalInterest Rate(Column L) MonthlyInterest Rate(Column M) 36 Month Investment(Column N) 60 Month Investment(Column O) Difference(Column P) Row 11 0.15957 =NOMINAL(K11,12) =L11/12 =PV(M11,36,-140000/36) =PV(M11,60,-160000/60) =N11-O11

There is an important assumption that exemplifies the need for caution. Nowhere does it say that the interest is an annual rate! This problem is stated in months, and the solution is a rate per month. That's how the PV() equation works. All time periods must be the same. If the reader interpreted this as annual rate, a significant error of 1.06% results.

There are 2 columns of interest here. Effective Annual Yield and Nominal Interest Rate. Excels Nominal() function "un-compounds" the annual rate into an equivalent monthly rate that can be correctly used in the PV() formula. The Goal Seek function was used to find the annual interest rate, which flows into the monthly rate.

## Compounded Interest Rates

As shown in Example 4, the time period and type of interest (compounded or simple) needs to be clearly defined. Use Excel to convert between an annual interest rate (yield) and the rate for the period of interest.

### Example 5

Show the monthly interest rates for a given annual interest rate. Conversely, show the effective annual interest rate for a monthly interest rate.

Displayed values:

 Annual Yield(Column K) Nominal Monthly Yield Monthly Yield(Column K) Effective Annual Yield 10% 9.57% 10% 10.47% 8% 7.72% 8% 8.30% 6% 5.84% 6% 6.17% 4% 3.93% 4% 4.07% 2% 1.98% 2% 2.02%

Cell formulas:

 Annual Yield(Column K) Nominal Monthly Yield Monthly Yield(Column N) Effective Annual Yield Row 22 0.1 =NOMINAL(K22,12) 0.1 =EFFECT(N22,12) Row 23 0.08 =NOMINAL(K23,12) 0.08 =EFFECT(N23,12) Row 24 0.06 =NOMINAL(K24,12) 0.06 =EFFECT(N24,12) Row 25 0.04 =NOMINAL(K25,12) 0.04 =EFFECT(N25,12) Row 26 0.02 =NOMINAL(K26,12) 0.02 =EFFECT(N26,12)

• Nominal, Microsoft help file for Nominal interest
• Effect, Microsoft help file for Effective annual rate

## Unequal Payments

The present value formula can only be used if all payments are the same amount. Suppose you have a series of payments that vary every period? The theory is the same- sum the present value of the individual cash flows (payments). The only difference is that you need to use a present value function for each payment.

### Example 6

Suppose you have the opportunity to purchase an investment which promises to make the following payments every quarter (3 months, see table below). If a 12% interest rate is desired, what is the most that you should pay for it?

You should pay at most \$8,212.79.

How:

First, note that neither the type of interest nor the period of time is stated. For this problem, it's 12 % per year, but the quarterly interest is defined as 3% = 12% / 4. This is simple interest, no compounding is used.

Since the payments vary every 3 month period, the present value of each payment is calculated. Then, the payments are summed to get the total present value. Excel already has a function that will do the individual calculations for you, called Net Present Value. Note that the answers match (Sum of Column F with Net Present Value in Cell G20).

Displayed values:

 Interest Rate(Quarterly) (Column C) 3 Month Period(Column D) Payment(Column E) Present Value(Column F) Net Present Value(Column G) Row 20 0.03 1 1000 \$970.87 \$8,212.79 Row 21 0.03 2 1200 \$1,131.12 Row 22 0.03 3 1500 \$1,372.71 Row 23 0.03 4 1700 \$1,510.43 Row 24 0.03 5 1800 \$1,552.70 Row 25 0.03 6 2000 \$1,674.97 Sum of Present Value: \$8,212.79

Cell formulas:

 Interest Rate (Quarterly)(Column C) 3 Month Period(Column D) Payment(Column E) Present Value(Column F) Net Present Value(Column G) Row 20 0.03 1 1000 =PV(C20,D20,0,-E20) =NPV(C20,E20:E25) Row 21 0.03 2 1200 =PV(C21,D21,0,-E21) Row 22 0.03 3 1500 =PV(C22,D22,0,-E22) Row 23 0.03 4 1700 =PV(C23,D23,0,-E23) Row 24 0.03 5 1800 =PV(C24,D24,0,-E24) Row 25 0.03 6 2000 =PV(C25,D25,0,-E25) Sum of Present Value: =SUM(F20:F25)

Refer to Microsoft's help file on NPV for additional information and another example. Enable browser cookies to view.

• NPV, Microsoft help file for Net Present Value
• PV, Microsoft help file for Present Value.

Both Present Value and Net Present Value assume that the interest rate is defined. Suppose that you only have the payments. How do you calculate the interest?

Not a problem. Just use the Internal Rate of Return function, IRR to find the yield (internal rate of return). Finding the solution is somewhat more complicated and best left to Excel. Refer to Microsoft's help for information on the IRR function. Pay special attention to setting up the problem. The sum of the income and payments must be zero for the function to work.

• IRR, Microsoft help file for Internal Rate of Return
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