The following formula is used to calculate TWRs:

Let’s assume that an investor had a portfolio with the following beginning and ending market values over five periods:

In order to calculate the TWR, we must first calculate the HPRs for each of the individual periods:

Once we know each of the individual HPRs, we can calculate the geometric average in order to determine the TWR:

Using an HP12C calculator, we can calculate the TWR using the following keystrokes:

[1.10][ENTER]
[1.1364][*]
[0.92][*]
[1.1478][*]
[0.9848][*]
[5][1/x][y^x]

The Excel model used to calculate TWRs can be found here.

We can solve for NPV using the following formula:

where:

Assume that a manufacturer was looking to expand production in order to meet the need for increases in product demand. The new machinery would have an initial cash investment of $10,000; additionally, management makes the following projections for the incremental increase in annual cash flows once the machine is running:

Further, management expects that the required rate of return is 7%. Using these assumptions what is the NPV of the project? Utilizing the table above, we can discount each of the cash flows by the required return as follows:

In order to calculate NPV, we simply add all of the present values together then subject from the total the initial cash outlay:

Generally speaking, projects that have a positive NPV add to shareholder wealth, while projects that have a negative NPV are detrimental to shareholder wealth.

Additionally, projects that have an NPV of $0 neither add or subtract to shareholder wealth and merely generate enough return to cover the costs of capital. The rate of return associated with an NPV of $0 is also referred to as the internal rate of return (IRR). In the world of fixed income investing, IRR is referred to as the yield-to-maturity (YTM).

Investment decisions may also be made by comparing IRR to the weighted average cost of capital (WACC). If the IRR is greater than the WACC, then management should move forward with the project. In instances where the decision made with IRR conflicts with NPV, then defer to NPV over IRR.

Using an HP12c, we can calculate the NPV of the project above using the following keystrokes:

[10000][CHS][CF0]
[3000][CFj]
[3250][CFj]
[3500][CFj]
[3750][CFj]
[4000][CFj]
[7][i][f][NPV]

The Excel model used to calculate NPV can be found here.

Typically, the geometric mean is used to calculate investments returns on a compounded annualized basis.

Let’s assume an investor held a security that had the following return series:

{10%, 5%, -10%, 7%}

We can calculate the geometric average of this series as follows:

Using an HP12C calculator, we can calculate the geometric mean using the series above with the following keystrokes:

[1.1][ENTER]
[1.05][*]
[.9][*]
[1.07][*]
[4][1/x][y^x]
[1][-]

The arithmetic average is the sum of all of the values in a data set divided by the total number of observations.

Assume that you are presented with the following mutual fund expense ratios:

• Fund One – 0.75%
• Fund Two – 0.08%
• Fund Three – 1.15%
• Fund Four – 0.90%
• Fund Five – 0.14%

Using these values, we can calculate the arithmetic average as follows:

Using an HP12C calculator, we can calculate the arithmetic average using the following keystrokes:

[.75][Σ+]
[.08][Σ+]
[.015][Σ+]
[.009][Σ+]
[.0014][Σ+]
[g][x-bar]

Let’s assume you have two portfolios, an Roth IRA and a Traditional IRA. The Roth IRA has an average expense ratio of 0.15% and a total portfolio value of $7,000, while the the Traditional IRA has an average expense ratio of 0.47% and a total portfolio value of $3,000.

Using these values we can calculate the weighted average expense ratio as follows:

Using an HP12C calculator, we can calculate the weighted average using the following keystrokes:

[.0015][ENTER]
[7000][Σ+]
[.0047][ENTER]
[3000][Σ+]
[g][x-bar w]

This formula is commonly used to calculate the weighted average expense ratio of a portfolio of investments.

Compared to an ordinary annuity, the present value of an annuity due can be calculate by modifying the formula above with the addition of the quantity
(1 + r) as follows:

Where:
PMT = payment
r = rate
n = periods

Assume an individual won the lottery and the prize was to be a series of $1,000 payments received at the beginning of each year, over a ten year period. The winner has the option of choosing between the stream of payments or a lump sum discounted at a required rate of 7%, we can calculate what the present value of the stream of payments is as follows:

Using an HP12C calculator, we can solve the equation above using the following keystrokes:

[g][BEG]
[1000][PMT]
[7][i]
[10][n][PV]

Compared to an ordinary annuity, payments for an annuity due are received at the beginning of each period. Due to this extra time that the payments have to compound, we can modify the ordinary annuity formula with the addition of the quantity (1 + r) to calculate the future value of an annuity due as follows:

Where:
PMT = payment
r = rate
n = periods

Assume that an individual was to invest $1,000 over a period of 10 years, in a security with a 7% rate of return, and the investment was made at the beginning of each year. We can calculate the future value of the investment as follows:

Using an HP12C calculator, we can calculate the future value of an annuity due using the variables above as follows:

[g][BEG]
[1000][PMT]
[7][i]
[10][n][FV]

Where:
PMT = payment
r = rate
n = periods

Assume you won the lottery and the prize is a $1,000 series of payments to be received over the next ten years, at the end of each year. As the winner you could choose either the $1,000 stream of payments or a lump sum discounted at a required rate of 7%. We can calculate the lump sum as follows:

Give the result above, as long as the lump sum is exactly $7,023.58, and assuming you could realize a 7% return over ten years, you should be indifferent to receiving a lump sum or the stream of payments. If the lump sum offered is less than $7,023.58, you should chose the income; however, if the lump sum offered is greater than $7,023.58 you should choose the lump sum over the payment stream.

Using an HP12C calculator, you can calculate the present value of an ordinary annuity with the variables above using the following keystrokes:

[1000][PMT]
[7][i]
[10][n][PV]

Where:
PMT = payment
r = rate
n = periods

Assume you were to invest $1,000 per year, in an investment that would grow at a 7% rate of return over ten years. What would the future value be at the end of the 10th year?

We can solver for the future value by plugging in the variables as follows:

You can calculate the value above with an HP12C using the following keystrokes:

[1000][PMT]
[7][i]
[10][n][FV]

Using Excel, we can model the growth of the investment at different PMTs and rates of growth over time.

Assuming the same variables above we can construct a table of values for each of the periods:

How would these values look if we reduced the return from 7% to 4%:

We can illustrate the two tables graphically as well:

Generally speaking, over longer periods of time, the higher the rate of return, or the larger the annual contributions, the larger the difference between the two ending values will become.

In the world of financial planning, this formula can be applied to determine the approximate amount of money you will have at retirement on a pre-tax basis.

The variables will be defined by the amount of money you are contributing into your employer sponsored 401(k) plan (a type of tax-deferred account), Roth IRA or Traditional IRA, on an annual basis, any company matching contributions you may receive, and the number of years until you reach your retirement age.

A copy of the Excel model used to calculate the future value of an annuity can be found here:

For instance, to determine the future value of an account that taxes interest or dividends on an annual basis at some specified rate would require you to calculate the future value interest factor using a formula that solves for such method of taxation.

Before the age of calculators and computers, solving future value and present value equations required the use of interest factor tables. Fortunately, solving for the factors is easier than in sounds.

The future value interest factor (FVIF) is simply the quantity that the present value is compounded by:

Let’s construct a future value interest factor table using an assumed annual rate of 7%:

Let’s assume you wanted to calculate the future value interest factor for an investment that will grow at 7% for ten years, plugging those values into the future value interest factor equation will yield the following:

Using the table above and finding the area where n = 10 and r = 7% intersect indicates that the FVIF is 1.9672. Any dollar amount multiplied by the FVIF of 1.9672 will yield the future value of an investment that grew at 7% for ten years.

Present value interest factors (PVIF) are simply the inverse of FVIFs:

We can construct a PVIF table in the same manner:

Notice, if you multiply any FVIF by its corresponding PVIF the product of the two numbers will equal one:

Using an HP12C calculator, we can solve for the FVIF in the example above using the following keystrokes:

[1][.][0][7][ENTER]
[10][y^x]

The Excel model for FVIF and PVIF table construction can be found here.