October 2020 – modviz

How to Calculate Net Wealth

Most people are familiar with the concept of net worth which is simply the sum of one’s assets less liabilities. Net worth is the amount reported on an individual’s traditional balance sheet.

Net wealth expands on the concept of net worth by taking into account human capital and the present value of future consumption needs. In other words, net wealth is the present value of all available marketable and non-marketable assets less the present value of all current and implied liabilities. Net wealth is the amount that is reported on the economic balance sheet, the formula is as follows:

net wealth formula

Let’s assume that an individual had the following assets on his traditional balance sheet:

Assets
Liquid Assets
Checking Account	$ 50,000.00
CDs	$ 250,000.00
Total Liquid Assets	$ 300,000.00

Investment Assets
Brokerage Account	$ 400,000.00
401(k)	$ 700,000.00
Cash value of life insurance	$ 32,000.00
Total Investment Assets	$ 1,132,000.00

Personal Property
House	$ 1,200,000.00
Cars	$ 50,000.00
House Contents	$ 200,000.00
Total Personal Property	$ 1,450,000.00

Total Assets	$ 2,882,000.00

traditional balance sheet assets

Based on this traditional balance sheet, this individual has $2,882,000 in traditional balance sheet assets. On the economic balance sheet, all of these entries would be consolidated into a single asset referred to as financial capital. From there, human capital and the present value of any pension assets would be added to financial capital in order to find the total dollar value of assets on the economic balance sheet.

Let’s assume that based on this individual’s profession, his human capital has a present value of $7,500,000 and the present value of future pension benefits is $500,000:

Assets
Financial Capital	$ 2,882,000.00

Human Capital	$ 7,500,000.00

PV Pension	$ 500,000.00

Total Assets	$ 10,882,000.00

economic balance sheet assets

Based on those assumptions, total assets on the economic balance sheet would amount to $10,882,000 compared to $2,882,000 in total traditional balance sheet assets.

Let’s also assume that this individual had the following liabilities on his traditional balance sheet:

Liabilities
Short-Term
Credit Cards	$ 15,000.00
Total Short Term	$ 15,000.00

Long Term
Mortgage	$ 400,000.00
HELOC	$ 125,000.00
Total Long Term	$ 525,000.00

Total Liabilities	$ 540,000.00

traditional balance sheet liabilities

Based on the traditional balance sheet, this individual’s net worth would be $2,882,000 – $525,000 = $2,342,000. Now, let’s calculate and compare the difference between net worth and net wealth.

On the economic balance sheet, the total dollar value of liabilities would be entered on the economic balance sheet as a single entry referred to as debt. In addition to debt, the economic balance sheet takes into account the present value of all future consumption needs.

Let’s assume that the present value of lifetime consumption needs amounts to $5,200,000, based on this individual’s lifestyle:

Liabilities
Debts	$ 540,000.00

PV Lifetime Consumption	$ 5,200,000.00

Total Liabilities	$ 5,740,000.00

economic balance sheet liabilities

Based on these assumptions, economic balance sheet assets amount to $5,740,000 compared to $540,000 in traditional balance sheet liabilities.

Given the numbers above, we can now calculate this individual’s net wealth, which amounts to $10,882,000 in economic balance sheet assets minus $5,740,000 in economic balance sheet liabilities, for a total net wealth of $5,142,000.

In short, give the assumptions above:

net worth = $2,342,000
net wealth = $5,142,000

Conceptually, two individuals could have the same exact net worth, but their economic net wealth could be vastly different after factoring in income potential and lifestyle needs. In essence, the difference in total net wealth between two individuals with identical net worth may result in different investment strategies, and tolerances and attitudes towards risk.

The Excel file used to calculate net worth and net wealth can be found here.

How to Calculate Human Capital

The concept of human capital can be thought of as the present value of an individual’s future earnings and wages. For most households, human capital represents the single largest asset on the economic balance sheet.

The formula used to calculate an individual’s human capital is as follows:

Where:

Depending on the profession, the wages used may be higher or lower and more or less sensitive to the business cycle. Additionally, the discount rates used in the model should be consistent with the risks of wage growth and consistency of the assumed profession.

Let’s assume an individual was 55 and planned on retiring when he or she reached the normal retirement age of 65. Further, this individual’s current salary is $100,000 and as a professor has consistently received a 3% cost-of-living adjustment (COLA) on an annual basis.

Since this individual has tenure, the discount rate assigned for occupational income volatility is 3%, and the risk free rate is currently 2%. What is the present value of human capital for this individual if he or she has an expected survival rate of 99% in the first year, declining at 1% thereafter on an annual basis?

First, let’s use Excel to model the future value of wage growth over the next ten years at a 3% annual COLA:

Year	FV Wages @ COLA
1	$103,000.00
2	$106,090.00
3	$109,272.70
4	$112,550.88
5	$115,927.41
6	$119,405.23
7	$122,987.39
8	$126,677.01
9	$130,477.32
10	$134,391.64

future value of wages

Next, we’ll need to discount the future value of wages in each year to the present period by the total discount rate, composed of the risk free rate and the discount rate assigned to occupational income volatility:

Risk-Free Rate	Income Volatility	Total Discount Rate
2.00%	3.00%	5.00%

rf rate + discount for occupational income volatility = total discount rate

Using the total discount rate of 5%, we can expand the table above as follows:

Year	FV Wages @ COLA	PV Wages
1	$103,000.00	$ 98,095.24
2	$106,090.00	$ 96,226.76
3	$109,272.70	$ 94,393.87
4	$112,550.88	$ 92,595.89
5	$115,927.41	$ 90,832.16
6	$119,405.23	$ 89,102.02
7	$122,987.39	$ 87,404.84
8	$126,677.01	$ 85,739.99
9	$130,477.32	$ 84,106.84
10	$134,391.64	$ 82,504.81

present value of future wages

Now, we will multiply the present value of wages in each year by the expected probability of survival in each given year:

Year	FV Wages @ COLA	PV Wages	p-survival	P-adjusted Wages
1	$103,000.00	$ 98,095.24	99%	$ 97,114.29
2	$106,090.00	$ 96,226.76	98%	$ 94,302.22
3	$109,272.70	$ 94,393.87	97%	$ 91,562.05
4	$112,550.88	$ 92,595.89	96%	$ 88,892.05
5	$115,927.41	$ 90,832.16	95%	$ 86,290.55
6	$119,405.23	$ 89,102.02	94%	$ 83,755.90
7	$122,987.39	$ 87,404.84	93%	$ 81,286.50
8	$126,677.01	$ 85,739.99	92%	$ 78,880.79
9	$130,477.32	$ 84,106.84	91%	$ 76,537.23
10	$134,391.64	$ 82,504.81	90%	$ 74,254.33
			HC	$852,875.90

human capital table

Multiplying the present value of wages by the probability of survival in each year, yields the product which represents the probability weighted present value of wages. The summation of each of these values indicates this individual’s human capital is $852,875.90 under the given assumptions.

In other words, if this individual were to pass away today and had dependents who were counting on this income for survival, a total of $852,875.90 of life insurance would be required to replace his income if no life insurance policies were currently in force.

Keep this formula and model in mind the next time an insurance agent tries to randomly assign an arbitrary face amount to a policy when attempting to sell you life insurance.

A copy of the Excel model used to calculate the present value of human capital can be found here.

How to Calculate the Geometric Mean

The formula used to calculate the geometric average is as follows:

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][-]

How to Calculate the Arithmetic Average

The formula used to calculate the arithmetic average is as follows:

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]

How to Calculate the Weighted Mean

The formula used to calculate the weighted mean is as follows:

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.

How to Calculate the Weighted Average Expense Ratio of Your Portfolio

The weighted average expense ratio of your portfolio is an important number to know. Most people know that expenses have a huge impact on your portfolio over time since they are a drag on investment returns.

If your portfolio returns 11% during a particular year before adjusting for expenses, and your portfolio has a weighted expense ratio of 1%, your portfolio returns after adjusting for expenses will be 10%.

Lets assume you have a portfolio of $10,000 invested in the following Fidelity index funds:

Fund	Ticker	Balance
Fidelity S&P 500 Index Fund	FXAIX	$ 5,000.00
Fidelity Large Cap Value Enhanced Index	FLVEX	$ 2,500.00
Fidelity Large Cap Growth Index	FSPGX	$ 2,500.00

hypothetical portfolio allocations

The expense ratios associated with each of the funds are as follows:

Fund	Exp Ratio
Fidelity S&P 500 Index Fund	0.015%
Fidelity Large Cap Value Enhanced Index	0.390%
Fidelity Large Cap Growth Index	0.035%

expense ratios

Adding the expense ratios together and dividing by three would yield an arithmetic average of 0.147%; however, that is not the actual average of the portfolio itself since there are different dollar amounts in each fund.

In order to calculate the weighted average, we need to multiply the dollar amount in each fund by that specific fund’s expense ratio:

Fund	Balance	Exp Ratio	Dollar Exp
Fidelity S&P 500 Index Fund	$ 5,000.00	0.015%	$ 0.75
Fidelity Large Cap Value Enhanced Index	$ 2,500.00	0.390%	$ 9.75
Fidelity Large Cap Growth Index	$ 2,500.00	0.035%	$ 0.88
TOTALS	$ 10,000.00		$ 11.38

fund expenses

Based on the table above, the Fidelity Large Cap Value Enhanced Index contributes $9.75 in total portfolio expenses on a weighted basis. This amount is calculated by multiplying $2,500 by the expense ratio of 0.39%.

Calculating the dollar contribution of expenses for every fund in the portfolio and summing them together gives you a total dollar expense of $11.38 on a weighted basis. This total dollar expense is divided by the total portfolio value of $10,000 to calculate the weighted average of 0.1138% which is less than the arithmetic average of 0.147%.

Fund	Balance	Exp Ratio	Dollar Exp	Weighted Avg
Fidelity S&P 500 Index Fund	$ 5,000.00	0.015%	$ 0.75
Fidelity Large Cap Value Enhanced Index	$ 2,500.00	0.390%	$ 9.75
Fidelity Large Cap Growth Index	$ 2,500.00	0.035%	$ 0.88
TOTALS	$ 10,000.00		$ 11.38	0.1138%

weighted average table

In this hypothetical scenario, the weighted average is actually lower than the arithmetic average; however, that may not always be the case. Depending on the expense ratios and dollar allocations in your portfolio, it is possible that the weighted average could be higher than the arithmetic average.

If you have a portfolio that includes both active and passive management, calculating the weighted average expense ratio will provide you with a clearer understanding of the impact investment costs may have on the future value of your portfolio over longer periods of time.

The Excel model used to calculate the weighted average can be found here.

How to Calculate the Present Value of an Annuity Due

Recall, that the present value of an ordinary annuity formula can be used to calculate the present value of a stream of payments received at the end of each year. The formula is as follows:

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]

How to Calculate the Future Value of an Annuity Due

Recall, that the future value of an ordinary annuity can be used to calculate the future value of a stream of payments that are received at the end of each year, using the following formula:

future value of an ordinary annuity formula

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]

How to Calculate the Present Value of an Ordinary Annuity

The present value of an ordinary annuity formula can be used to calculate the present value of a stream of income payments, the formula is as follows:

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]

How to Calculate the Future Value of an Ordinary Annuity

The future value of an ordinary annuity formula can be used to calculate the future value of a stream of payments over time, the formula is as follows:

future value of ordinary annuity formula

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:

Period	PV	rate	FV	PMT
1				$ 1,000.00
2	$ 1,000.00	7.00%	$ 1,070.00	$ 2,070.00
3	$ 2,070.00	7.00%	$ 2,214.90	$ 3,214.90
4	$ 3,214.90	7.00%	$ 3,439.94	$ 4,439.94
5	$ 4,439.94	7.00%	$ 4,750.74	$ 5,750.74
6	$ 5,750.74	7.00%	$ 6,153.29	$ 7,153.29
7	$ 7,153.29	7.00%	$ 7,654.02	$ 8,654.02
8	$ 8,654.02	7.00%	$ 9,259.80	$ 10,259.80
9	$ 10,259.80	7.00%	$ 10,977.99	$ 11,977.99
10	$ 11,977.99	7.00%	$ 12,816.45	$ 13,816.45

FV of an annuity table; PMT = $1,000, r = 0.07

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

Period	PV	rate	FV	PMT
1				$ 1,000.00
2	$ 1,000.00	4.00%	$ 1,040.00	$ 2,040.00
3	$ 2,040.00	4.00%	$ 2,121.60	$ 3,121.60
4	$ 3,121.60	4.00%	$ 3,246.46	$ 4,246.46
5	$ 4,246.46	4.00%	$ 4,416.32	$ 5,416.32
6	$ 5,416.32	4.00%	$ 5,632.98	$ 6,632.98
7	$ 6,632.98	4.00%	$ 6,898.29	$ 7,898.29
8	$ 7,898.29	4.00%	$ 8,214.23	$ 9,214.23
9	$ 9,214.23	4.00%	$ 9,582.80	$ 10,582.80
10	$ 10,582.80	4.00%	$ 11,006.11	$ 12,006.11

FV of an annuity table; PMT = $1,000, rate = 0.04

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: