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.

If an investor started with a $1,000 portfolio, that is now worth $1,100, we could calculate the HPR as follows:

HPRs can also be adjusted to take into account dividends or interest received during the period. Assume in the example above the portfolio generated $25 in income:

In this instance, the HPR increases from 10% to 12.5%.

In real life, it is often common for portfolio managers to make additional investments throughout the year. Let’s assume that a PM created a portfolio with an initial investment of $1,500,000. Throughout the course of the year the PM made three additional deposits of $60,000 each.

When calculating HPRs given the assumptions above, the beginning and ending values must be taken into account before and after the deposit was made. We can see how the $60,000 deposits affected the beginning and ending market values on the day the deposit was received in the table below:

Let’s examine the effect the deposits had in the month of January. On January 15th, the portfolio had an ending market value of $1,550,000, the deposit of $60,000 received on January 16th is credited to the ending market value on the 15th, for a total beginning market value of $1,610,000 on the 16th.

If the ending market value on January 31st was $1,615,000, in order to calculate the HPR for the month, we must calculate the HPR for each of these two sub-periods and geometrically link them together as follows:

Based on the table above, the HPR return after geometrically linking the sub-periods of 1/1/20 – 1/15/20 and 1/6/20 to 1/31/20 is 3.65%. Geometrically linking returns can be done as follows:

Let’s use the same methodology of geometrically linking returns to calculate the HPRs of this portfolio over the eight month period:

Now, we can geometrically link each of the months together for a total HPR of 4.66% for the period of January 1st to August 28th.

The Excel model used to calculate multiple HPRs 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.

where:

r-sub-f = risk free rate
i = inflation premium
d = default risk premium
L = liquidity premium
m = maturity premium

Suppose you wanted to purchase a $1,000 bond, what is the required return that you should expect on the bond given the individual components above?

At the very least, you should expect to earn the nominal risk-free rate which is the product of the risk-free rate and the expected rate of inflation. Assume that short term treasuries are yielding 2%, and the expected rate of inflation is 3%. What is the nominal risk-free rate?

On an additive basis, we can calculate the nominal risk free rate as follows:

However, the convention used by the CFA Institute is multiplicative rather than additive. Let’s calculate the nominal risk-free rate using this method:

Based on the numbers above, the market has determined that the nominal risk-free rate, given the current yield on risk-free bonds and the projected rate of inflation, should be 5.06%.

From here, values need to be assigned for default risk, liquidity risk, and maturity.

Default risk adds a premium based on the probability of default of the borrower. These probabilities are typically reflected by the credit ratings of the borrower. Credit rating agencies assign credit ratings which classify bonds as being either investment grade, or high yield (junk). The further you go down in credit rating the higher the premium for default risk should be.

Liquidity risk is determined by how quickly or how long it would take to liquidate the bond on the open market prior to maturity. Thinly traded bonds with low volume would require a higher liquidity premium.

Lastly, the maturity premium is determined by the amount of time that needs to elapse before the bond matures and you receive you principal back. The longer the maturity, the greater the maturity premium should be.

If we assume that the premiums assigned for default risk, liquidity risk, and maturity are 0.20%, 0.50%, and 2.00% respectively, we can now calculate the total required rate of return or discount rate as follows:

Understanding the components of interest rates components is critically important when attempting to determine the required rate of return on fixed income securities.

Typically, bond rates are quoted as a nominal risk-free rate plus some spread. In this case, the spread represents the default, liquidity, and maturity risks.

where:

CF = the periodic cash flow of the perpetuity
i = the discount rate

Assume an investor wanted to purchase a preferred stock that paid an annual dividend of $3.50, using a discount rate of 7%, what is the value of the preferred stock?

We can calculate the present value using the formula above as follows:

In this particular scenario, the present value of the stream of dividend payments for this particular security would be $50.00.

The Information Ratio can be calculated using the following formula:

where:

Let’s assume an investor wants to compare two large cap value managers. Manager A & B’s portfolios have the following characteristics:

Given the numbers above, let’s calculate the information ratio for Manager A:

Next, we’ll calculate the information ratio for Manager B:

On the surface, it would appear that Manager B’s portfolio is superior to Manager A’s portfolio based solely on the absolute level of investment returns; however, Manager A’s portfolio is superior if looking at absolute returns on a risk adjusted basis.

Since Manager A’s information ratio of 1.00 is greater than Manager B’s information ratio of 0.667, we can make the determination that Manager A has better risk adjusted returns, all else being equal, since both of these managers are creating portfolios with a large cap value mandate and their returns are adjusted using the same benchmark.

Generally speaking, information ratios near one are good, above one are great, and above zero are passable. It is important to note, that no information can be gleaned from information ratios that are negative.

where:

Let’s assume that an investor purchases a security that has a project rate of return of 7%, if the risk free rate of return is 3% and the standard deviation of the asset is 15%, what is the Sharpe Ratio of the asset?

We can calculate the Sharpe Ratio as follows:

A Sharpe Ratio of 0.266 can be interpreted as the amount of return the asset produces for each given unit of risk. In other words, for each 1% increase in standard deviation, this particular asset produces 0.26% in return. If you multiple the standard deviation of 0.15 by 0.266, the resulting product of the two numbers is 0.07, which is asset’s assumed rate of return.

Portfolio managers typically compare the Sharpe Ratios of different portfolios and assets in order to determine which portfolio or asset has a higher risk adjusted rate of return. If two portfolios have similar investment characteristics, the portfolio with the higher Sharpe Ratio should be considered over the one with the lower ratio, all else equal.

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:

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

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:

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:

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:

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.

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:

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:

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

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

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.

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][-]