When measuring returns over a single period the holding period return (HPR) is used. HPRs can be calculated using the following formula:
holding period return formula
If an investor started with a $1,000 portfolio, that is now worth $1,100, we could calculate the HPR as follows:
v1 = $1,100, v0 = $1,000
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:
v1 = $1,100, v0 = $1,000, income = $25
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:
Sub-Period
Cf
Beg MV
End MV
Jan 1, 2020 to Jan 31, 2020
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
Feb 1, 2020 to Feb 29, 2020
$ –
$ 1,615,000.00
$ 1,650,000.00
Mar 1, 2020 to Mar 31, 2020
$ –
$ 1,650,000.00
$ 1,625,000.00
Apr 1, 2020 to Apr 30, 2020
Apr 1, 2020 to Apr 7, 2020
$ –
$ 1,625,000.00
$ 1,630,000.00
Apr 8, 2020 to Apr 30, 2022
$ 60,000.00
$ 1,690,000.00
$ 1,685,000.00
May 1, 2020 to May 31, 2020
$ –
$ 1,685,000.00
$ 1,700,000.00
Jun 1, 2020 to Jun 30, 2020
$ –
$ 1,700,000.00
$ 1,710,000.00
Jul 1, 2020 to Jul 31, 2020
Jul 1, 2020 to Jul 5, 2020
$ –
$ 1,710,000.00
$ 1,712,000.00
Jul 6, 2020 to Jul 31, 2020
$ 60,000.00
$ 1,772,000.00
$ 1,760,000.00
Aug 1, 2020 to Aug 28, 2020
$ –
$ 1,760,000.00
$ 1,750,000.00
initial portfolio value and subsequent cash flows
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:
Sub-Period
Cf
Beg MV
End MV
Monthly Ret
HPR
Jan 1, 2020 to Jan 31, 2020
3.65%
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
3.33%
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
0.31%
January HPR calculation
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:
geometric HPR
Let’s use the same methodology of geometrically linking returns to calculate the HPRs of this portfolio over the eight month period:
Sub-Period
Cf
Beg MV
End MV
Monthly Ret
HPR
Jan 1, 2020 to Jan 31, 2020
3.65%
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
3.33%
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
0.31%
Feb 1, 2020 to Feb 29, 2020
$ –
$ 1,615,000.00
$ 1,650,000.00
2.17%
2.17%
Mar 1, 2020 to Mar 31, 2020
$ –
$ 1,650,000.00
$ 1,625,000.00
-1.52%
-1.52%
Apr 1, 2020 to Apr 30, 2020
0.01%
Apr 1, 2020 to Apr 7, 2020
$ –
$ 1,625,000.00
$ 1,630,000.00
0.31%
Apr 8, 2020 to Apr 30, 2022
$ 60,000.00
$ 1,690,000.00
$ 1,685,000.00
-0.30%
May 1, 2020 to May 31, 2020
$ –
$ 1,685,000.00
$ 1,700,000.00
0.89%
0.89%
Jun 1, 2020 to Jun 30, 2020
$ –
$ 1,700,000.00
$ 1,710,000.00
0.59%
0.59%
Jul 1, 2020 to Jul 31, 2020
-0.56%
Jul 1, 2020 to Jul 5, 2020
$ –
$ 1,710,000.00
$ 1,712,000.00
0.12%
Jul 6, 2020 to Jul 31, 2020
$ 60,000.00
$ 1,772,000.00
$ 1,760,000.00
-0.68%
Aug 1, 2020 to Aug 28, 2020
$ –
$ 1,760,000.00
$ 1,750,000.00
-0.57%
-0.57%
Jan 1, 2020 to Aug 28, 2020 HPR
4.66%
HPR table
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.
The net present value (NPV) is the present value of a series of cash flows over a specified period of time. In the world of corporate finance, NPV is used to determine whether or not investment decisions in machinery or projects will add or subtract from shareholder wealth.
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:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
initial cash outlay and projected cash flows
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:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
PV
$ (10,000.00)
$ 2,803.74
$ 2,838.68
$ 2,857.04
$ 2,860.86
$ 2,851.94
present values
In order to calculate NPV, we simply add all of the present values together then subject from the total the initial cash outlay:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
PV
$ (10,000.00)
$ 2,803.74
$ 2,838.68
$ 2,857.04
$ 2,860.86
$ 2,851.94
NPV
$ 4,212.26
NPV = $4,212.26
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:
The interest rate “r” is usually referred to most commonly as the required rate of return or the discount rate. Interest rates can be broken down into five major components using the following formula:
interest rate components
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.
A perpetuity is a series of cash flow payments occurring in equal amounts forever. The formula used to calculate the present value of a perpetuity is as follows:
present value of a perpetuity formula
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:
where; CF =$3.50, i = 0.07
In this particular scenario, the present value of the stream of dividend payments for this particular security would be $50.00.
The information ratio is one component of the Fundamental Law of Active management and is a measure of risk adjusted returns relative to a stated benchmark.
The Information Ratio can be calculated using the following formula:
information ratio formula
where:
Let’s assume an investor wants to compare two large cap value managers. Manager A & B’s portfolios have the following characteristics:
Manager A
Manager B
Benchmark
Return
8.00%
9.00%
7.00%
Std Dev
11.00%
13.00%
10.00%
manager and benchmark characteristics
Given the numbers above, let’s calculate the information ratio for Manager A:
manager A’s information ratio
Next, we’ll calculate the information ratio for Manager B:
manager b’s information ratio
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.
The Sharpe Ratio is used to help investors calculate the risk adjusted return relative to the risk free rate of return, the formula is as follows:
sharpe ratio formula
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.
The formula used to calculate the weighted mean is as follows:
weighted average formula
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:
weighted average calculation
Using an HP12C calculator, we can calculate the weighted average using the following keystrokes:
