Think & Done

Annual holding period return & yield is explained and illustrated with example calculation using the HPR,AHPR, HPY and AHPY formula. We will discuss the topics of holding period return, holding period yield, annual holding period return and annual holding period yield. Illustrations will be provided that show calculation of these returns using the mathematical and Excel formula.

Rates of Return and Risks

Investors need as much information as they can gather about return on their investments and risk associated with their investments to contemplate whether to invest or not to invest. Rate of returns and risks are related as when risks are greater, investors will want a higher rate of return on the investment.

Risk Free Rate

If an investor was to put aside an amount of money for a fixed period of time, she will be expecting a larger sum of money in the future. Say, for example, Judith has a spare $1000 and she is willing to invest it for five years but she expects to collect an amount of $1500 at the end of 5 years. Assumption is that inflation stays at the current rate and doesn't increase during the next five years. This excess amount of $500 collected at the end of fifth year illustrates pure time value of money concept. The rate of return for this investment is 8.45% and is called the risk free rate

Nominal Risk Free Rate

But seldom do the prices stay static, with passage of time inflation grows. Thus, Judith would want to offset the short fall in her investment due to inflation with a higher rate of return. Assume inflation were to grow by 2% in the next five years, thus Judith would demand a rate of return of 10.45% so that her final savings gives her the same amount of money as she would otherwise had if inflation wasn't a factor. This new rate is called Nominal Risk Free Rate.

Risk Premium

All investments carry an inherent risk such as default, thus to augment such risks, investors would want even a higher rate of return. In our example if there was a 2% risk factor of the company unable to meet its promised return, Judith would want 2% added to the nominal rate of return thus her rate of return would be 12.45%. Such a rate of 2% ,in this case, is known as risk premium .

HPR or Holding Period Return

As an investor, you would be willing to forego an immediate use of your money in exchange for a higher sum that you could use at a later date. Your money will be tied up for a fixed duration of time, your investment will comprise of an initial amount at the beginning of period and a final amount at the end of period. An HPR or Holding Period Return is a measure that shows how your investment has grown or shrunk over a period of time.

Formula for Holding Period Return

HPR = Investment Amount at the End / Investment Amount at the Start

Calculation for Holding Period Retrun

Let us take the example we discussed earlier, where Judith invested an amount of $1000 for 5 years which resulted in a closing balance of $1500. So what is the HPR or Holding Period Return for Judith's investment

HPR = 1500/1000
HPR = 1.5

Thus an Holding Period Return of 1.5 means that Judith's investment has grown by a multiple of 1.5

HPY or Holding Period Yield

Rather than using a multiple, we can find a percentage rate by which an investment has grown or shrunk. It is called HPY or Holding Period Yield. To find HPY we subtract 1 from HPR, resulting in a percentage rate.

Formula for Holding Period Yield

HPY = Investment Amount at the End / Investment Amount at the Start - 1
HPY = HPR - 1

Calculation for Holding Period Yield

Staying with Judith's investment, we see that HPY or Holding period yield is 0.5 or 50% as calculated below.

HPY = 1.5 - 1
HPY = 0.5
HPY = 50%

AHPR or Annual Holding Period Return

An HPR reflects the overall return on an investment across the duration of investment, we usually quote returns and rates on annual terms. An annual holding period return is computed by finding the nth root of the HPR as metioned below:

Formula for Annual Holding Period Return

Annual Holding Period Return = (HPR)^1/n
where n is the duration of the investment.

Calculation of Annual Holding Period Return

In our example, Judith's investment has an Annual Holding Period Return of 1.0845 , as calculated below

Annual Holding Period Return = (1.5)^1/5
Annual Holding Period Return = (1.5)^0.2
Annual Holding Period Return = 1.0845

AHPY or Annual Holding Period Yield

Just as we quoted holding period return on an annual basis, we do the same for holding period yield thus resulting in Annual holding period yield which is calculated by subtracting 1 from annual holding period return.

Formula for Annual Holding Period Yield (AHPY)

Annual Holding Period Yield = AHPR-1

Calculation of Annual Holding Period Yield

Annual Holding Period Yield = 1.0845-1
Annual Holding Period Yield = 0.0845
Annual Holding Period Yield = 8.45%

How do you derive the Formula for Annual Holding Period Yield

We need to consider four variables when computing Annual holding period yield, the beginning investment amount (PV) , the ending investment amount (FV), the term of the investment (n) and the Annual holding period yield or the rate (i). If you look closely at these variable you would notice that it is nothing more than the equation for finding Future value or present value of an amount which is listed below.

PV = FV/(1+i)ⁿ

Our task is to separate i from this equation and bring it to the left hand side of the equation, let us do that with algebraic manipulation as follows:

(1+i)ⁿ = FV/PV
1+i= (FV/PV)^1/n
i= (FV/PV)^1/n-1

Plugging in the values for PV of $1000, FV of $1500 and n of 5, we have

i= (1500/1000)^1/5-1
i= (1.5)^0.2-1
i= 1.0844717711976986137456099224112-1
i= 0.0844717711976986137456099224112
i= 0.0845
i= 8.45%

IRR as Annual Holding Period Yield (AHPY)

It would seem that an annual holding period yield is nothing more than investor's required rate of return or what we call IRR, as we have an initial cash outlay followed with series of benefits albeit the interim benefits are in amounts of zero followed by a terminal value. We will attempt to illustrate that the AHPY is in fact an IRR.

	A	B
1	Year	Cash Flows
2	0	($1000)
3	1	$0
4	2	$0
5	3	$0
6	4	$0
7	5	$1500
8		=IRR(B2:B7)
9	IRR	8.45%

AHPY or Annual Holding Period Yield with MS Excel

You may download this MS Excel Template for find HPR, HPY, AHPR and AHPY

We can use MS Excel to find Holding Period Return, Holding Period Yield, Annual Holding Period Return and Annual Holding Period Yeild. The RATE function in MS Excel may be used to find the Annual Holding Period Yield as illustrated below.
=RATE(NPER,PMT,PV,FV,0)
=RATE(5,0,-1000,1500,0)
0.0845
8.45%