A perpetuity is like a bond, but with no fixed term (no fixed maturity date).  If a corporation issues a perpetuity to an investor, the perpetuity will continue making payments to this investor indefinitely^[1]. Common examples of perpetuities are scholarship funds, perpetual trusts and war bonds, like the British Treasury Bonds issued for World War 1^[2]. Finally, some business investments can also be treated like perpetuities^[3].

There are several different types of perpetuities — see the sections below for the key formulas, tips and examples related to perpetuity calculations.

payments for Ordinary Perpetuities

If the payments for a perpetuity are withdrawn at the end of each interval, we call this an ordinary perpetuity.  The payment (PMT) for an ordinary perpetuity can be given by the following formula:

[latex]\textrm{PMT}= \textrm{PV} \times i[/latex]

where PV is the initial value of the perpetuity and i is the periodic rate (the rate per period).

We can also use the BAII Plus to calculate the PMT. There are a few ‘tricks’ to know when using the BAII Plus:

• B/E = “END” for ordinary perpetuities
• N = 1000 × P/Y. Use total # years = 1,000 for perpetuities^[4].
• PV = the initial balance of the perpetuity.
• FV = 0. The funds are never withdrawn. The amount withdrawn equals 0 for this reason.
• CPT PMT. Calculate the value of the payment (PMT).

Example 5.8.1

Sasha, a wealthy BCIT Alumnus, just donated $100,000 to create a scholarship at BCIT for first year business students. The scholarship will be awarded semi-annually and the fund will earn 5% compounded semi-annually. The first scholarship will be awarded in 6 months. What will be the size of the semi-annual scholarships awarded?

Using the formula:

[latex]\begin{align*} \textrm{PMT}&= \textrm{PV} × i = $100,000 × \frac{0.05}{2}\\ &= $100,000 × 0.025 = $2,500 \end{align*}[/latex]

Using the BAII Plus:

Conclusion: BCIT can pay out a scholarship of $2,500 every 6 months.

the Present Value for Ordinary Perpetuities

There are also two ways to calculate the present value of an ordinary annuity. We can use the following formula:

[latex]\textrm{PV}= \frac{\textrm{PMT}}{i}[/latex]

or we can, again, use the BAII Plus and use the same tricks as when calculating the payment (PMT).

Let us look again at Sasha’s example but instead, let us figure out the amount Sasha needs to donate.

Example 5.8.2

How much does Sasha (the wealthy BCIT alumnus) need to donate if he wants BCIT to give out $3,000 semi-annual scholarships with the first scholarship awarded in 6 months. Assume the fund still earns 5% compounded semi-annually.

Using the formula:

[latex]\textrm{PV}= \frac{\textrm{PMT}}{i} = \frac{\$3,000}{0.025}= \$120,000[/latex]

Using the BAII Plus:

Conclusion: Sasha will need to donate $120,000 to the scholarship fund at BCIT.

the Present Value for Perpetuitues Due

It is possible to have a perpetuity that gives out the first payment immediately.  We call this a perpetuity due.  An example of this is a scholarship fund where the first scholarship is awarded as soon as the fund has been created. ^[5]

If payments are withdrawn from a perpetuity fund at the start of each payment interval (perpetuity due), there will need to be slightly more in the perpetuity fund to account for the initial balance in the account dropping in value right at the start of the perpetuity.  That “slightly more” amount will be the value of the payment^[6]:

[latex]\text{PV}_{due} = \frac{PMT}{i} + PMT[/latex]

Let us now revisit Sasha’s scholarship example and determine the size of Sasha’s donation if the scholarship fund is a perpetuity due.

Example 5.8.3

Suppose that Sasha would like to donate enough money such that the semi-annual scholarships are still $3,000 but the first scholarship will be awarded immediately.  Assume the scholarship fund still earns 5% compounded semi-annually.  How much more will Sasha need to donate?

Using the formula and i = 0.025 and PMT = 3,000 gives:

[latex]PV_{due} = \frac{$3,000}{0.025} + \$3,000 = $123,000[/latex]

Using the BAII Calculator gives:

Sasha will need to donate $123,000 if the first scholarship is awarded immediately. Taking the difference between the required donation amount donated for Example 2 (an ordinary perpetuity) and Example 3 (a perpetuity due) gives:

[latex]\textrm{Difference} = \$123,000 - \$120,000 = \$3,000[/latex]

Conclusion: Sasha will need to donate $3,000 more if the first scholarship is awarded today instead of 6 months from now. It is no coincidence that the difference is equal to the payment size.

the Present Value for Deferred Perpetuitues

It is possible to defer the payments received from a perpetuity.  A first example is if a non-profit or charity receives a donation to cover future costs for the organization^[7].  Another example of a perpetuity is a business decision where there is an initial amount invested to start the business (PV₁) and then it takes several years for the business to start earning income^[8].

When calculating payment sizes for deferred perpetuities or the initial amount needed to be deposited at the start of the perpetuity, it is important to remember that there are actually two parts to the deferred perpetuity problem:

Part 1: The initial balance gathers interest.  There are no payments nor withdrawals in part 1 (the deferral period).  The only money being added to the balance is the interest being earned (or charged).  This problem is a compound interest problem (Chapter 4):

Part 2: Payments are now being withdrawn. These payments exactly equal to the interest earned on the current balance in the perpetuity account. The starting balance for the perpetuity (PV₂) equals to the ending balance from the deferral period (FV₁) .

Example 5.8.4

What if Sasha’s scholarship fund (from Examples 1 to 3) doesn’t give out its first scholarship for one year?  How much would Sasha need to donate if the semi-annual scholarships are still $3,000 and the fund still earns 5% compounded semi-annually?

Let us first draw out the timeline for this problem:

Because we know the payments for part 2, start there. We can either use the formula or the BAII Plus to do the calculations for this problem. Let us start by using the BAII Plus:

Part 2: Perpetuity (using the BAII Plus)

Let us ‘start’ part 2 exactly when the first scholarship is awarded. For this reason, B/E will be set to BGN:

Notice that PV₂ equals to the amount Sasha would need to donate in Example 3. We will now enter that value into FV₁ and calculate the amount donated initially (PV₁).

Part 1: Deferral Period (using the BAII Plus)

There are no payments nor withdrawals for part 1 so enter ‘––’ for B/E. Also, because the deferral period lasts for 1 year, N₁ =1×2 = 2.

Conclusion: Sasha will only need to donate $117,073.17 if the first scholarship is awarded in one year.

Method 2 — Using Formulas:

We will again start with part 2. Let us rewrite the timeline, slightly differently, however:

Notice that we will only run the deferral period for 6 months and start the perpetuity (part 2) 6 months before the first scholarship is withdrawn. This makes the calculation easier for part 2:

Part 2: Perpetuity (using Formulas)

We now have ‘started’ part 2 one payment period earlier. This now makes the perpetuity in part 2 an ordinary annuity. We use the following formula to calculate its present value:

[latex]\text{PV}_2= \frac{\text{PMT}_2}{i} = \frac{$3,000}{0.025}=$120,000[/latex]

There will need to be $120,000 in the scholarship fund in 6 months. We can find the present value of this amount using the compound interest formula from Chapter 4.

Part 1: Deferral Period (using Formulas)

[latex]\textrm{PV}_{1}= \frac{\textrm{FV}_{1}}{(1+i)^{n}} = \frac{$120,000}{(1+0.025)^{1}}=$117,073.17[/latex]

Conclusion: Again, we find that Sasha will need to donate $117,073.17 now if the first scholarship is awarded in 1 year.

payments for Deferred Perpetuities

It is possible to use formulas to calculate the payment size for deferred perpetuities. We will however, just use the BAII Plus method for this section.

Let us examine a similar donation example. In this example, Ezra, a wealthy philanthropist will make a large donation to help out a medical center that will not open for several years.

Example 5.8.5

Ezra, a wealthy philanthropist, donates $1,000,000 to the Moshi Medical Diagnostics Center, located in Moshi, Tanzania. The Center is due to open in 3 years. The Center will use the donated funds to cover their annual equipment and testing costs. They will invest the funds at 4.25%, effective, into a perpetuity.  They will withdraw the first perpetuity payment in exactly 3 years when the center opens.  Calculate the size of the annual withdrawals.

Let us first draw out a timeline for this problem:

Next, determine which part to start with. In this case, because we know the initial deposit (PV₁), start with part 1 (the deferral period).

Part 1 (the deferral period):

It does not matter what we enter in B/E above because there are no payments. Let us now enter the ending value for the deferral period (FV₁) into the starting value for the perpetuity (PV₂).

Part 2 (the perpetuity):

Notice that BGN is on in part 2. This is because we started part 2 (the perpetuity) right when the first payment was withdrawn.

Conclusion: The Moshi Medical Center will receive $46,189.27 annually to cover equipment and testing costs starting in 3 years when the center opens.

Key Takeaways for Perpetuities

Your Own Notes

• Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
• These notes are for you only (they will not be stored anywhere)
• Make sure to download them at the end to use as a reference

The Footnotes