The most common example of a deferred annuity is a retirement fund where the investor is not yet ready to retire. They defer their withdrawals (payments) until they retire. In the mean time, the fund earns interest. The fund continues to earn interest as the investor withdraws money from the fund.

Other possible examples of deferred annuities are student loans^[1] and in-store credit cards that offer “pay nothing for 18 months” promotions on their in-store credit cards. These are both less than perfect examples of deferred annuities in that they both come with exceptions.

In the case of the student loan, the student defers repaying the loan until after they have finished school^[2]. What is special about this case is that the Government does not charge them any interest during the deferment. If the loan is granted by a bank or other financial institution, then the student gets charged interest during the deferral period.

In the case of “pay nothing for 18 months” promotion, the credit card holder defers payments (possibly waits for 18 months to start making payments) on the card. These cards and promotions come with a lot of fine print and exceptions. The credit card example becomes like a deferred annuity if the credit card holder fails to pay off the entire balance on the card before the 18 months is up. They get back-charged interest on the amount borrowed. If, after this happens, they set up a payment schedule to repay the amount owing on the card with regular, equal-sized payments, then the credit card example becomes a deferred annuity example.

If you search the web for ‘deferred annuity’ — the savings deferred annuity related to planning for retirement will come up in at least the top 5 searches. It is the most common type of deferred annuity. When people plan for their retirement, they start saving before they retire. They usually do not need to withdrawn money from a retirement fund until they retire. In other words, they defer the withdrawals until retirement. Let us look at an example of this to better understand this type of deferred annuity.

Example 5.5.1: Deferred Withdrawals from a Retirement Fund with BGN Off

Ann and Sue sell their recreation property. They make $245,000 on the sale and deposit it immediately into a retirement fund. The fund earns 2.45%, compounded monthly, for the entire time.   Exactly ten years after the sale, Ann retires. They now want to start supplementing their income with payments from the retirement fund. They want the payments to start one month after Ann retires.  They want these payments to last for 5 years. How much can they withdraw per month from the retirement fund?

Let us first organize this information into a time diagram:

• The deferral period (part 1) occurs first.
• The deposit of $245,000 occurs at the very beginning of this deferral period.
• No payments nor withdrawals are made for the first 10 years (only interest is added for these 10 years).
• After the 10 years is complete, part 2, the annuity, begins.
• One month after the start of part 2 (annuity), the first withdrawal occurs.
• These withdrawals repeat for 5 years until no money is left in the annuity.

The tricky part with these types of problems with two parts is determining which part to start with. Start where the “known” money is. In this example, we know that Ann and Sue deposit $245,000 immediately (PV₁ = 245,000). For this reason, we will start with part 1 because PV₁ is known.

Part 1: Let us now organize the information for Part 1 into a BAII Plus table:

• Since there are no payments, B/E doesn’t matter (that is why ‘––‘  is set for B/E).
• Because the interest rate is 2.45% compounding monthly, I/Y = 2.45 and C/Y = 12.
• Because there are no payments, match P/Y to C/Y (P/Y = 12).
• The deferral period lasts 10 years, so N₁ = 10×12=120.
• Remember, the $245,000 deposit occurs at the very beginning so PV₁ = +245,000.
• There are no payments for the deferral period so PMT₁ = 0.
• CPT  FV₁ = −312,939.05. Ann and Sue’s initial deposit grows to $312,939.05 after 10 years due to the interest earned on the deposit.
• The $312,939.05 (FV₁) will become the starting value for the annuity in part 2 (PV₂).

Part 2: Let us now organize the information for Part 2 into a BAII Plus table:

• The payments start one month after Ann retires (in one payment interval), so B/E = END.
• Because the interest rate is still 2.45% compounding monthly, I/Y = 2.45 and C/Y = 12.
• Ann and Sue receive monthly payments so P/Y = 12.
• They want these payments to last 5 years so, N₂ = 5×12=60.
• Use the (positive) value of FV₁ for PV₂. So, PV2 = +312,939.05.
• There is no money left at the end of 5 years so FV₂ = 0.
• Compute PMT₂ to get the size of Ann and Sue’s monthly withdrawals.

Conclusion: Ann and Sue can withdraw $5,546.95 per month from their retirement fund.

Example 5.5.2: Deferred Withdrawals from a Retirement Fund with BGN On

What would change in Example 1 if Ann and Sue made the first withdrawal exactly 10 years after the sale of their property?  Ie: what if their first withdrawal occurred the day that Ann retired?

Let us first look at the timeline for this problem:

• Part 1 is the same as in Example 1, so we do not need to redo those calculations.
• Part 2 has changed. The first withdrawal (PMT₂) now occurs exactly at the start (beginning) of that annuity.
• This means we set the calculator to BGN when calculating the annuity payments in part 2 (PMT₂):

Conclusions:

• The amount withdrawn by Ann and Sue decreases slightly from $5,546.95 to $5,535.64 when they make the first withdrawal on the day that Ann retires.
• Withdrawing the first payment one month earlier means that Ann and Sue will miss out on the interest that would have been earned on that payment.
• This is because the money in the annuity continues to earn interest as the withdrawals are being made.
• As a result, Sue and Ann withdraw $11.31 less per month if they make the first withdrawal exactly 10 years after the sale of their property.

Again we use same interest formula:

$\begin{align*} \textrm{Interest Earned} &= \textrm{Money Out} - \textrm{Money In} = \textrm{\$ OUT} - \textrm{\$ IN} \end{align*}$

We need to be careful when calculating money in and money out for deferred annuities.

• What money was deposited in the account? Only the initial deposit (PV₁) is considered money in ($ IN).
• What money do we take out (withdraw)? Only the regular withdrawals (PMT₂) and final withdrawal (FV₂) in part 2 are considered to be money out ($ OUT).
• Because FV₁ does not get withdrawn but instead becomes the starting balance for the annuity (PV₂), it is not included in our interest calculation. The money is neither being deposited nor withdrawn.

This gives us the following equation for interest earned:

$\begin{align*} \textrm{Interest Earned} &= \textrm{\$ OUT} - \textrm{\$ IN}\\ &=( \textrm{Regular Withdrawals}+\textrm{Final Withdrawal}) - \textrm{Initial Deposit}\\ &= ( \textrm{PMT}_2\times\textrm{N}_2+\textrm{FV}_2)-\textrm{PV}_1 \end{align*}$

It is quite common to defer payments when purchasing items, especially in retail.  A retailer purchases their goods from a supplier.  If cash flow is an issue for the retailer, they can often choose to defer the payments on the shipment until a later date.  It is also common when consumers purchase items using “in-store” credit cards.  Some stores have “Pay Nothing” promotions for their customers.  If a customer purchases goods from the store on the in-store credit card, they can choose to make no payments on their purchase for a certain amount of time (6 months, 12 months, 18 months,…).

In the above cases, the retailer or customer owes the value of the amount of goods they purchased.  They make no payments for a certain amount of time (they defer their payments).  After the deferral period has passed, they make payments to repay the value of the goods plus the interest charged during the deferral period.  See the timeline describing this process:

Part 1: The initial amount owing gathers interest.  There are no payments in part 1 (the deferral period).  The only money being added to the balance is the interest being charged.  This problem is a compound interest problem (Chapter 4):

Part 2: Payments are now being made on the balance owing.  This is the ‘annuity’ part of the problem.

Example 5.5.3: Deferred Repayment of Money Owed with BGN ON

Luis is renovating his kitchen.  He takes advantage of a promotion offered by Home Depot: “Pay nothing for 18 months.”  He purchases $7,500 of materials on his Home Depot card.  Home Depot charges him 28.8% compounded monthly.  18 months after the purchase, Luis makes his first of 6 monthly payments to pay off the credit card.  What is the size of Luis’s monthly payments?

Let us first look at the timeline for this problem:

From there, fill in the exercise below to figure out the size of Luis’ payments.

Again we use same interest formula:

$\begin{align*} \textrm{Cost of Financing} &= \textrm{Money Out} - \textrm{Money In} = \textrm{\$ OUT} - \textrm{\$ IN} \end{align*}$

We need to be careful when calculating money in and money out for deferred annuities.

• What is the money in ($ IN)? Only the initial amount borrowed (PV₁) is considered money in ($ IN).
• What is the money out ($ OUT)? Only the regular payments (PMT₂) are considered to be money out ($ OUT):
• You should note that because FV₁ is not a payment but instead becomes the starting balance for the annuity (PV₂), it is not included in our interest calculation.

This gives us the following equation for cost of financing:

$\begin{align*} \textrm{Cost of Financing} &=\textrm{\$ OUT} -\textrm{\$ IN}\\ &=\textrm{Regular Payments} - \textrm{Amount Borrowed}\\ &=PMT_{2}×N_{2}-PV_{1} \end{align*}$

It is possible that the initial amount borrowed (PV₁) for a deferred annuity is unknown.

In the case, we must know the payment size in part 2 (PMT₂). For this reason, we start with part 2 (the annuity) in these cases. See Jessa’s example below where she knows the size of the monthly payments required to finance a bedroom set at Ikea.

Example 5.5.4: Deferred Repayment with the Initial Amount Owed Unknown

Jessa is shopping at Ikea. She sees a sign for a beautiful bedroom furniture set. The sign reads the following. “Make no payments for 6 months followed by 12 easy payments of $129.99 for this entire bedroom set.” Jessa has just moved into her new apartment and needs new bedroom furniture but she’s a bit tight on cash right now. She is interested in the furniture and curious about how much the actual bedroom set costs. She reads the fine-print on the sign that states that Ikea charges an effective interest rate of 28.8% on financed purchases. What is the cost of financing that Jessa will pay if she purchases the bedroom set this way?

Let us first look at the timeline for this problem:

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