12.1: Deferred Annuities
- Page ID
- 22138
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The power of compound interest, as you have already seen, is amazing. Your investment realizes exponential growth, and over long periods of time the results are spectacular.
Taking advantage of this principle, many parents (and grandparents too) invest large amounts of money when their children are young to have enough money to pay for their college or university education. A simple investment of $10,000 at birth with no further contributions could sustain approximately four years’ worth of $1,200 monthly payments to the child starting at age 18. That is over $55,000 toward the child's education!
Many students now stay at home longer with their parents while they pursue postsecondary education. Some accumulate sums of wealth at an early age through part-time earnings when they have little or no expenditure. Assume a 21 year old student had accumulated $50,000 and instead of using that money on a new car or backpacking across Europe, she invested it into her RRSP. By age 65 and without contributing another dime, she could have a retirement fund that sustains 20 years of $15,000 monthly income, or approximately $3.6 million!
Compare this to someone who makes $300 monthly payments every month from age 21 to age 65, therefore contributing 528 payments totalling $158,400 out-of-pocket over the years. Under equal conditions, this person will receive about $12,000 monthly for 20 years, or approximately $2.9 million. See the difference? This person not only put $108,400 more into the RRSP but receives $0.7 million less in income during retirement.
Although investing the $50,000 into an RRSP instead of buying a new car is not quite as sexy, can you imagine never having to contribute to your RRSP again and being secure in your retirement before you even start your career? The financial freedom you will experience for the rest of your life would be enviable.
This section explores the concept of investing single payments today with the goal of using the maturity value to sustain an annuity afterwards. This is more commonly referred to as a deferred annuity.
What Is a Deferred Annuity?
A deferred annuity is a financial transaction where annuity payments are delayed until a certain period of time has elapsed. Usually the annuity has two stages, as depicted in this figure.
- Accumulation Stage. A single payment is allowed to earn interest for a specified duration. There are no annuity payments during this period of time, which is commonly referred to as the period of deferral.
- Payments Stage. The annuity takes the form of any of the four annuity types and starts at the beginning of this stage as per the financial contract. Note that the maturity value of the accumulation stage is the same as the principal for the payments stage.
The interest rate on deferred annuities can be either variable or fixed. However, since deferred annuities are commonly used to meet a specific need, fixed interest rates are more prevalent since they allow for certainty in the calculations.
The Formula
For a deferred annuity, you apply a combination of formulas that you have already used throughout this book. The accumulation stage is not an annuity, so it uses the various single payment compound interest formulas from Chapter 9. The payments stage is an annuity, so it uses the various annuity formulas from Chapter 11.
How It Works
For deferred annuities, the most common unknown variables are either the present value, the length of the period of deferral, the annuity payment amount, or the number of annuity payments that are sustainable for a fixed income payment. Follow this sequence of steps for each of these variables:
| Solving for the Present Value | Solving for the Period of Deferral | Solving for the Annuity Payment Amount | Solving for the Number of Annuity Payments |
|---|---|---|---|
| Step 1: Draw a timeline and identify the variables that you know, along with the annuity type. | |||
| Step 2: Starting at the end of your timeline, calculate the present value of the annuity using the steps from Section 11.3 (Formulas 11.4 or 11.5). Round your answer to two decimals. | Step 2: Starting at the beginning of the timeline, calculate the future value of the single payment using the steps from Section 9.2 (Formula 9.3). Round your answer to two decimals. | ||
| Step 3: Take the principal of the annuity, and using the steps from Section 9.3 (Formula 9.3) calculate the present value for the single amount | Step 3: Solve for the number of compounding periods using the applicable steps from Section 9.7 (Formula 9.3). The single payment investment is the present value, and the principal of the annuity is the future value. | Step 3: Calculate the annuity payment amount using steps from Section 11.4 (Formula 11.4 or 11.5). | Step 3: Calculate the number of annuity payments using steps from Section 11.5 (Formula 11.4 or 11.5). |
Important Notes
Rounding. The maturity value of the single payment or the present value of the annuity is always rounded to two decimals. Since an accumulation fund is different from a payment annuity, logistically the money is transferred between different bank accounts, which means that only two decimals are carried either forwards or backwards through this step of the required calculations.
Things To Watch Out For
Avoid these three common sources of error when you work with deferred annuities:
- Combining the Deferral Period and the Annuity Term. It is an error to treat the period of deferral and the term of the annuity as simultaneous time periods. For example, if a deferred annuity has a three-year period of deferral and a 10-year annuity term, this is sometimes interpreted, mistakenly, as an annuity ending 10 years from today. These time segments are separate and consecutive on the timeline! The correct interpretation is that the annuity term ends 13 years from today, since the 10-year term does not start until the three-year deferral terminates.
- Incorrect Timing between Stages. A common mistake is to determine incorrectly when the period of deferral ends and the annuity starts. This error usually results from forgetting that the payments on ordinary annuities start one payment interval after the annuity starts, whereas annuity due payments start immediately. Thus, if the first quarterly payment on an ordinary annuity is to be paid 6¾ years from today, then the period of deferral is 6½ years. If the deferral is for an annuity due, then the period of deferral is 6¾ years.
- Confusing \(N\). A deferred annuity requires different calculations of \(N\) using either Formula 9.2 or Formula 11.1. In the accumulation stage, recall that \(N\) must represent the number of compound periods calculated by Formula 9.2. In the payments stage, the \(N\) must represent the number of annuity payments calculated by Formula 11.1.
- If a deferred annuity has a four-year period of deferral and a seven-year annuity term, how many years from today will the term of the annuity end?
- If an ordinary deferred annuity makes its first monthly payment 25 months from now, how long is the period of deferral? What if it were a deferred annuity due?
- Answer
-
- 11 years
- 24 months; 25 months
Frasier is 33 years old and just received an inheritance from his parents' estate. He wants to invest an amount of money today such that he can receive $5,000 at the end of every month for 15 years when he retires at age 65. If he can earn 9% compounded annually until age 65 and then 5% compounded annually when the fund is paying out, how much money must he invest today?
Solution
Calculate the single payment that must be invested today. This is the present value (\(PV\)) of the deferred annuity.
What You Already Know
Step 1:
The deferred annuity has monthly payments at the end with an annual interest rate. Therefore, this is an ordinary general annuity.
The timeline for the deferred annuity appears below.
Ordinary General Annuity: \(FV\) = $0, \(IY\) = 5%, \(CY\) = 1, \(PMT\) = $5,000, \(PY\) = 12, Years = 15
Period of Deferral: \(FV = PV_{ORD}\), \(IY\) = 9%, \(CY\) = 1, Years = 32
How You Will Get There
Step 2:
Calculate the periodic interest rate (\(i\), Formula 9.1), number of annuity payments (\(N\), Formula 11.1), and present value of the ordinary general annuity (\(PV_{ORD}\), Formula 11.4).
Step 3:
Discount the principal of the annuity back to today. Calculate the periodic interest rate (\(i\), Formula 9.1), number of single payment compound periods (\(N\), Formula 9.2), and present value of a single payment (\(PV\), Formula 9.3 rearranged).
Perform
Step 2:
\(i=5 \% / 1=5 \% ; N=12 \times 15=180\) payments
\[PV_{ORD}=\$ 5,000\left[\dfrac{1-\left[\dfrac{1}{(1+0.05)^{\frac{1}{12}}}\right]^{180}}{(1+0.05)^{\frac{1}{12}}-1}\right]=\$ 636,925.79 \nonumber \]
Step 3:
\(i=9 \% / 1=9 \% ; N=1 \times 32=32\) compounds
\[\begin{align*} \$ 636,925.79 &= PV(1+0.09)^{32} \\ PV &= \$ 636,925.79\div(1.09)^{32}\\ &= \$ 40,405.54 \end{align*} \nonumber \]
Calculator Instructions
| Stage | Mode | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|---|
| Payments | END | 180 | 5 | Answer: -636,925.79 | 5000 | 0 | 12 | 1 |
| Accumulation | \(\surd\) | 32 | 9 | Answer: 40,405.53861 | 0 | 636925.79 | 1 | 1 |
If Frasier invests $40,405.54 today, he will have enough money to sustain 180 withdrawals of $5,000 in retirement.
Bashir wants an annuity earning 4.3% compounded semi-annually to pay him $2,500 at the beginning of every month for 10 years. To achieve his goal, how far in advance of the start of the annuity does Bashir need to invest $50,000 at 8.25% compounded quarterly? Assume 91 days in a quarter.
Solution
Calculate the amount of time between today and the start of the annuity. This is the period of deferral, or \(N\).
What You Already Know
Step 1:
The deferred annuity has monthly payments at the beginning with a semi-annual interest rate. Therefore, this is a general annuity due.
The timeline for the deferred annuity appears below.
General Annuity Due: \(FV\) = $0, \(IY\) = 4.3%, \(CY\) = 2, \(PMT\) = $2,500, \(PY\) = 12, Years = 10
Period of Deferral: \(PV\) = $50,000, \(FV = PV_{DUE}\), \(IY\) = 8.25%, \(CY\) = 4
How You Will Get There
Step 2:
Calculate the periodic interest rate (\(i\), Formula 9.1), number of annuity payments (\(N\), Formula 11.1), and present value of the ordinary general annuity (\(PV_{ORD}\), Formula 11.4).
Step 3:
Determine the number of compounds during the accumulation stage. Calculate the periodic interest rate (\(i\), Formula 9.1) followed by the number of single payment compound periods (\(N\), Formula 9.3 rearranged).
Perform
Step 2:
\(i=4.3 \% / 2=2.15 \% ; N=12 \times 10=120 \) payments
\[PV_{DUE}=\$ 2,500\left[\dfrac{1-\left[\dfrac{1}{\left.(1+0.0215)^{\frac{2}{12}}\right]}\right.}{(1+0.0215)^{\frac{2}{12}}-1}\right] \times(1+0.0215)^{\frac{2}{12}}=\$ 244,780.93 \nonumber \]
Step 3:
\[\begin{array}{c}{i=8.25 \% / 4=2.0625 \%} \\ {\$ 244,780.93=\$ 50,000(1+0.020625)^{N}} \\ {4.895618=1.020625^{N}} \\ {\ln (4.895618)=N \times \ln (1.020625)} \\ {\ln (4.895618)=N \times \ln (1.020625)} \\ {\quad N=\dfrac{1.588340}{0.020415}=77.801923 \text { quarterly compounds }} \\ {\text { Years }=77 \div 4=19.25=19 \text { years, } 3 \text { months }} \\ {0.801923 \times 91 \text { days }=72.975105=73 \text { days }}\end{array} \nonumber \]
Calculator Instructions
| Stage | Mode | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|---|
| Payments | BGN | 120 | 4.3 | Answer: -244,780.9336 | 2500 | 0 | 12 | 2 |
| Accumulation | END | Answer: 77.801924 | 8.25 | -5000 | 0 | 244780.93 | 4 | 4 |
To achieve his goal, Bashir needs to invest the $50,000 19 years, 3 months and 73 days before the annuity starts.
On the day of their granddaughter's birth, Henri and Frances deposited $3,000 into a trust fund for her future education. The fund earns 6% compounded monthly. When she turns 18, they then want it to make payments at the end of every quarter for five years. If the income annuity can earn 4.5% compounded quarterly, what is the amount of each annuity payment to the granddaughter?
Solution
Calculate the amount of the annuity payment (\(PMT\)) during the income payments stage of the deferred annuity.
What You Already Know
Step 1:
The deferred annuity has quarterly payments at the end with a quarterly interest rate. Therefore, this is an ordinary simple annuity.
The timeline for the deferred annuity appears below.
Period of Deferral: \(PV\) = $3,000, \(IY\) = 6%, \(CY\) = 12, Years = 18
Ordinary Simple Annuity: \(PV_{ORD} = FV\) after deferral, \(FV\) = $0, \(IY\) = 4.5%, \(CY\) = 4, \(PY\) = 4, Years = 5
How You Will Get There
Step 2:
Calculate the future value of the single payment investment. Calculate the periodic interest rate (\(i\), Formula 9.1), number of single payment compound periods (\(N\), Formula 9.2), and future value of a single payment amount (\(FV\), Formula 9.3).
Step 3:
Work with the ordinary simple annuity. First, calculate the periodic interest rate (\(i\), Formula 9.1), number of annuity payments (\(N\), Formula 11.1), and finally the annuity payment amount (\(PMT\), Formula 11.4).
Perform
Step 2:
\(i=6 \% / 12=0.5 \% ; N=12 \times 18=216\) compounds
\[FV=\$ 3,000(1+0.005)^{216}=\$ 8,810.30 \nonumber \]
Step 3:
\(i=4.5 \% / 4=1.125 \% ; N=4 \times 5=20 \) payments
\[\$ 8,810.30=PMT\left[\dfrac{1-\left[\dfrac{1}{(1+0.01125)^{\frac{4}{4}}}\right]^N}{(1+0.01125)^{\frac{4}{4}}-1}\right] \nonumber \]
\[PMT=\dfrac{\$ 8,810.30}{\left [\dfrac{1-\left[\dfrac{1}{(1+0.01125)^{\frac{4}{4}}}\right]^N}{(1+0.01125)^{\frac{4}{4}}-1} \right ]}= \dfrac{\$ 8,810.30}{\dfrac{0.200480}{0.01125}} = \$494.39 \nonumber \]
Calculator Instructions
| Stage | Mode | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|---|
| Payments | END | 216 | 6 | -3000 | 0 | Answer: 8,810.297916 | 12 | 12 |
| Accumulation | \(\surd\) | 20 | 4.5 | -8810.30 | Answer: 494.392721 | 0 | 4 | 4 |
The granddaughter will receive $494.39 at the end of every quarter for five years starting when she turns 18. Since the payment is rounded, the very last payment is a slightly different amount, which could be determined exactly using techniques discussed in Chapter 13.
Emile received a $25,000 one-time bonus from his employer today, and he immediately invested it at 8% compounded annually. Fourteen years from now, he plans to withdraw $2,300 at the beginning of every month to use as his retirement income. If the income annuity can earn 3.25% compounded semi-annually, what is the term of the annuity before it is depleted (including the smaller final payment)?
Solution
Figure out how long the income annuity is able to sustain the income payments. This requires you to calculate the number of annuity payments, or \(N\).
What You Already Know
Step 1:
The deferred annuity has monthly payments at the beginning with a semi-annual interest rate. Therefore, this is a general annuity due.
The timeline for the deferred annuity appears below.
Period of Deferral: \(PV\) = $25,000, \(IY\) = 8%, \(CY\) = 1, Years = 14
General Annuity Due: \(PV_{DUE} = FV\) after deferral, \(FV\) = $0, \(IY\) = 3.25%, \(CY\) = 2, \(PMT\) = $2,300, \(PY\) = 12
How You Will Get There
Step 2:
Calculate the future value of the single deposit. Calculate the periodic interest rate (\(i\), Formula 9.1), number of single payment compound periods (\(N\), Formula 9.2), and future value of a single payment amount (\(FV\), Formula 9.3).
Step 3:
Work with the general annuity due. Calculate the periodic interest rate (\(i\), Formula 9.1) and the number of annuity payments (\(N\), Formula 11.5 rearranged for \(N\)). Finally substitute into the annuity payments Formula 11.1 to solve for Years.
Perform
Step 2:
\(i=8 \% / 1=8 \% ; N=1 \times 14=14 \) compounds
\[FV=\$ 25,000(1+0.08)^{14}=\$ 73,429.84 \nonumber \]
Step 3:
\(i=3.25 \% / 2=1.625 \%\)
\[\$ 73,429.84=\$ 2,300\left[\dfrac{1-\left[\dfrac{1}{(1+0.01625)^{\frac{2}{12}}}\right]^N}{(1+0.01625)^{\frac{2}{12}}-1}\right] \times(1+0.01625)^{\frac{2}{12}} \nonumber \]
\[\begin{array}{l}{31.840361=\dfrac{1-0.997317^{N}}{0.002690}} \\ {0.085656=1-0.99730} \\ {\ln (0.997317)=N \times \ln (0.914343)} \\ {N=33.332019=34 \text { payments }} \\ {34=12 \times \text { Years }}\end{array} \nonumber \]
\[\text { Years }=2.8 \overline{3}=2 \text { years, } 10 \text { months } \nonumber \]
Calculator Instructions
| Stage | Mode | N | I/Y | PV | PMT | FV | P/Y | C/Y |
|---|---|---|---|---|---|---|---|---|
| Payments | END | 14 | 8 | -25000 | 0 | Answer: 73,429.84061 | 1 | 1 |
| Accumulation | BGN | Answer: 33.332019 | 3.25 | -73429.84 | 2300 | 0 | 12 | 2 |
The annuity will last 2 years and 10 months before being depleted.