12.S: Summary
- Page ID
- 28584
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Concepts
12.1: Deferred Annuities
- The stages of deferred annuities
- The four common unknown variables and how to solve for them
12.2: Constant-Growth Annuities
- The concept of constant growth and the modifications required to annuity formulas
- The four new annuity formulas
- How to solve constant growth scenarios
12.3: Perpetuities
- An explanation of perpetuities
- Ordinary and due types of perpetuities
- The two perpetuity formulas
- How to solve perpetuity scenarios
12.4: Leases
- An explanation of lease characteristics and how leases operate
- Capitalized lease liabilities explained
- How to solve leasing scenarios
12.5: Application - How To Purchase A Vehicle
- Four financial choices available to make payments on a vehicle
- The basis and procedure upon which the selection of the vehicle ownership choice is made
- Key considerations to keep in mind when purchasing a vehicle
12.6: Application - Planning Your RRSP
- The three components of retirement income planning
- The procedure for retirement income planning
The Language of Business Mathematics
- capitalized lease liability
-
The present value of the remaining lease payments and residual value on a capital good using a discount rate equivalent to the interest rate the business would have had to pay if it had purchased the asset instead.
- constant growth annuity
-
An annuity in which each annuity payment is increased by a fixed percentage.
- deferred annuity
-
A financial transaction where annuity payments are delayed until a certain period of time has elapsed.
- down payment
-
A portion of the purchase price required up front.
- lease
-
A contract by which the owner of an asset gives another party an exclusive right to possess and use the asset under specified conditions for a specific period of time in return for agreed-upon payments.
- lessee
-
The borrower of a leased asset.
- lessor
-
The owner of a leased asset.
- net rate
-
The growth in a constant growth annuity that is attributable solely to the interest rate and not to the growth in the annuity payment.
- period of deferral
-
The time segment of a deferred annuity where the single payment earns interest and no contributions are made to the investment.
- perpetuity
-
A special type of annuity that has fixed, regular payments continuing indefinitely.
- residual value
-
The projected value of an asset at the end of its lease term.
- zero growth annuity formula
-
Any annuity formula where the growth rate is 0%. All formulas presented in Chapter 11 incorporate zero growth
The Formulas You Need to Know
Symbols Used
\(∆\%\) = the constant growth rate per payment interval (percent change)
\(CY\) = compounding frequency
\(FV_{DUE}\) = future value of an annuity due
\(FV_{ORD}\) = future value of an ordinary annuity
\(i\) = periodic interest rate
\(N\) = number of annuity payments
\(PMT\) = annuity payment amount
\(PV_{DUE}\) = present value of an annuity due
\(PV_{ORD}\) = present value of an ordinary annuity
\(PY\) = payment frequency
Formulas Introduced
Formula 12.1 Future Value of a Constant Growth Ordinary Annuity:
\[FV_{ORD}=PMT(1+\Delta \%)^{N-1} \left [ \dfrac{\left [ \dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%} \right ]^N - 1}{\dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%}-1} \right ] \nonumber \]
Formula 12.2 Future Value of a Constant Growth Annuity Due:
\[FV_{DUE}=PMT(1+\Delta \%)^{N-1} \left [ \dfrac{\left [ \dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%} \right ]^N - 1}{\dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%}-1} \right ]\times (1+i)^{\frac{CY}{PY}} \nonumber \]
Formula 12.3 Present Value of a Constant Growth Ordinary Annuity:
\[PV_{ORD}=\dfrac{PMT}{1+\Delta \%} \left [ \dfrac{1 - \left [ \dfrac{1+\Delta \%}{(1+i)^{\frac{CY}{PY}}} \right ]^N }{\dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%}-1} \right ]\times (1+i)^{\frac{CY}{PY}} \nonumber \]
Formula 12.4 Present Value of a Constant Growth Annuity Due:
\[PV_{DUE}=\dfrac{PMT}{1+\Delta \%} \left [ \dfrac{1 - \left [ \dfrac{1+\Delta \%}{(1+i)^{\frac{CY}{PY}}} \right ]^N }{\dfrac{(1+i)^{\frac{CY}{PY}}}{1+\Delta \%}-1} \right ]\times (1+i)^{\frac{CY}{PY}} \nonumber \]
Formula 12.5 Ordinary Perpetuity Present Value:
\[PV_{ORD}=\dfrac{PMT}{(1+i)^{\frac{CY}{PY}-1}} \nonumber \]
Formula 12.6 Perpetuity Due Present Value:
\[PV_{DUE}=PMT\left(\dfrac{1}{(1+i)^{\frac{CY}{PY}}-1}+1\right) \nonumber \]