2 Annuities

2.1 Suplementary Concepts

An annuity is a financial product or arrangement designed to pay out a series of fixed or variable cash flows over a specified period. In this course, we will consider only fixed annuities, also known as annuities-certain.
If cash flows are constant throughout a specified period, geometric series can be applied in the calculation of present and future values.
The following formulae refer the present value of cash flows at year-end (annuity-immediate) and cash flows at the beginning of the year (annuity-due), respectively: \[ \boxed{ \begin{array}{rl} \\ \quad a_{\enclose{actuarial}{n}} &= v+v^2+\cdots +v^n &= \cfrac{1-v^n}{1-v} & = \cfrac{1-v^n}{i} \quad \\ \quad \ddot{a}_{\enclose{actuarial}{n}} &= 1+v^2+\cdots +v^{n-1} &= \cfrac{1-v^n}{1-v}(1+i) & = \cfrac{1-v^n}{d} \quad \\ \\ \end{array} } \] \(\\\)
The following formulae refer the future value of cash flows at year-end (annuity-immediate) and cash flows at the beginning of the year (annuity-due), respectively: \[ \boxed{ \begin{array}{rl} \\ \quad s_{\enclose{actuarial}{n}} &= (1+i)^{n-1} + (1+i)^{n-2} + \cdots + 1 &= \cfrac{(1+i)^n - 1}{i} \quad \\ \quad \ddot{s}_{\enclose{actuarial}{n}} &= (1+i)^n + (1+i)^{n-1} + \cdots + (1+i) &= \cfrac{(1+i)^n - 1}{d} \quad \\ \\ \end{array} } \]
Other useful formulae applicable when dealing with nominal rates of interest are: \[ \boxed{ \begin{array}{rl} \\ \quad a_{\enclose{actuarial}{n}}^{(m)} &= \cfrac{1-v^n}{i^{(m)}} \quad \\ \quad \ddot{a}_{\enclose{actuarial}{n}}^{(m)} &= \cfrac{1-v^n}{d^{(m)}} \quad \\ \\ \quad s_{\enclose{actuarial}{n}}^{(m)} &= \cfrac{(1+i)^n-1}{i^{(m)}} \quad \\ \quad \ddot{s}_{\enclose{actuarial}{n}}^{(m)} &= \cfrac{(1+i)^n-1}{d^{(m)}} \quad \\ \\ \end{array} } \]
The present or future value of sequences of cash flows that follow an arithmetic series result in a combination of a geometric and arithmetic series, because the \(k^{th}\) cash flow will be multiplied by either \(v^k\) or \((1+i)^k\). Below, some well-known formulae for the present and future values of increasing annuities. \[ \boxed{ \begin{array}{rl} \\ \quad (Ia)_{\enclose{actuarial}{n}} &= \cfrac{\ddot{a}_{\enclose{actuarial}{n}}-nv^n}{i} \quad \\ \\ \quad (Is)_{\enclose{actuarial}{n}} &= \cfrac{s_{\enclose{actuarial}{n}}-(n+1)}{i} \quad \\ \\ \end{array} } \]
A general formula for the present value of varying annuities is the following, where \(P\) represents the initial cash flow, and \(Q\) the constant adjustment to that cash flow. For the future value \(n\) years hence, simply multiply \(A\) by \((1+i)^n\). \[ \boxed{ \\ \begin{array}{rl} \\ \quad A &= P \, a_{\enclose{actuarial}{n}} + Q \cfrac{a_{\enclose{actuarial}{n}}-nv^n}{i} \quad \\ \\ \end{array} \\ } \]

2.2 Solved Exercises

Before opening the solutions, take a moment to work through each problem—-your mastery grows most when you engage actively with the mathematics.

Special annuities

An annuity pays \(500\) at the end of each month for the first \(10\) years, \(750\) at the end of each month for the next five years, and \(1{,}000\) at the end of the next five years.

If the nominal rate of interest convertible monthly is \(9\%\), compute the annuity’s value at time \(t=10\).

Solution

Since the cash flows are monthly, let’s use the effective monthly rate, \(j\), and set “months” as the unit of time.

Then \(j=\cfrac{i{(12)}}{12}-1 = \cfrac{0.09}{4}-1=0.0075\).

Let \(B_{10}\) be the book value at \(t=10\).

Then:

\(B_{10}=500 \, s_{\enclose{actuarial}{120}} + 750 \, a_{\enclose{actuarial}{60}} + 1000 \, (1.0075)^{-60} \, a_{\enclose{actuarial}{60}} \approx 163{,}655\)

Perpetuity

A perpetuity pays \(300\) at the end of each quarter for the first \(10\) years.

Thereafter, quarterly payments increase by \(1.5\%\) each.

If the nominal rate of interest convertible quarterly is \(8\%\), compute the perpetuity’s present value at time \(t=0\).

Solution

Since the cash flows are quarterly, let’s use the effective quarterly rate, \(j\), and set “quarters” as the unit of time.

Then \(j=\cfrac{i{(4)}}{4}-1 = \cfrac{0.08}{4}-1=0.02\).

Let \(PV\) be the present value of the perpetuity at \(t=0\).

Recall that \(a_{\enclose{actuarial}{\infty}}=\cfrac{1}{i}\).

Therefore,

\[\begin{align} PV &= \cfrac{300}{0.02} + 300 \times [ v^{41} \times 1.015 + v^{42} \times 1.015^2 + \cdots \,] \\ \\ &= \cfrac{300}{0.02} + 300 \times v^{41} \times 1.015 \times [1 + v \times 1.015 + \cdots \,] \\ \\ &= 300 \times \left[\cfrac{1}{j} + v^{41} \times 1.015 \times \left( \cfrac{1}{1 - v \times 1.015} \right) \right] \\ \\ & \approx 42{,}581 \end{align}\]

\(\\\)

Increasing/decreasing annuity

An annuity-immediate pays at the end of each year for \(20\) years.

The payment in year \(k\) is \(100k\), \(k=1, 2, ..., 15\).

Thereafter, payments decrease by \(200\) each.

The effective annual interest rate is \(7\%\).

Compute the accumulated value at time \(t=20\).

Solution

Recall the general formula for the present value of arithmetic increasing or decreasing annuities.

\(PV=P a_{\enclose{actuarial}{n}}+Q \cfrac{a_{\enclose{actuarial}{n}}-nv^n}{i}\), where \(P\) is the initial cash flow, and \(Q\) is the periodic change.

Let’s break this annuity into two annuities:

Let \(PV_1\) be the present value of the first annuity (i.e., years \(1-15\)). In this case, \(P=100\) and \(Q=100\).

Then,

\(PV_1 = 100 a_{\enclose{actuarial}{15}}+100 \cfrac{a_{\enclose{actuarial}{15}}-15v^{15}}{0.07} \approx 6{,}155.40\)

Let \(PV_2\) be the present value of the second annuity (i.e., years \(16-20\)). In this case, \(P=1500\) and \(Q=-200\).

Then,

\(PV_2 = (1.07)^-{15} \times \left[ 1500 \, a_{\enclose{actuarial}{5}}-200 \cfrac{a_{\enclose{actuarial}{5}}-5v^{5}}{0.07} \right] \approx 1{,}921.30\)

However, we are asked to supply the future value, \(FV\), at \(t=20\).

Then

\(FV = (PV_1 + PV_2) \times 1.07^{20} \approx 29{,}150.44\)

\(\\\)

2.3 Supplementary Exercises

An individual wishes to accumulate \(50\,000\) in a retirement fund at the end of \(20\) years.

If the individual deposits \(1\,000\) in the fund at the end of each of the first \(10\) years and \(1\,000 + X\) at the end of each of the second \(10\) years, find \(X\) if the fund earns \(7\%\) effective.

The present value of a seven-year immediate annuity of \(1\) is \(5.153\). If the term were \(11\), the present value would change to \(7.036\). If the term were \(18\), the present value would change to \(9.180\).

Find \(i\).

Find the present value of payments of \(200\) every six months starting immediately and continuing through four years from the present, and \(100\) every six months thereafter through ten years from the present, if \(i^{(2)}=0.06\).

A worker aged \(40\) wishes to accumulate a retirement fund by depositing \(1\,000\) at the beginning of each year for \(25\) years. Starting at age \(65\) the worker plans to make \(15\) annual withdrawals at the beginning of each year.

Find the amount of each withdrawal starting at age \(65\), if the effective rate of interest is \(8\%\) during the first \(25\) years but only \(7\%\) thereafter.

Find the present value of payments of \(1\,000\) at the beginning of each of \(8\) years, if the effective rate of discount is \(10\%\).

Deposits of \(1\,000\) are placed into a fund at the beginning of each year for the next \(20\) years. After \(30\) years annual payments commence and continue forever, with the first payment at the end of the thirtieth year.

Find the amount of each payment if \(i=0.055\).

A loan of \(1\,000\) is to be repaid by annual amounts of \(100\) to commence at the end of the \(5^{th}\) year and to continue thereafter for as long as necessary.

Find the time and amount of the final payment, if the final payment is to be larger than the regular payments.

Assume \(i=0.045\).

A fund of \(2\,000\) is to be accumulated by nine annual payments of \(50\) made at the end of each the year, followed by nine annual payments of \(100\) made at the end of each year, plus a smaller final payment made one year after the last regular payment.

If the effective rate of interest is \(4.5\%\), find the amount of the final irregular payment.

Annuity \(M\) pays \(4\) at the end of each year for \(36\) years. Annuity \(N\) pays \(5\) at the end of each year for \(18\) years. The present value of \(M\) and \(N\) are equal at an effective rate of interest \(i\).

An amount invested at the same rate \(i\) will double in \(n\) years.

Find \(n\).

A borrower has the following two options for repaying a loan:

Sixty monthly payments of \(100\) at the end of each month
A single payment of \(6000\) at the end of \(K\) months

Interest is at the nominal annual rate of \(12\%\) convertible monthly.

The two options have the same present value.

Find \(K\).

A loan of \(1\,000\) is to be repaid with annual payments at the end of each year for the next \(20\) years. For the first \(5\) years the payments are \(k\) per year; the second \(5\) years, \(2k\) per year; the third \(5\) years, \(3k\) per year; and the fourth \(5\) years, \(4k\) per year.

Find \(k\), if \(i=0.08\).

Find the accumulated value of a \(10\)-year annuity-immediate of \(100\) per year if the effective rate of interest is \(5\%\) for the first six years and \(4\%\) for the last four years.

Find the accumulated value \(18\) years after the first payment is made of an annuity on which there are eight payments of \(2\,000\) each made at two-year intervals.

The nominal rate of interest convertible semiannually is \(7\%\).

A sum of \(100\) is placed into a fund at the beginning of every other year for eight years.

If the fund balance a the end of eight years is \(520\), find the rate of simple interest earned by the fund.

Find the present value of an annuity-due of \(600\) per annum payable semiannually for \(10\) years if \(d^{(12)}=0.09\).

The present value of a perpetuity paying \(1\,000\) at the end of every three years is \(1\,373.63\). Find \(i\).

There is \(40\,000\) in a fund which is accumulating at \(4\%\) per annum convertible continuously.

If money is withdrawn continuously at the rate of \(2\,400\) per annum, how long will the fund last?

If \(X\) is the present value of a perpetuity of \(1\) per year with the first payment at the end of the second year, and \(20X\) is the present value of a series of annual payments \(1, 2, 3, ...\) with the first payment at the end of the third year, find \(d\).

Annual deposits are made into a fund at the beginning of each year for \(10\) years.

The first five deposits are \(1000\) each, and deposits increase by \(5\%\) per year thereafter.

If the fund earns \(8\%\) effective, find the accumulated value at the end of \(10\) years.

Find the present value of a \(20\)-year annuity with annual payments which pays \(600\) immediately, and each subsequent payment is \(5\%\) greater than the preceding payment.

The annual effective rate of interest is \(10.25\%\).

A perpetuity has payments at the end of each four-year period. The first payment at the end of four years is \(1\,000\)

Each subsequent payment is \(5\,000\) more than the previous payment. It is known that \(v^{4}=0.75\).

Calculate the present value of this perpetuity.

A family wishes to provide an annuity of \(100\) at the end of each month to their daughter now entering college.

The annuity will be paid for only nine months each year for four years.

Find the present value one month before the first payment, if \(i=0.04\).

Charlie deposits \(1\,000\) into an account at the beginning of each four-year period for \(40\) years.

The account credits interest at an annual effective interest rate \(i\).

The accumulated amount in the account at the end of \(40\) years is \(X\), which is five times the accumulated value in the account at the end of \(20\) years.

Calculate \(X\).

A factory buys a robot for \(30\,000\) by paying \(5\,000\) down and \(5\,000\) at the end of each year.

If the annual effective rate of interest is \(10\%\), what final payment, one year after the last full payment, will be necessary?

One thousand are deposited into Fund \(A\), which earns an annual effective rate of \(6\%\). At the end of each year, the interest earned plus an additional \(100\) is withdrawn from the fund. At the end of the tenth year, the fund is depleted.

The annual withdrawals of interest and principal are deposited into Fund \(B\), which earns an annual effective rate of \(9\%\).

Determine the accumulated value of Fund \(B\) at the end of year \(10\).

A certain stock is expected to pay a dividend of \(4\) at the end of each quarter for an indefinite period.

If an investor wishes to realize an annual effective yield of \(12\%\), how much should she pay for the stock?

To accumulate \(8\,000\) at the end of \(3n\) years, deposits of \(98\) are made at the end of each of the first \(n\) years, and \(196\) at the end of each of the next \(2n\) years.

The annual effective rate of interest is \(i\).

You are given that \((1+i)^{n}=2\).

Determine \(i\).

An actuarial student is presented with two offers:

Buy a car for \(9\,800\) and after three years trade it in for \(2\,000\)
Rent a car for \(250\) a month payable at the end of each month for three years.

If all additional expenses are identical for both alternatives, and the effective annual interest rate is \(8\%\), how much will the student save by renting the car?

An insurance company has an obligation to pay medical costs for a claimant.

Average annual claims costs today are \(5\,000\), and medical inflation is expected to be \(7\%\) per year.

The claimant is expected to live an additional \(20\) years.

Claim payments are made at yearly intervals, with the first claim payment to be made one year from today.

Find the present value of the obligation if the annual interest rate is \(5\%\).

30.

A couple pays \(3\,000\) at the end of each month on their home mortgage.

Find the equivalent semiannual payments at \(10\%\) per annum compounded semiannually.

31.

Find the discounted vaue of \(11\) payments made at the end of each year at an effective annual rate \(i=7\%\), if the successive payments are \(100\), \(200\), \(300\), \(400\), \(500\), \(600\), \(500\), \(400\), \(300\), \(200\), and \(100\).

Find the discounted value one year before the first payment of a series of \(20\) annual payments, the first of which is \(5\,000\).

The rate of inflation is \(6\%\) annual effective, and the rate of interest is \(8\%\) annual effective.

Charlie has deposited \(1000\) at the end of each year into a retirement savings plan for the last \(10\) years.

His deposits earned interest at the following annual effective rates: \(8\%\) for the first three years; \(10.25\%\) for the next four years; and \(9\%\) for the last three years.

What is the total interest earned for the \(10\) years?

An actuarial student buys a computer by paying \(150\) down and \(18.25\) a month for three years.

If the annual effective rate of interest is \(18\%\), what was the total amount of interest paid?

To settle a debt with interest at \(12\%\) compounded semiannually, a bank customer agrees to make \(15\) payments of \(400\) at the end of each half-year, and a final payment of \(292.39\) six months later.

Find the amount of debt.

A car is sold for \(8\,000\) down and six semiannual payments of \(3\,000\), the first due at the end of two years.

Find the price of the car if money is worth \(6\%\) compounded daily.

A \(40\,000\) mortgage is to be paid off by monthly payments at an annual effective rate of interest of \(12\%\).

The borrower can either repay the mortgage in \(20\) or \(30\) years.

Find the amount if interest saved by the borrower if she repays the mortgage in \(20\) years instead of \(30\) years.

A one-year deferred continuous varying annuity is payable for \(13\) years.

The rate of payments at time \(t\) is \(t^2-1\) per annum, and the force of interest at time \(t\) is \((1+t)^{-1}\).

Find the present value of the annuity.

A perpetuity is payable continuously at the annual rate of \(100\,(1+t)\) at time \(t\).

If \(\delta=0.05\), fund the present value of the perpetuity.

An annuity immediate has semiannual payments of \(800, 750, 700, \dots, 350\) at \(i^{(2)}=0.16\).

Find the present value of the annuity.

An annuity pays \(250\) at the beginning of every three months for \(12\) years.

Calculate the present value of the annuity if \(i^{(3)}=0.12\).