An annuity

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INTRODUCTION

An annuity is a sequence of equal payments made at regular intervals of time. For example, as the proud parent of a newborn daughter, you decide to save for her college education by depositing 200 at the end of each month into a savings account paying 2.7% interest compounded monthly. Eighteen years from now, after you make the last of 216 payments, the account will contain 55,547.79.

Other Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments.

Annuity certain is the number of payments that is known in advance.

A perpetuity annuityis an annuity that has no end, or a stream of cash payments that continues forever.

An annuity due is an annuity whose payments are made at the beginning of each period.

Annuity immediate is the annuity that payments are made at the end of the time periods so that interest is accumulated before the payment.

MAIN BODY

The following are the types of annuities;

Annuities can be classified by the frequency of payment dates. The payments or deposits may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.Annuities may be classified in several ways as follows;

Timing of payments, Payments of an annuity immediate are made at the end of payment periods so that interest grows between the issue of the annuity and the first payment. Payments of an annuity due are made at the beginning of payment periods so a payment is made immediately on issueter.

Contingency of payments, annuities that provide payments that will be paid over a period known in advance are annuities certain or guaranteed annuities. Annuities paid only under certain circumstances are contingent annuities. Certain and life annuities are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.

 

Variability of payments

ü  Fixed annuities, these are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission.

ü  Variable annuities, they allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits.

ü  Equity indexed annuities, Annuities with payments linked to an index. Typically, the minimum payment will be 0% and the maximum will be predetermined. The performance of an index determines whether the minimum, the maximum or something in between is credited to the customer.

Deferral of payments, an annuity which begins payments only after a period is a deferred annuity. An annuity which begins payments without a deferral period is an immediate annuity.

Valuation of an annuity

Valuation of an annuity entails calculation of the present value of the future annuity payments. The valuation of an annuity entails concepts such as time value of money, interest rate, and future value.

Present value (PV)is the value of an expected income stream determined as of the date of valuation.

The standard formula is:PV = FV            

                                        (1+r) n

Whereby,

Often, (1+r) n is referred to as the Present Value Factor

FVis the future amount of money that must be discounted.

nis the number of compounding periods between the present date and the date where the sum is worth FV.

ris the interest rate for one compounding period

 

 

For example

If you are to receive Tsh. 1,000/= in five years, and the effective annual interest rate during this period is 10%, then what is the present value of this amount?

Solutions:

Given that:

FV= 1,000

n = 5 years

r= 10% = 0.1

PV = FV / (1+r) n

= 1000 / (1+0.1)5

=1000/ (1.1)5

=1000/1.61051

PV =620.92

 

An ordinary annuity is a series of equal payments, with all payments being made at the end of each successive period. The present value calculation for an ordinary annuity is used to determine the total cost of an annuity if it were to be paid right now.

The formula for calculating the present value of an ordinary annuity is:

 

P = PMT [(1 - (1 / (1 + r) n)) / r]

Where:

P = the present value of the annuity stream to be paid in the future

PMT = the amount of each annuity payment

r = the interest rate

n = the number of periods over which payments are to be made

For example, ABC International has committed to a legal settlement that requires it to pay Tsh.50, 000 per year at the end of each of the next ten years. What would it cost ABC if it were to instead settle the claim immediately with a single payment, assuming an interest rate of 5%? The calculation is:

Solution

P = 50, 000 [(1 - (1/ (1+.0.05)10))/.0.05]

   =50,000[(1 - (1/ (1.05)10))/.0.05]

   = 50,000[(1 - (1/ (1.628894627)/.0.05]

  = 50,000[(1 –0.613913253/0.05]

  = 50,000[0.386086747/0.05]

  = 50,000[7.72173494]

P = Tsh.386, 087

The present value of an annuity due is used to derive the current value of a series of cash payments that are expected to be made on predetermined future dates and in predetermined amounts.

The formula for calculating the present value of an annuity due (where payments occur at the beginning of a period) is:

P = (PMT [(1 - (1 / (1 + r) n)) / r]) x (1+r)

Where:

P = the present value of the annuity stream to be paid in the future

PMT = the amount of each annuity payment

r = the interest rate

n = the number of periods over which payments are made.

For example, PES International is paying a third party Tsh.100, 000 at the beginning of each year for the next eight years in exchange for the rights to a key patent.  What would it cost PES if it were to pay the entire amount immediately, assuming an interest rate of 5%? The calculation is:

P = (Tsh.100, 000 [(1 - (1 / (1 + 0.05)8)) / 0.05]) x (1+0.05)

   = (Tsh.100, 000 [(1 - (1 / (1.05)8)) / 0.05]) x (1+0.05)

   = (Tsh.100, 000 [(1 - (1 /1.477455444) / 0.05]) x (1+0.05)

   = (Tsh.100, 000 [(1 - (0.676839361) / 0.05]) x (1.05)

   = (Tsh.100, 000 [0.323160639 / 0.05]) x (1.05)

   = (646,321.278) x (1.05)

P = Tsh.678, 637

Future value (FV) is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is worth at a specified time in the future assuming a certain interest rate, more generally and rate of return.

                                i.            To determine future value (FV) using simple interest

FV=PV (1+rt)

                              ii.            To determine future value (FV) using compound interest

FV= PV (1+i) n

The future value of annuity due can be determined by growing the future value of an ordinary annuity for one additional period:

FV of Annuity Due = R × (1 + i) n − 1× (1 + i)

i

In the above formulas,

   i is the periodic interest rate which equals annual percentage rate divided by periods per year;

n are the number of compounding periods.

R is the fixed periodic payment.

Example

Mr. A deposited Tsh. 700/= at the end of each month of calendar year 2020 in an investment account of 9% annual interest rate. Calculate the future value of the annuity on Dec 31, 2020. Compounding is done on monthly basis.

Solution

We have,

Periodic Payment       R = 700

Number of Periods      n = 12

Interest Rate          i = 9%/12 = 0.75%

Future Value          FV = 700 × {(1+0.75%)^12-1}/0.75%

= 700 × {1.0075^12-1}/0.0075

  = 700 × (1.0938069-1)/0.0075

= 700 × 0.0938069/0.0075

= 700 × 12.5076

FV = 8,755

Therefore, the formula for the future value of an ordinary annuity refers to the value on a specific future date of a series of periodic payments, where each payment is made at the end of a period.

The formula for calculating the future value of an ordinary annuity is:

P = PMT [((1 + r) n - 1) / r]

Where:

P = the future value of the annuity stream to be paid in the future

PMT = the amount of each annuity payment

r = the interest rate

n = the number of periods over which payments are made.

For example, the treasurer of AC International expects to invest Tsh.100,000 of the firm's funds in a long-term investment vehicle at the end of each year for the next five years. He expects that the company will earn 7% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as:

P = 100, 000 [((1 + 0.07)5 - 1) / 0.07]

= 100, 000 [(1.402551731- 1) / 0.07]

= 100, 000 [0.40255173/ 0.07]

= 100, 000 [5.750739]

P = Tsh.575, 074

 

CONCLUSION

Annuities are one of several tools used in estate and financial planning. Like all tools used they have advantages and disadvantages. Like Annuities are a saving tool, they have better rates of return but the surrender period and charges are greater. Unlike a certificate of deposit during the accumulation phase the taxation of growth inside the annuity is deferred.

 

REFERENCE

1.      Education 2020 Homeschool Console, class "Economic Math"; definition of FUTURE VALUE: "Future value is the value of an asset at a specific date."

2.      Samuel A. Broverman (2010). Mathematics of Investment and Credit, 5th Edition. ACTEX Academic Series. ACTEX Publications. ISBN 978-1-56698-767-7.

3.      Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000). Fundamentals of corporate finance. Boston: Irwin/McGraw-Hill. p. 175. ISBN 0-07-231289-0.

 

 

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