Annuities are insurance products that allow you to save money for any reason tax deferred and create an income stream for life. Think of annuities like savings accounts but with insurance companies and not with banks. Below you will see the math involved with annuities so enjoy and contact us with any questions.
In finance theory, an annuity is a terminating “stream” of fixed payments, i.e., a collection of payments to be periodically received over a specified period of time. The valuation of such a stream of payments entails concepts such as the time value of money, interest rate, and future value.
Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time.
An annuity is a series of payments made at fixed intervals of time. If the number of payments is known in advance, the annuity is an annuity-certain. If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid.
The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future. The present value is given in actuarial notation by:
The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by:
Future and present values are related as:
An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.
|Because each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated through the formula:|
Future and present values for annuities due are related as:
An annuity-due with n payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity with one payment more, minus the last payment. Thus we have:
- (value at the time of the first of n payments of 1)
- (value one period after the time of the last of n payments of 1)
A perpetuity is an annuity for which the payments continue forever. Since:
Proof of Annuity Formula
To calculate present value, the k-th payment must be discounted to the present by dividing by the interest, compounded by k terms. Hence the contribution of the k-th payment R would be R/(1+i)^k. Just considering R to be one, then:
If an annuity is for repaying a debt P with interest, the amount owed after n payments is:
Also, this can be thought of as the present value of the remaining payments:
Formula for Finding the Periodic payment(R), Given A:
R = A/(1+〖(1-(1+(j/m) )〗^(-(n-1))/(j/m))
- Find the periodic payment of an annuity due of $70000, payable annually for 3 years at 15% compounded annually.
- R = 70000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))
- R = 70000/2.625708885
- R = $26659.46724
- Find the periodic payment of an annuity due of $250700, payable quarterly for 8 years at 5% compounded quarterly.
- R= 250700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
- R = 250700/26.5692901
- R = $9435.71
Finding the Periodic Payment(R), Given S:
R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1)
- Find the periodic payment of an accumulated value of $55000, payable monthly for 3 years at 15% compounded monthly.
- R=55000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
- R = 55000/45.67944932
- R = $1204.04
- Find the periodic payment of an accumulated value of $1600000, payable annually for 3 years at 9% compounded annually.
- R=1600000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
- R = 1600000/3.573129
- R = $447786.80
- Fixed annuities – These are annuities with fixed payments. They are primarily used for low risk investments like government securities or corporate bonds. Fixed annuities offer a fixed rate but are not regulated by the Securities and Exchange Commission. This type can be adversely affected by high inflation.
- Variable annuities – Unlike fixed annuities, these are regulated by the SEC. They allow you to invest in portions of money markets.
- Equity-indexed annuities – Lump sum payments are made to an insurance company. Can be implemented with a Call option.