Yang–Baxter equation

An annuity is a series of payments made at fixed intervals of time.

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.

Valuation

The valuation of an annuity entails concepts such as me time value of money, interest rate, and future value.

Annuity-immediate

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.

	↓	↓	...	↓	payments
———	———	———	———	—
0	1	2	...	n	periods

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:

$a_{\bar{n}|i}=\frac{1-{(1+i)}^{-n}}{i},$

where $n$ is the number of terms and $i$ is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent $R$ is:

$PV(i,n,R)=R\times a_{\bar{n}|i}$

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest $I$ is stated as a nominal interest rate, and $i=I/12$.

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:

$s_{\bar{n}|i}=\frac{(1+i{)}^n-1}{i}$

where $n$ is the number of terms and $i$ is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent $R$ is:

$FV(i,n,R)=R\times s_{\bar{n}|i}$

Example: The present value of a 5 year annuity with nominal annual interest rate 12% and monthly payments of $100 is:

$PV(0.12/12,5\times 12,\$100)=\$100\times a_{\overline{60}|0.01}=\$4,495.50$

The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.

Future and present values are related as:

$s_{\bar{n}|i}=(1+i{)}^n\times a_{\bar{n}|i}$

and

$\frac{1}{a_{\bar{n}|i}}-\frac{1}{s_{\bar{n}|i}}=i$

Annuity-due

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.

↓	↓	...	↓		payments
———	———	———	———	—
0	1	...	n-1	n	periods

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:

${\ddot{a}}_{\overline{n|}i}=(1+i)\times a_{\overline{n|}i}=\frac{1-{(1+i)}^{-n}}{d}$

and

${\ddot{s}}_{\overline{n|}i}=(1+i)\times s_{\overline{n|}i}=\frac{(1+i{)}^n-1}{d}$

where $n$ are the number of terms, $i$ is the per term interest rate, and $d$ is the effective rate of discount given by $d=i/(i+1)$.

Future and present values for annuities due are related as:

${\ddot{s}}_{\bar{n}|i}=(1+i{)}^n\times {\ddot{a}}_{\bar{n}|i}$

and

$\frac{1}{{\ddot{a}}_{\bar{n}|i}}-\frac{1}{{\ddot{s}}_{\bar{n}|i}}=d$

Example: The final value of a 7 year annuity-due with nominal annual interest rate 9% and monthly payments of $100:

$F{V}_{due}(0.09/12,7\times 12,\$100)=\$100\times {\ddot{a}}_{\overline{84}|0.0075}=\$11,730.01.$

Note that in Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.

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. Thus we have:

${\ddot{a}}_{\overline{n|}i}=a_{\bar{n}|i}(1+i)=a_{\overline{n-1|}i}+1$ (value at the time of the first of n payments of 1)

${\ddot{s}}_{\overline{n|}i}=s_{\bar{n}|i}(1+i)=s_{\overline{n+1|}i}-1$ (value one period after the time of the last of n payments of 1)

Perpetuity

A perpetuity is an annuity for which the payments continue forever. Since:

$\underset{n\to \mathrm{\infty }}{\lim }PV(i,n,R)=\frac{R}{i}$

even a perpetuity has a finite present value when there is a non-zero discount rate. The formula for a perpetuity are:

$a_{\bar{\mathrm{\infty }}|i}=1/i;{\ddot{a}}_{\bar{\mathrm{\infty }}|i}=1/d.$

where $i$ is the interest rate and $d=i/(1+i)$ is the effective discount rate.

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:

$a_{\bar{n}|i}=\sum _{k=1}^n\frac{1}{(1+i{)}^k}=(\frac{1}{1+i}-\frac{1}{(1+i{)}^{n+1}})\sum _{k=0}^{\mathrm{\infty }}\frac{1}{(1+i{)}^k}$

We notice that the second factor is an infinite geometric progression of the form,

$\sum _{k=0}^{\mathrm{\infty }}{\kappa }^k=\frac{1}{1-\kappa }$

therefore,

$a_{\bar{n}|i}=\left(\frac{(1+i{)}^n-1}{(1+i{)}^{n+1}}\right)\left(\frac{1}{1-1/(1+i)}\right)=\left(\frac{1-(1+i{)}^{-n}}{1+i}\right)\left(\frac{1+i}{i}\right)=\frac{1-(1+i{)}^{-n}}{i}.$

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (n−1) years. Therefore,

$s_{\bar{n}|i}=1+(1+i)+(1+i{)}^2+\cdots +(1+i{)}^{n-1}=(1+i{)}^n{a}_{\bar{n}|i}=\frac{(1+i{)}^n-1}{i}$

Amortization calculations

If an annuity is for repaying a debt P with interest, the amount owed after n payments is:

$\frac{R}{i}-{(1+i)}^n(\frac{R}{i}-P)$

because the scheme is equivalent with borrowing the amount $R/i$ to create a perpetuity with coupon $R$, and putting $R/i-P$ of that borrowed amount in the bank to grow with interest $i$.

Also, this can be thought of as the present value of the remaining payments:

$R[\frac{1}{i}-\frac{(i+1{)}^{n-N}}{i}]=R\times a_{\overline{N-n}|i}$

See also fixed rate mortgage.

Example calculations

Formula for Finding the Periodic payment(R), Given A:

R = A/(1+〖(1-(1+(j/m) )〗^(-(n-1))/(j/m))

Examples:

1. 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
2. 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)

Examples:

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
2. 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

Other types

• Fixed annuities – These are annuities with fixed payments. The insurance company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission.

• Variable annuities – Registered products that are regulated by the SEC in the United States of America. 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.

• 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.

See also

• Amortization calculator
• Fixed rate mortgage
• Life annuity
• Perpetuity
• Time value of money

References

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