KNOWLEDGE FOR ALL: Annuity (finance theory)

Annuity (finance theory)

From Wikipedia, the free encyclopedia

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.

Contents

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5 Proof of annuity formula

6 Amortization calculations

7 Example calculations

8 Other types

9 See also

10 References

Valuation[edit]

The valuation of an annuity entails concepts such as me time value of money, interest rate, and future value.^[1]

Annuity-immediate[edit]

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:

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

where

is the number of terms and

is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent

is:

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

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest

is stated as a nominal interest rate, and

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:

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

where

is the number of terms and

is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent

is:

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

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:

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

and

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

Annuity-due[edit]

An annuity-due is an annuity whose payments are made at the beginning of each period.^[2] 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:

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

and

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

where

are the number of terms,

is the per term interest rate, and

is the effective rate of discount given by

Future and present values for annuities due are related as:

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

and

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

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

$FV_{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:

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

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

Perpetuity[edit]

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

$\lim_{n\,\rightarrow\,\infty}\,PV(i,n,R)\,=\,\frac{R}{i}$

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

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

where

is the interest rate and

is the effective discount rate.

Proof of annuity formula[edit]

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:

$a_{\overline n|i} = \sum_{k=1}^n \frac{1}{(1+i)^k} = \left( \frac{1}{1+i} - \frac{1}{(1+i)^{n+1}}\right) \sum_{k=0}^\infty \frac{1}{(1+i)^k}$

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

$\sum_{k=0}^\infty \kappa^k = \frac{1}{1-\kappa}$

therefore,

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

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,