The word annuity comes from the Latin annuus meaning yearly, and it means yearly payments, but it can be either yearly payments one makes, as when one is saving for retirement, or yearly payments one receives, such as a retirement pension. Although retirement pensions are usually for life, we shall only consider payouts for a fixed period.

Accumulation annuities

If one saves $100 a year, with continuous compounding at 5%, how much will one have in 10 years? In order to be specific (and consistent with the text), we will assume that a deposit is made at the end of each year, so the first deposit has nine years' interest and the last deposit has no interest. Thus the total value is $100×(1.05)^9 +$100×(1.05)^8 +$100×(1.05)^7 + ... + $100×(1.05)^1 + $100. There is a formula:

1 + a + a^2 + a^3 + ... + a^(n-1) = (1 - a^n)/(1 - a)

which is readily verified by multiplying both sides by 1-a (most of the terms cancel out). Letting R denote the value of the annual payment and V the value of the accumulation at the end (r is the annual interest rate and n is the number of years),

V = R(1 - (1+r)^n)/(1 - (1+r)) = R((1+r)^n -1)/(r).

(The text presents a more general formula which allows for multiple payments per year).

Thus, in the case of $100 per year (at the end of the year) at 5% interest with continuous compounding, the value at the end of ten years will be $100(1.05^10 - 1)/(.05) = $1257.79; note that $1000 of this is the sum of the payments, and $257.79 is the interest.

Exercise: What will be the value of an annuity after 12 years if $250 is deposited each year and the annual interest rate is 8% (compounded annually)?
What will the value be if $250 is deposited each year for 24 years?
What will the value be if $500 is dposited each year for 12 years?

Payout annuities

The question is that if one has a lump sum of money, how much will a bank be willing to pay him each year for a fixed period of time. Although retirement annuities are usually until death, one might want to provide one's child with an income until age 21. Letting R again denote the regular payment (in this case, you receive it) and P denote the original principal, R can be calculated in a manner similar to the above. The present value of R t years in the future is R/(1+r)^t = P. Summing this over all the payments gives P = R(1 - (1/(1+r))^n)/(1 - (1/(1+r))) where P is the sum of the present values associated with each payment. This can be simplified to P = R(1+r)((1+r)^n - 1)/(r(1+r)^n) or R = P(r(1+r)^n)/((1+r)((1+r)^n - 1)). The formula in the text which differs by a factor of (1+r):

R = Pr(1+r)^n/((1+r)^n - 1)

is obtained by making payments at the end instead of the beginning of the year (so you get your first payment one year after you pay for the annuity).

How large a payment can you receive for 10 years if you pay $25000 for an annuity at 7% annual interest? R = $25000(.07)(1.07^10)/((1.07^10) - 1) = $3559.44.Note that this entails a total of $35594.40 in payments, of which $25000 is the price of the annuity and $10594.40 is interest.

Note that a payout annuity is what you are giving a bank when you take out a mortgage on your house (or a loan on your car, or ...). They give you a large sum of money, and you make regular payments for a fixed period (you generally make monthly payments, which is why that is the form of the formula in the text).

Mortgages are often presented to us as the the amount we borrow, the monthly payment, and the number of years we must make payments. The interest rate is also stated, but that merely tells us how the interest is calculated, not how much interest we pay. For example, if you borrow $10,000 for fifteen years with a monthly payment of $90, the interest rate is 7%. The total interest can be calculated by multiplying the size of the payment by the number of payments, and subtracting the principal: $90 × 12 × 15 - $10,000 = $16200 - $10000 = $6,200, or 62% of your principal. This is large because you are paying interest for several years, almost $700 the first year when you have repaid very little of the principal, but less than $70 the last year when you have already repaid most of the principal to the bank. Although calculations are difficult for most of the payments, it is easy to calculate that your first payment includes 7% simple interest for one month on $10,000, or $10,000 × .07/12 = $58.33 in interest (hence the principal is only reduced by $31.67). The last payment is just the future value of the remaining principal: $90 = (1+.07/12)P or the principal portion of the last payment is $89.48, hence the interest portion of the last payment is only $.52.

Competency: How much will you accumulate if you save$300 a year for 12 years at 10% interest?
How much will you accumulate if you save$300 a year for 24 years at 10% interest?
How much will you accumulate if you save $600 a year for 12 years at 10% interest?
How much do you need to save per year at 10% interest if you want to have $10,000 at the end of 12 years?

What will your annual payment be if you borrow $250,000 for 15 years at 6% annual interest?
What will your annual payment be if you borrow $250,000 for 30 years at 6% annual interest?
What will your annual payment be if you borrow $125,000 for 15 years at 6% annual interest?
How much can you borrow at 6% annual interest if you can afford an annual payment of $6000?

If you borrow $20.000 for five years at 13.5%, your monthly payment is $460.19
What is the total amount you must repay?
How much interest do you pay?
How much of your first payment is interest? principal?
How much of your last payment is interest? principal?


Challenge:If the interest rate is 6%, how much must you save per month for 35 years, in order to receive an annual payment of $10,000 per year for 25 years?

16 DE 2004

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