© 2000 John Petroff  

2)- Anuitas

Seperti rumus umum mengenai nilai yang dijelaskan di Bagian A, penaksiran dapat dilakukan dengan menjumlahkan rangkaian jumlah nilai masa depan. Jika semua nilai masa depan sama, maka mereka aakan dinamakan anuitas. Matematika Akturial memberikan hasil yang cepat dan mudah dalam mencari anuitas. Nilai masa depan dari anuitas dapat ditulis sebagai.

FVA = A(1+i) + A(1+i)2 + ... + A(1+i)n

or FVA = A [ (1+i) + (1+i)2 + ... + (1+i)n ]

where the sum of terms within the summation brackets can be simplified (see Appendix) and gives

FVA = A ((1+i)n - 1)/i

The present value of an annuity is written as

PVA = A/(1+i) + A/(1+i)2 + ... + A/(1+i)n

or PVA = A [ 1/(1+i) + 1/(1+i)2 + ... + 1/(1+i)n ]

where the sum of terms within the summation brackets can also be simplified (see Appendix) and gives

PVA = A/ (1 - (1+i)-n)/i

Actuarial tables give conveniently precalculated factors for both future value annuity factors

FVAF = ((1+i)n - 1)/i

and present value annuity factors

PVAF = (1 - (1+i)-n)/i

Lets verify the use of these formulas with simple examples. What amount will be available to a saver who puts $500 each year for 20 years into a savings account earning 8% per year annually? The answer is

FVA = 500 ((1 + 0.8)20 - 1) / .08 = 500 x 45.762 = $22,881 

The future value annuity factor for 8% and 20 years is 45.762, and gives naturally the same result

FVA = 500 x 45.762 = $22.881

A bond pays a coupon C which is an annuity, and a principal P at maturity. Its value BP is the sum of the present values of the two

BP = C ((1 - (1 + i)-n) / i) + P (1 + i)-n

 In 1995, a bondholder of ATT 6 3/4, 2005 considers selling her $10,000 bond. What should be the selling price if the current yield on comparable bonds is 7%, the coupon of $675 (or 10000 x 0.0675) is paid annually, the first coupon is paid exactly one year from now, and the 10th and last coupon exatly 10 years from now together with the principal? The bond selling price BP is

BP = 675((1 - (1 + 0.07)-10 ) / 0.07) + 10000 / (1 + 0.07)10

= 675 x 7.0236 + 10000 x 0.5083 = $9,823.93

Using the factors is fast and convenient when interest rates are round numbers, annuities are paid once a year and exactly one year from now. Otherwise additional steps are necessary. When the interest rate is not a round number, factors may be obtained from actuarial books available in some libraries. Interpolation of the factor using factors for interest rates above and below the given interest rate, can give an approximate result. But, for most analysts, a financial calculator becomes an investment with immediate very tangible returns and the best alternative. If the first annuity takes place less than one year from now, two steps are necessary: calculating present or future values of annuities, then add the accrued interest for the number of days since the last coupon date.

Intraannual compounding can also be used in the case of these formulas: as before, just divide the interest rate by the number of times m interest is compounded during the year, and multiply the number of years by m. The price of a bond BP is

BP = C ((1 - (1 + (i / m))-mn) / (i / m)) + P (1 + i / m)-mn

Most bonds pay coupons semi-annually, m equals 2, and bond price BP is

BP = C ((1 - (1 + (i / 2))-2n) / (i / 2)) + P (1 + i / 2)-2n

For an example, see the bond value calculation in Chapter 3 Section A-1.

When an annuity is assumed to go on forever, then the present value of the annuity can be simplified (see Appendix for proof of simplification)

PVA = A / i

 Take the example of a consol (which is bond issued in perpetuity, and which is rather rare, but still exists in the U.K.) with an annual coupon of $40 and a return on comparable issues of 2.5%, the value of such consol is

CV = 40 / 0.025 = $1,600

Finally, if the future amounts received each year are assumed to grow at a constant rate g, then the present value of that series of amounts is

PVA = A / (i - g)

While actuarial mathematics gives answers to many other problems of finance, the remainder of this book will use primarily the simplified formulas presented here. As already suggested in initial general remarks, there is so much approximation in estimates of future sales revenues, costs, as well as discount rate, that the imprecision of these estimates makes the precision available from actuarial mathematics almost redundant in most financial analysis. But this is true only at the analytical stage. Once each of the elements of a valuation (that is, the interest rate charged or earned, the dates and the amounts of every payment) is known or agreed upon, for instance in a loan contract, or in the selling of a previously issued note, then the two parties (i.e. borrower and lender, or buyer and seller) will determine the amount or amounts to be paid with great precision, down to a penny, using the actuarial formulas presented above.

See review questions Q-2C2.1 through Q-2C2.15.

See research assignments R-2C.1 and R-2C.2.

 Previous: 1-Discounting single amount

Last modified: Jun/01/01
 Next: Interest rate