268:, or stock price appreciation). The interest rate is the change, expressed as a percentage, in the amount of money during one compounding period. A compounding period is the length of time that must transpire before interest is credited, or added to the total. For example, interest that is compounded annually is credited once a year, and the compounding period is one year. Interest that is compounded quarterly is credited four times a year, and the compounding period is three months. A compounding period can be any length of time, but some common periods are annually, semiannually, quarterly, monthly, daily, and even continuously.
149:, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to
153:. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds (the present value) is less than the total amount of money paid to the lender.
256:, and can be thought of as rent that is required of a borrower in order to use money from a lender. For example, when an individual takes out a bank loan, the individual is charged interest. Alternatively, when an individual deposits money into a bank, the money earns interest. In this case, the bank is the borrower of the funds and is responsible for crediting interest to the account holder. Similarly, when an individual invests in a company (through
25:
2845:
initial outlay, as well as some surplus (for example, interest, or future cash flows). An investor can decide which project to invest in by calculating each projects’ present value (using the same interest rate for each calculation) and then comparing them. The project with the smallest present value – the least initial outlay – will be chosen because it offers the same return as the other projects for the least amount of money.
184:, and more. These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times, since time and dates must be consistent in order to make comparisons between values. When deciding between projects in which to invest, the choice can be made by comparing respective present values of such projects by means of discounting the expected income streams at the corresponding project interest rate, or
2854:
annum, a deal might be struck at "20 years' purchase", which would value the lease at 20 * $ 10,000, i.e. $ 200,000. This equates to a present value discounted in perpetuity at 5%. For a riskier investment the purchaser would demand to pay a lower number of years' purchase. This was the method used for example by the
English crown in setting re-sale prices for manors seized at the
201:. Time preference can be measured by auctioning off a risk free security—like a US Treasury bill. If a $ 100 note with a zero coupon, payable in one year, sells for $ 80 now, then $ 80 is the present value of the note that will be worth $ 100 a year from now. This is because money can be put in a bank account or any other (safe) investment that will return interest in the future.
2791:
of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate. The full
Laplace transform is the curve of all present values, plotted as a function of interest rate. For discrete time, where payments are separated by large time periods, the transform reduces
1231:
The interest rates per period might not be the same. The cash flow must be discounted using the interest rate for the appropriate period: if the interest rate changes, the sum must be discounted to the period where the change occurs using the second interest rate, then discounted back to the present
2844:
An investor, the lender of money, must decide the financial project in which to invest their money, and present value offers one method of deciding. A financial project requires an initial outlay of money, such as the price of stock or the price of a corporate bond. The project claims to return the
2143:
Where, as above, C is annuity payment, PV is principal, n is number of payments, starting at end of first period, and i is interest rate per period. Equivalently C is the periodic loan repayment for a loan of PV extending over n periods at interest rate, i. The formula is valid (for positive n, i)
1616:
Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules; payments of the same amount at regular time intervals. Such an arrangement
1431:
2739:
The purchase price is equal to the bond's face value if the coupon rate is equal to the current interest rate of the market, and in this case, the bond is said to be sold 'at par'. If the coupon rate is less than the market interest rate, the purchase price will be less than the bond's face value,
761:
A cash flow is an amount of money that is either paid out or received, differentiated by a negative or positive sign, at the end of a period. Conventionally, cash flows that are received are denoted with a positive sign (total cash has increased) and cash flows that are paid out are denoted with a
335:
Spreadsheets commonly offer functions to compute present value. In
Microsoft Excel, there are present value functions for single payments - "=NPV(...)", and series of equal, periodic payments - "=PV(...)". Programs will calculate present value flexibly for any cash flow and interest rate, or for a
331:
The operation of evaluating a present sum of money some time in the future is called a capitalization (how much will 100 today be worth in five years?). The reverse operation—evaluating the present value of a future amount of money—is called discounting (how much will 100 received in five years be
243:
It follows that if one has to choose between receiving $ 100 today and $ 100 in one year, the rational decision is to choose the $ 100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least $ 105 in one year so
2853:
The traditional method of valuing future income streams as a present capital sum is to multiply the average expected annual cash-flow by a multiple, known as "years' purchase". For example, in selling to a third party a property leased to a tenant under a 99-year lease at a rent of $ 10,000 per
1437:
The interest rate must necessarily coincide with the payment period. If not, either the payment period or the interest rate must be modified. For example, if the interest rate given is the effective annual interest rate, but cash flows are received (and/or paid) quarterly, the interest rate per
2804:
There are mainly two flavors of
Present Value. Whenever there will be uncertainties in both timing and amount of the cash flows, the expected present value approach will often be the appropriate technique. With Present Value under uncertainty, future dividends are replaced by their conditional
2067:
The above formula (1) for annuity immediate calculations offers little insight for the average user and requires the use of some form of computing machinery. There is an approximation which is less intimidating, easier to compute and offers some insight for the non-specialist. It is given by
2179:
The formula can, under some circumstances, reduce the calculation to one of mental arithmetic alone. For example, what are the (approximate) loan repayments for a loan of PV = $ 10,000 repaid annually for n = ten years at 15% interest (i = 0.15)? The applicable approximate formula is C ≈
2376:
Again there is a distinction between a perpetuity immediate – when payments received at the end of the period – and a perpetuity due – payment received at the beginning of a period. And similarly to annuity calculations, a perpetuity due and a perpetuity immediate differ by a factor of
1219:
2831:
if there are no risks involved in the project. The rate of return from the project must equal or exceed this rate of return or it would be better to invest the capital in these risk free assets. If there are risks involved in an investment this can be reflected through the use of a
244:
that the two options are equivalent (either receiving $ 100 today or receiving $ 105 in one year). This is because if $ 100 is deposited in a savings account, the value will be $ 105 after one year, again assuming no risk of losing the initial amount through bank default.
2740:
and the bond is said to have been sold 'at a discount', or below par. Finally, if the coupon rate is greater than the market interest rate, the purchase price will be greater than the bond's face value, and the bond is said to have been sold 'at a premium', or above par.
793:
For example, if a stream of cash flows consists of +$ 100 at the end of period one, -$ 50 at the end of period two, and +$ 35 at the end of period three, and the interest rate per compounding period is 5% (0.05) then the present value of these three Cash Flows are:
239:
is called a capitalization (how much will $ 100 today be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of money—is called a discounting (how much will $ 100 received in 5 years—at a lottery for example—be worth today?).
2836:. The risk premium required can be found by comparing the project with the rate of return required from other projects with similar risks. Thus it is possible for investors to take account of any uncertainty involved in various investments.
219:
which corresponds to the minimum guaranteed rate provided by a bank's saving account for example, assuming no risk of default by the bank to return the money to the account holder on time. To compare the change in purchasing power, the
2364:
2473:
1239:
1232:
using the first interest rate. For example, if the cash flow for period one is $ 100, and $ 200 for period two, and the interest rate for the first period is 5%, and 10% for the second, then the net present value would be:
1825:
1976:
2136:
1034:
2537:, and maturity date which in turn yields the number of periods until the debt matures and must be repaid. A bondholder will receive coupon payments semiannually (unless otherwise specified) in the amount of
948:
868:
664:
1022:
196:
If offered a choice between $ 100 today or $ 100 in one year, and there is a positive real interest rate throughout the year, a rational person will choose $ 100 today. This is described by economists as
1537:
2812:– in this approach a single set of estimated cash flows and a single interest rate (commensurate with the risk, typically a weighted average of cost components) will be used to estimate the fair value.
204:
An investor who has some money has two options: to spend it right now or to save it. But the financial compensation for saving it (and not spending it) is that the money value will accrue through the
2183:
The overall approximation is accurate to within ±6% (for all n≥1) for interest rates 0≤i≤0.20 and within ±10% for interest rates 0.20≤i≤0.40. It is, however, intended only for "rough" calculations.
500:
is the interest rate for one compounding period (the end of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily). The interest rate,
790:, of a stream of cash flows consists of discounting each cash flow to the present, using the present value factor and the appropriate number of compounding periods, and combining these values.
2251:
2195:
refers to periodic payments, receivable indefinitely, although few such instruments exist. The present value of a perpetuity can be calculated by taking the limit of the above formula as
407:
1601:
579:
591:
For example, if you are to receive $ 1000 in five years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is
211:
Therefore, to evaluate the real value of an amount of money today after a given period of time, economic agents compound the amount of money at a given (interest) rate. Most
2734:
2174:
2687:
1724:
788:
2593:
2407:
2256:
Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly by summing the present value of the payments
2056:
2030:
2004:
1759:
1696:
1674:
1652:
1567:
1458:
740:
718:
696:
520:
498:
476:
454:
432:
2262:
669:
The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $ 1000 in five years, or $ 620.92 today.
2558:
2818:– in this approach multiple cash flows scenarios with different/expected probabilities and a credit-adjusted risk free rate are used to estimate the fair value.
2535:
2515:
141:, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has
1426:{\displaystyle NPV=100\,(1.05)^{-1}+200\,(1.10)^{-1}\,(1.05)^{-1}={\frac {100}{(1.05)^{1}}}+{\frac {200}{(1.10)^{1}(1.05)^{1}}}=\$ 95.24+\$ 173.16=\$ 268.40}
762:
negative sign (total cash has decreased). The cash flow for a period represents the net change in money of that period. Calculating the net present value,
2415:
1767:
1836:
1214:{\displaystyle NPV=PV_{1}+PV_{2}+PV_{3}={\frac {100}{(1.05)^{1}}}+{\frac {-50}{(1.05)^{2}}}+{\frac {35}{(1.05)^{3}}}=95.24-45.35+30.23=80.12,}
2076:
2180:
10,000*(1/10 + (2/3) 0.15) = 10,000*(0.1+0.1) = 10,000*0.2 = $ 2000 pa by mental arithmetic alone. The true answer is $ 1993, very close.
874:
800:
597:
954:
1729:
An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ by a factor of
1467:
1228:
The periods might not be consecutive. If this is the case, the exponents will change to reflect the appropriate number of periods
252:
Interest is the additional amount of money gained between the beginning and the end of a time period. Interest represents the
2560:, until the bond matures, at which point the bondholder will receive the final coupon payment and the face value of a bond,
89:
3035:
Youings, Joyce, "Devon
Monastic Lands: Calendar of Particulars for Grants 1536–1558", Devon & Cornwall Record Society,
61:
3075:
2205:
3062:
3007:
2971:
2927:
108:
68:
3067:
309:
354:
2855:
75:
46:
42:
3107:
2793:
264:), the company is borrowing funds, and must pay interest to the individual (in the form of coupon payments,
2760:
for further discussion. These calculations must be applied carefully, as there are underlying assumptions:
57:
2787:(In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the
1572:
528:
2486:
208:
that he or she will receive from a borrower (the bank account in which he has the money deposited).
169:
2691:
2147:
1632:
There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate,
456:
is the number of compounding periods between the present date and the date where the sum is worth
188:. The project with the highest present value, i.e. that is most valuable today, should be chosen.
2828:
297:
291:
216:
35:
3022:
Swingler, D. N., (2014), "A Rule of Thumb approximation for time value of money calculations",
1569:
is the nominal annual interest rate, compounded quarterly, and the interest rate per quarter is
2769:
2497:, an interest earning debt security, to an investor to raise funds. The bond has a face value,
2604:
1830:
The present value of an annuity immediate is the value at time 0 of the stream of cash flows:
2780:
1701:
765:
585:
225:
2563:
2359:{\displaystyle PV=\sum _{k=1}^{\infty }{\frac {C}{(1+i)^{k}}}={\frac {C}{i}},\qquad i>0,}
2768:, or alternatively, that the cost of inflation is incorporated into the interest rate; see
2380:
2039:
2013:
1987:
1732:
1679:
1657:
1635:
1545:
1441:
723:
701:
679:
503:
481:
459:
437:
415:
82:
2598:
The present value of a bond is the purchase price. The purchase price can be computed as:
8:
2757:
1438:
quarter must be computed. This can be done by converting effective annual interest rate,
253:
146:
2540:
3057:
3053:
2520:
2500:
319:
313:
221:
3081:
3071:
3003:
2967:
2923:
2887:
2867:
2788:
345:
279:
212:
205:
2775:
That the likelihood of receiving the payments is high — or, alternatively, that the
2468:{\displaystyle PV_{\text{perpetuity due}}=PV_{\text{perpetuity immediate}}(1+i)\,\!}
1698:
payments are received (or paid) at the beginning of each period, at times 0 through
2749:
2370:
1726:. This subtle difference must be accounted for when calculating the present value.
1626:
753:), then it allows the present values in the discount future to have higher values.
673:
3102:
720:
years into the future, can be computed with the same formula, where in this case
303:
285:
198:
2877:
2494:
743:
257:
185:
181:
1820:{\displaystyle PV_{\text{annuity due}}=PV_{\text{annuity immediate}}(1+i)\,\!}
1654:
payments are received (or paid) at the end of each period, at times 1 through
3096:
3085:
2997:
2872:
2833:
2776:
1611:
236:
177:
173:
157:
2792:
to a sum, but when payments are ongoing on an almost continual basis, the
1971:{\displaystyle PV=\sum _{k=1}^{n}{\frac {C}{(1+i)^{k}}}=C\left,\qquad (1)}
2882:
294:, the effective equivalent compared to multiple compound interest periods
522:, is given as a percentage, but expressed as a decimal in this formula.
2922:(12 ed.). Winsted: South-Western Publishing Co. pp. 147–498.
2192:
2858:
in the early 16th century. The standard usage was 20 years' purchase.
2131:{\displaystyle C\approx PV\left({\frac {1}{n}}+{\frac {2}{3}}i\right)}
2765:
2753:
2062:
1622:
229:
165:
122:
24:
2998:
Ross, Stephen; Randolph W. Westerfield; Bradford D. Jordan (2010).
272:
265:
142:
2917:
306:, an inverse interest rate when performing calculations in reverse
1618:
150:
126:
282:, interest that increases exponentially over subsequent periods,
943:{\displaystyle PV_{2}={\frac {-\$ 50}{(1.05)^{2}}}=-\$ 45.35\,}
863:{\displaystyle PV_{1}={\frac {\$ 100}{(1.05)^{1}}}=\$ 95.24\,}
756:
659:{\displaystyle PV={\frac {\$ 1000}{(1+0.10)^{5}}}=\$ 620.92\,}
300:, the simple annual interest rate of multiple interest periods
1621:. The expressions for the present value of such payments are
1017:{\displaystyle PV_{3}={\frac {\$ 35}{(1.05)^{3}}}=\$ 30.23\,}
261:
161:
1532:{\displaystyle (1+i)=\left(1+{\frac {i^{4}}{4}}\right)^{4}}
2144:
for ni≤3. For completeness, for ni≥3 the approximation is
344:
The most commonly applied model of present valuation uses
336:
schedule of different interest rates at different times.
2918:
Moyer, Charles; William
Kretlow; James McGuigan (2011).
145:-earning potential, a characteristic referred to as the
3002:(9 ed.). New York: McGraw-Hill. pp. 145–287.
1460:, to nominal annual interest rate compounded quarterly:
434:
is the future amount of money that must be discounted,
2694:
2607:
2566:
2543:
2523:
2503:
2418:
2383:
2265:
2208:
2150:
2079:
2058:= effective periodic interest rate or rate of return.
2042:
2016:
1990:
1839:
1770:
1735:
1704:
1682:
1660:
1638:
1575:
1548:
1470:
1444:
1242:
1037:
957:
877:
803:
768:
726:
704:
682:
600:
531:
506:
484:
462:
440:
418:
357:
235:
The operation of evaluating a present value into the
2839:
2063:
An approximation for annuity and loan calculations
49:. Unsourced material may be challenged and removed.
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2681:
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2509:
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2401:
2358:
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2050:
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401:
271:There are several types and terms associated with
2464:
1816:
3094:
2246:{\displaystyle PV\,=\,{\frac {C}{i}}.\qquad (2)}
2186:
316:of an interest rate with a period of zero time.
2764:That it is not necessary to account for price
1605:
339:
2966:. Winsted: ACTEX Publishers. pp. 4–229.
2779:is incorporated into the interest rate; see
2756:is the sum of each one's present value. See
581:is referred to as the Present Value Factor
2957:
2955:
2953:
2951:
2949:
2947:
2945:
2943:
2941:
2939:
1224:There are a few considerations to be made.
757:Net present value of a stream of cash flows
402:{\displaystyle PV={\frac {C}{(1+i)^{n}}}\,}
2822:
288:, additive interest that does not increase
156:Present value calculations, and similarly
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2961:
2463:
2219:
2215:
2047:
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2017:
1995:
1991:
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109:Learn how and when to remove this message
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2487:Bond valuation#Present value approach
1028:Thus the net present value would be:
2964:Mathematics of Investment and Credit
2900:
2743:
749:If we are using lower discount rate(
47:adding citations to reliable sources
18:
2980:
2794:mathematics of continuous functions
2752:. The present value of a bundle of
13:
3046:
2848:
2810:Traditional Present Value Approach
2796:can be used as an approximation.)
2291:
1596:{\displaystyle {\frac {i^{4}}{4}}}
1417:
1408:
1399:
1007:
977:
933:
900:
853:
823:
649:
613:
574:{\displaystyle v^{n}=\,(1+i)^{-n}}
14:
3119:
3063:Concise Encyclopedia of Economics
3000:Fundamentals of Corporate Finance
2920:Contemporary Financial Management
2840:Present value method of valuation
247:
3068:Library of Economics and Liberty
310:Continuously compounded interest
160:calculations, are used to value
23:
2816:Expected Present Value Approach
2478:
2343:
2233:
1958:
322:, which accounts for inflation.
34:needs additional citations for
3066:(2nd ed.). Indianapolis:
3029:
3016:
2856:Dissolution of the Monasteries
2827:The interest rate used is the
2714:
2701:
2662:
2649:
2582:
2570:
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2448:
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2384:
2315:
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1965:
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985:
915:
908:
838:
831:
676:in today's money of an amount
634:
621:
559:
546:
386:
373:
326:
1:
2893:
2187:Present value of a perpetuity
191:
3026:, Vol. 13, Issue 2, pp.57-61
2781:Corporate bond#Risk analysis
2729:{\displaystyle +F(1+i)^{-n}}
2169:{\displaystyle C\approx PVi}
1676:, while for an annuity due,
586:formula for the future value
584:This is also found from the
7:
3024:Journal of Personal Finance
2861:
1606:Present value of an annuity
348:. The standard formula is:
340:Present value of a lump sum
10:
3124:
2962:Broverman, Samuel (2010).
1609:
2682:{\displaystyle PV=\left}
1612:Annuity § Valuation
139:present discounted value
2829:risk-free interest rate
2823:Choice of interest rate
2493:A corporation issues a
2032:= amount of cash flows,
1719:{\displaystyle \,n-1\,}
783:{\displaystyle \,NPV\,}
298:Nominal annual interest
292:Effective interest rate
217:risk-free interest rate
2770:Inflation-indexed bond
2730:
2683:
2642:
2589:
2588:{\displaystyle F(1+r)}
2554:
2531:
2511:
2469:
2403:
2360:
2295:
2247:
2170:
2132:
2052:
2026:
2000:
1972:
1869:
1821:
1755:
1720:
1692:
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864:
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575:
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494:
472:
450:
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403:
232:rate) should be used.
2731:
2684:
2622:
2590:
2555:
2532:
2512:
2470:
2404:
2402:{\displaystyle (1+i)}
2361:
2275:
2248:
2199:approaches infinity.
2171:
2133:
2053:
2051:{\displaystyle \,i\,}
2027:
2025:{\displaystyle \,C\,}
2001:
1999:{\displaystyle \,n\,}
1973:
1849:
1822:
1756:
1754:{\displaystyle (1+i)}
1721:
1693:
1691:{\displaystyle \,n\,}
1671:
1669:{\displaystyle \,n\,}
1649:
1647:{\displaystyle \,n\,}
1598:
1564:
1562:{\displaystyle i^{4}}
1534:
1455:
1453:{\displaystyle \,i\,}
1428:
1216:
1019:
945:
865:
785:
742:is an assumed future
737:
735:{\displaystyle \,i\,}
715:
713:{\displaystyle \,n\,}
693:
691:{\displaystyle \,C\,}
661:
576:
517:
515:{\displaystyle \,i\,}
495:
493:{\displaystyle \,i\,}
473:
471:{\displaystyle \,C\,}
451:
449:{\displaystyle \,n\,}
429:
427:{\displaystyle \,C\,}
404:
226:nominal interest rate
215:calculations use the
3108:Mathematical finance
2692:
2605:
2564:
2541:
2521:
2501:
2444:perpetuity immediate
2416:
2381:
2263:
2206:
2148:
2077:
2040:
2014:
2006:= number of periods,
1988:
1837:
1768:
1733:
1702:
1680:
1658:
1636:
1573:
1546:
1468:
1442:
1240:
1035:
955:
875:
801:
766:
724:
702:
680:
598:
588:with negative time.
529:
504:
482:
460:
438:
416:
355:
43:improve this article
3054:Henderson, David R.
2800:Variants/approaches
2758:time value of money
254:time value of money
147:time value of money
2726:
2679:
2585:
2553:{\displaystyle Fr}
2550:
2527:
2507:
2465:
2399:
2356:
2243:
2166:
2128:
2048:
2022:
1996:
1968:
1817:
1751:
1716:
1688:
1666:
1644:
1593:
1559:
1529:
1450:
1423:
1211:
1014:
940:
860:
780:
732:
710:
688:
656:
571:
512:
490:
468:
446:
424:
399:
320:Real interest rate
314:mathematical limit
222:real interest rate
2888:Net present value
2868:Capital budgeting
2789:Laplace transform
2748:Present value is
2744:Technical details
2530:{\displaystyle r}
2510:{\displaystyle F}
2445:
2429:
2338:
2325:
2228:
2118:
2105:
1949:
1899:
1797:
1796:annuity immediate
1781:
1591:
1516:
1394:
1350:
1182:
1154:
1121:
1002:
925:
848:
644:
396:
346:compound interest
280:Compound interest
206:compound interest
137:), also known as
119:
118:
111:
93:
3115:
3089:
3040:
3033:
3027:
3020:
3014:
3013:
2995:
2978:
2977:
2959:
2934:
2933:
2915:
2735:
2733:
2732:
2727:
2725:
2724:
2688:
2686:
2685:
2680:
2678:
2674:
2673:
2672:
2641:
2636:
2594:
2592:
2591:
2586:
2559:
2557:
2556:
2551:
2536:
2534:
2533:
2528:
2516:
2514:
2513:
2508:
2474:
2472:
2471:
2466:
2447:
2446:
2443:
2431:
2430:
2427:
2408:
2406:
2405:
2400:
2371:geometric series
2365:
2363:
2362:
2357:
2339:
2331:
2326:
2324:
2323:
2322:
2297:
2294:
2289:
2252:
2250:
2249:
2244:
2229:
2221:
2175:
2173:
2172:
2167:
2137:
2135:
2134:
2129:
2127:
2123:
2119:
2111:
2106:
2098:
2057:
2055:
2054:
2049:
2031:
2029:
2028:
2023:
2005:
2003:
2002:
1997:
1977:
1975:
1974:
1969:
1954:
1950:
1945:
1944:
1943:
1912:
1900:
1898:
1897:
1896:
1871:
1868:
1863:
1826:
1824:
1823:
1818:
1799:
1798:
1795:
1783:
1782:
1779:
1760:
1758:
1757:
1752:
1725:
1723:
1722:
1717:
1697:
1695:
1694:
1689:
1675:
1673:
1672:
1667:
1653:
1651:
1650:
1645:
1627:geometric series
1602:
1600:
1599:
1594:
1592:
1587:
1586:
1577:
1568:
1566:
1565:
1560:
1558:
1557:
1538:
1536:
1535:
1530:
1528:
1527:
1522:
1518:
1517:
1512:
1511:
1502:
1459:
1457:
1456:
1451:
1432:
1430:
1429:
1424:
1395:
1393:
1392:
1391:
1376:
1375:
1356:
1351:
1349:
1348:
1347:
1328:
1323:
1322:
1303:
1302:
1277:
1276:
1220:
1218:
1217:
1212:
1183:
1181:
1180:
1179:
1160:
1155:
1153:
1152:
1151:
1135:
1127:
1122:
1120:
1119:
1118:
1099:
1094:
1093:
1078:
1077:
1062:
1061:
1023:
1021:
1020:
1015:
1003:
1001:
1000:
999:
983:
975:
970:
969:
949:
947:
946:
941:
926:
924:
923:
922:
906:
895:
890:
889:
869:
867:
866:
861:
849:
847:
846:
845:
829:
821:
816:
815:
789:
787:
786:
781:
741:
739:
738:
733:
719:
717:
716:
711:
697:
695:
694:
689:
674:purchasing power
665:
663:
662:
657:
645:
643:
642:
641:
619:
611:
580:
578:
577:
572:
570:
569:
541:
540:
521:
519:
518:
513:
499:
497:
496:
491:
477:
475:
474:
469:
455:
453:
452:
447:
433:
431:
430:
425:
408:
406:
405:
400:
397:
395:
394:
393:
368:
114:
107:
103:
100:
94:
92:
51:
27:
19:
16:Economic concept
3123:
3122:
3118:
3117:
3116:
3114:
3113:
3112:
3093:
3092:
3078:
3058:"Present Value"
3049:
3047:Further reading
3044:
3043:
3034:
3030:
3021:
3017:
3010:
2996:
2981:
2974:
2960:
2937:
2930:
2916:
2901:
2896:
2864:
2851:
2849:Years' purchase
2842:
2825:
2802:
2746:
2717:
2713:
2693:
2690:
2689:
2665:
2661:
2637:
2626:
2621:
2617:
2606:
2603:
2602:
2565:
2562:
2561:
2542:
2539:
2538:
2522:
2519:
2518:
2517:, coupon rate,
2502:
2499:
2498:
2481:
2442:
2438:
2426:
2422:
2417:
2414:
2413:
2382:
2379:
2378:
2330:
2318:
2314:
2301:
2296:
2290:
2279:
2264:
2261:
2260:
2220:
2207:
2204:
2203:
2189:
2149:
2146:
2145:
2110:
2097:
2096:
2092:
2078:
2075:
2074:
2065:
2041:
2038:
2037:
2015:
2012:
2011:
1989:
1986:
1985:
1936:
1932:
1913:
1911:
1907:
1892:
1888:
1875:
1870:
1864:
1853:
1838:
1835:
1834:
1794:
1790:
1778:
1774:
1769:
1766:
1765:
1734:
1731:
1730:
1703:
1700:
1699:
1681:
1678:
1677:
1659:
1656:
1655:
1637:
1634:
1633:
1614:
1608:
1582:
1578:
1576:
1574:
1571:
1570:
1553:
1549:
1547:
1544:
1543:
1523:
1507:
1503:
1501:
1494:
1490:
1489:
1469:
1466:
1465:
1443:
1440:
1439:
1387:
1383:
1371:
1367:
1360:
1355:
1343:
1339:
1332:
1327:
1315:
1311:
1295:
1291:
1269:
1265:
1241:
1238:
1237:
1175:
1171:
1164:
1159:
1147:
1143:
1136:
1128:
1126:
1114:
1110:
1103:
1098:
1089:
1085:
1073:
1069:
1057:
1053:
1036:
1033:
1032:
995:
991:
984:
976:
974:
965:
961:
956:
953:
952:
918:
914:
907:
896:
894:
885:
881:
876:
873:
872:
841:
837:
830:
822:
820:
811:
807:
802:
799:
798:
767:
764:
763:
759:
725:
722:
721:
703:
700:
699:
681:
678:
677:
637:
633:
620:
612:
610:
599:
596:
595:
562:
558:
536:
532:
530:
527:
526:
505:
502:
501:
483:
480:
479:
461:
458:
457:
439:
436:
435:
417:
414:
413:
389:
385:
372:
367:
356:
353:
352:
342:
332:worth today?).
329:
286:Simple interest
258:corporate bonds
250:
199:time preference
194:
115:
104:
98:
95:
58:"Present value"
52:
50:
40:
28:
17:
12:
11:
5:
3121:
3111:
3110:
3105:
3091:
3090:
3077:978-0865976658
3076:
3048:
3045:
3042:
3041:
3028:
3015:
3008:
2979:
2972:
2935:
2928:
2898:
2897:
2895:
2892:
2891:
2890:
2885:
2880:
2878:Lifetime value
2875:
2870:
2863:
2860:
2850:
2847:
2841:
2838:
2824:
2821:
2820:
2819:
2813:
2801:
2798:
2785:
2784:
2773:
2745:
2742:
2737:
2736:
2723:
2720:
2716:
2712:
2709:
2706:
2703:
2700:
2697:
2677:
2671:
2668:
2664:
2660:
2657:
2654:
2651:
2648:
2645:
2640:
2635:
2632:
2629:
2625:
2620:
2616:
2613:
2610:
2584:
2581:
2578:
2575:
2572:
2569:
2549:
2546:
2526:
2506:
2491:
2490:
2480:
2477:
2476:
2475:
2462:
2459:
2456:
2453:
2450:
2441:
2437:
2434:
2428:perpetuity due
2425:
2421:
2398:
2395:
2392:
2389:
2386:
2367:
2366:
2355:
2352:
2349:
2346:
2342:
2337:
2334:
2329:
2321:
2317:
2313:
2310:
2307:
2304:
2300:
2293:
2288:
2285:
2282:
2278:
2274:
2271:
2268:
2254:
2253:
2242:
2239:
2236:
2232:
2227:
2224:
2218:
2214:
2211:
2188:
2185:
2165:
2162:
2159:
2156:
2153:
2141:
2140:
2139:
2138:
2126:
2122:
2117:
2114:
2109:
2104:
2101:
2095:
2091:
2088:
2085:
2082:
2064:
2061:
2060:
2059:
2046:
2034:
2033:
2020:
2008:
2007:
1994:
1979:
1978:
1967:
1964:
1961:
1957:
1953:
1948:
1942:
1939:
1935:
1931:
1928:
1925:
1922:
1919:
1916:
1910:
1906:
1903:
1895:
1891:
1887:
1884:
1881:
1878:
1874:
1867:
1862:
1859:
1856:
1852:
1848:
1845:
1842:
1828:
1827:
1814:
1811:
1808:
1805:
1802:
1793:
1789:
1786:
1777:
1773:
1750:
1747:
1744:
1741:
1738:
1714:
1711:
1708:
1686:
1664:
1642:
1607:
1604:
1590:
1585:
1581:
1556:
1552:
1540:
1539:
1526:
1521:
1515:
1510:
1506:
1500:
1497:
1493:
1488:
1485:
1482:
1479:
1476:
1473:
1462:
1461:
1448:
1434:
1433:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1390:
1386:
1382:
1379:
1374:
1370:
1366:
1363:
1359:
1354:
1346:
1342:
1338:
1335:
1331:
1326:
1321:
1318:
1314:
1310:
1307:
1301:
1298:
1294:
1290:
1287:
1283:
1280:
1275:
1272:
1268:
1264:
1261:
1257:
1254:
1251:
1248:
1245:
1234:
1233:
1229:
1222:
1221:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1178:
1174:
1170:
1167:
1163:
1158:
1150:
1146:
1142:
1139:
1134:
1131:
1125:
1117:
1113:
1109:
1106:
1102:
1097:
1092:
1088:
1084:
1081:
1076:
1072:
1068:
1065:
1060:
1056:
1052:
1049:
1046:
1043:
1040:
1026:
1025:
1012:
1009:
1006:
998:
994:
990:
987:
982:
979:
973:
968:
964:
960:
950:
938:
935:
932:
929:
921:
917:
913:
910:
905:
902:
899:
893:
888:
884:
880:
870:
858:
855:
852:
844:
840:
836:
833:
828:
825:
819:
814:
810:
806:
778:
775:
772:
758:
755:
744:inflation rate
730:
708:
686:
667:
666:
654:
651:
648:
640:
636:
632:
629:
626:
623:
618:
615:
609:
606:
603:
568:
565:
561:
557:
554:
551:
548:
544:
539:
535:
510:
488:
466:
444:
422:
410:
409:
392:
388:
384:
381:
378:
375:
371:
366:
363:
360:
341:
338:
328:
325:
324:
323:
317:
307:
301:
295:
289:
283:
249:
248:Interest rates
246:
193:
190:
186:rate of return
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3120:
3109:
3106:
3104:
3101:
3100:
3098:
3087:
3083:
3079:
3073:
3069:
3065:
3064:
3059:
3055:
3051:
3050:
3039:, Vol.1, 1955
3038:
3032:
3025:
3019:
3011:
3009:9780077246129
3005:
3001:
2994:
2992:
2990:
2988:
2986:
2984:
2975:
2973:9781566987677
2969:
2965:
2958:
2956:
2954:
2952:
2950:
2948:
2946:
2944:
2942:
2940:
2931:
2929:9780538479172
2925:
2921:
2914:
2912:
2910:
2908:
2906:
2904:
2899:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2873:Current yield
2871:
2869:
2866:
2865:
2859:
2857:
2846:
2837:
2835:
2830:
2817:
2814:
2811:
2808:
2807:
2806:
2805:expectation.
2797:
2795:
2790:
2782:
2778:
2774:
2771:
2767:
2763:
2762:
2761:
2759:
2755:
2751:
2741:
2721:
2718:
2710:
2707:
2704:
2698:
2695:
2675:
2669:
2666:
2658:
2655:
2652:
2646:
2643:
2638:
2633:
2630:
2627:
2623:
2618:
2614:
2611:
2608:
2601:
2600:
2599:
2596:
2579:
2576:
2573:
2567:
2547:
2544:
2524:
2504:
2496:
2489:
2488:
2483:
2482:
2457:
2454:
2451:
2439:
2435:
2432:
2423:
2419:
2412:
2411:
2410:
2393:
2390:
2387:
2374:
2372:
2369:which form a
2353:
2350:
2347:
2344:
2340:
2335:
2332:
2327:
2319:
2311:
2308:
2305:
2298:
2286:
2283:
2280:
2276:
2272:
2269:
2266:
2259:
2258:
2257:
2237:
2230:
2225:
2222:
2216:
2212:
2209:
2202:
2201:
2200:
2198:
2194:
2184:
2181:
2177:
2163:
2160:
2157:
2154:
2151:
2124:
2120:
2115:
2112:
2107:
2102:
2099:
2093:
2089:
2086:
2083:
2080:
2073:
2072:
2071:
2070:
2069:
2044:
2036:
2035:
2018:
2010:
2009:
1992:
1984:
1983:
1982:
1962:
1955:
1951:
1946:
1940:
1937:
1929:
1926:
1923:
1917:
1914:
1908:
1904:
1901:
1893:
1885:
1882:
1879:
1872:
1865:
1860:
1857:
1854:
1850:
1846:
1843:
1840:
1833:
1832:
1831:
1809:
1806:
1803:
1791:
1787:
1784:
1775:
1771:
1764:
1763:
1762:
1745:
1742:
1739:
1727:
1712:
1709:
1706:
1684:
1662:
1640:
1630:
1628:
1624:
1620:
1617:is called an
1613:
1603:
1588:
1583:
1579:
1554:
1550:
1524:
1519:
1513:
1508:
1504:
1498:
1495:
1491:
1486:
1480:
1477:
1474:
1464:
1463:
1446:
1436:
1435:
1420:
1414:
1411:
1405:
1402:
1396:
1388:
1380:
1372:
1364:
1357:
1352:
1344:
1336:
1329:
1324:
1319:
1316:
1308:
1299:
1296:
1288:
1281:
1278:
1273:
1270:
1262:
1255:
1252:
1249:
1246:
1243:
1236:
1235:
1230:
1227:
1226:
1225:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1176:
1168:
1161:
1156:
1148:
1140:
1132:
1129:
1123:
1115:
1107:
1100:
1095:
1090:
1086:
1082:
1079:
1074:
1070:
1066:
1063:
1058:
1054:
1050:
1047:
1044:
1041:
1038:
1031:
1030:
1029:
1010:
1004:
996:
988:
980:
971:
966:
962:
958:
951:
936:
930:
927:
919:
911:
903:
897:
891:
886:
882:
878:
871:
856:
850:
842:
834:
826:
817:
812:
808:
804:
797:
796:
795:
791:
776:
773:
770:
754:
752:
747:
745:
728:
706:
684:
675:
670:
652:
646:
638:
630:
627:
624:
616:
607:
604:
601:
594:
593:
592:
589:
587:
582:
566:
563:
555:
552:
549:
542:
537:
533:
523:
508:
486:
464:
442:
420:
390:
382:
379:
376:
369:
364:
361:
358:
351:
350:
349:
347:
337:
333:
321:
318:
315:
311:
308:
305:
304:Discount rate
302:
299:
296:
293:
290:
287:
284:
281:
278:
277:
276:
274:
269:
267:
263:
260:, or through
259:
255:
245:
241:
238:
233:
231:
227:
223:
218:
214:
209:
207:
202:
200:
189:
187:
183:
179:
175:
174:sinking funds
171:
167:
163:
159:
154:
152:
148:
144:
140:
136:
132:
131:present value
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3061:
3036:
3031:
3023:
3018:
2999:
2963:
2919:
2852:
2843:
2834:risk premium
2826:
2815:
2809:
2803:
2786:
2777:default risk
2747:
2738:
2597:
2492:
2484:
2479:PV of a bond
2375:
2368:
2255:
2196:
2190:
2182:
2178:
2142:
2066:
1980:
1829:
1728:
1631:
1615:
1541:
1223:
1027:
1024:respectively
792:
760:
750:
748:
671:
668:
590:
583:
524:
411:
343:
334:
330:
270:
251:
242:
237:future value
234:
210:
203:
195:
178:perpetuities
158:future value
155:
138:
134:
130:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2883:Liquidation
1780:annuity due
327:Calculation
3097:Categories
3037:New Series
2894:References
2754:cash flows
2193:perpetuity
1623:summations
1610:See also:
698:of money,
192:Background
99:March 2012
69:newspapers
3086:237794267
2766:inflation
2719:−
2667:−
2624:∑
2292:∞
2277:∑
2155:≈
2084:≈
1938:−
1918:−
1851:∑
1710:−
1418:$
1409:$
1400:$
1317:−
1297:−
1271:−
1191:−
1130:−
1008:$
978:$
934:$
931:−
901:$
898:−
854:$
824:$
650:$
614:$
564:−
266:dividends
230:inflation
213:actuarial
170:annuities
166:mortgages
123:economics
3056:(2008).
2862:See also
2750:additive
273:interest
143:interest
1981:where:
1619:annuity
525:Often,
275:rates:
127:finance
83:scholar
3103:Income
3084:
3074:
3006:
2970:
2926:
1542:Here,
1421:268.40
1412:173.16
653:620.92
412:Where
312:, the
228:minus
85:
78:
71:
64:
56:
2485:See:
1403:95.24
1206:80.12
1200:30.23
1194:45.35
1188:95.24
1011:30.23
937:45.35
857:95.24
262:stock
182:bonds
162:loans
90:JSTOR
76:books
3082:OCLC
3072:ISBN
3004:ISBN
2968:ISBN
2924:ISBN
2495:bond
2348:>
1381:1.05
1365:1.10
1337:1.05
1309:1.05
1289:1.10
1263:1.05
1169:1.05
1141:1.05
1108:1.05
989:1.05
912:1.05
835:1.05
672:The
631:0.10
617:1000
151:rent
125:and
62:news
1625:of
1358:200
1330:100
1282:200
1256:100
1101:100
827:100
121:In
45:by
3099::
3080:.
3070:.
3060:.
2982:^
2938:^
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2595:.
2409::
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2191:A
2176:.
1761::
1629:.
1162:35
1133:50
981:35
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478:,
180:,
176:,
172:,
168:,
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135:PV
129:,
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3012:.
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1005:=
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972:=
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928:=
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909:(
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73:·
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39:.
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