Present Value of a Perpetuity
A perpetuity starts as an ordinary annuity (first cash flow is one period from today) but has no end and continues indefinitely with level, sequential payments. Perpetuities are more a product of the CFA world than the real world - what entity would obligate itself making to payments that will never end? However, some securities (such as preferred stocks) do come close to satisfying the assumptions of a perpetuity, and the formula for PV of a perpetuity is used as a starting point to value these types of securities.
The formula for the PV of a perpetuity is derived from the PV of an ordinary annuity, which at N = infinity, and assuming interest rates are positive, simplifies to:
| Formula 2.2 PV of a perpetuity = annuity payment A interest rate r |
Therefore, a perpetuity paying $1,000 annually at an interest rate of 8% would be worth:
PV = A/r = ($1000)/0.08 = $12,500
Future Value and Present Value of a SINGLE SUM OF MONEY
PV = A/r = ($1000)/0.08 = $12,500
Future Value and Present Value of a SINGLE SUM OF MONEY
If we assume an annual compounding of interest, these problems can be solved with the following formulas:
| Formula 2.3 (1) FV = PV * (1 + r)N (2) PV = FV * { 1 } (1 + r)N Where: FV = future value of a single sum of money, PV = present value of a single sum of money, R = annual interest rate, and N = number of years |
Example: Present Value
At an interest rate of 8%, we calculate that $10,000 five years from now will be:
FV = PV * (1 + r)N = ($10,000)*(1.08)5 = ($10,000)*(1.469328)
FV = $14,693.28
At an interest rate of 8%, we calculate today's value that will grow to $10,000 in five years:
PV = FV * (1/(1 + r)N) = ($10,000)*(1/(1.08)5) = ($10,000)*(1/(1.469328))
PV = ($10,000)*(0.680583) = $6805.83
Example: Future Value
An investor wants to have $1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:
A. $100,000
B. $117,459
C. $148,644
D. $161,506
Answer:
The problem asks for a value today (PV). It provides the future sum of money (FV) = $1,000,000; an interest rate (r) = 10% or 0.1; yearly time periods (N) = 20, and it indicates annual compounding. Using the PV formula listed above, we get the following:
At an interest rate of 8%, we calculate that $10,000 five years from now will be:
FV = PV * (1 + r)N = ($10,000)*(1.08)5 = ($10,000)*(1.469328)
FV = $14,693.28
At an interest rate of 8%, we calculate today's value that will grow to $10,000 in five years:
PV = FV * (1/(1 + r)N) = ($10,000)*(1/(1.08)5) = ($10,000)*(1/(1.469328))
PV = ($10,000)*(0.680583) = $6805.83
Example: Future Value
An investor wants to have $1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:
A. $100,000
B. $117,459
C. $148,644
D. $161,506
Answer:
The problem asks for a value today (PV). It provides the future sum of money (FV) = $1,000,000; an interest rate (r) = 10% or 0.1; yearly time periods (N) = 20, and it indicates annual compounding. Using the PV formula listed above, we get the following:
PV = FV *[1/(1 + r) N] = [($1,000,000)* (1/(1.10)20)] = $1,000,000 * (1/6.7275) = $1,000,000*0.148644 = $148,644
Using a calculator with financial functions can save time when solving PV and FV problems. At the same time, the CFA exam is written so that financial calculators aren't required. Typical PV and FV problems will test the ability to recognize and apply concepts and avoid tricks, not the ability to use a financial calculator. The experience gained by working through more examples and problems increase your efficiency much more than a calculator.
Using a calculator with financial functions can save time when solving PV and FV problems. At the same time, the CFA exam is written so that financial calculators aren't required. Typical PV and FV problems will test the ability to recognize and apply concepts and avoid tricks, not the ability to use a financial calculator. The experience gained by working through more examples and problems increase your efficiency much more than a calculator.
FV and PV of an Ordinary Annuity and an Annuity Due
To solve annuity problems, you must know the formulas for the future value annuity factor and the present value annuity factor.
To solve annuity problems, you must know the formulas for the future value annuity factor and the present value annuity factor.
| Formula 2.4 Future Value Annuity Factor =(1 + r)N - 1 r |
| Formula 2.5 Present Value Annuity Factor = 1 - 1 (1 + r)N r Where r = interest rate and N = number of payments |
FV Annuity Factor
The FV annuity factor formula gives the future total dollar amount of a series of $1 payments, but in problems there will likely be a periodic cash flow amount given (sometimes called the annuity amount and denoted by A). Simply multiply A by the FV annuity factor to find the future value of the annuity. Likewise for PV of an annuity: the formula listed above shows today's value of a series of $1 payments to be received in the future. To calculate the PV of an annuity, multiply the annuity amount A by the present value annuity factor.
The FV and PV annuity factor formulas work with an ordinary annuity, one that assumes the first cash flow is one period from now, or t = 1 if drawing a timeline. The annuity due is distinguished by a first cash flow starting immediately, or t = 0 on a timeline. Since the annuity due is basically an ordinary annuity plus a lump sum (today's cash flow), and since it can be fit to the definition of an ordinary annuity starting one year ago, we can use the ordinary annuity formulas as long as we keep track of the timing of cash flows. The guiding principle: make sure, before using the formula, that the annuity fits the definition of an ordinary annuity with the first cash flow one period away.
Example: FV and PV of ordinary annuity and annuity due
An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9% interest compounded annually. The amount of money in the account of the end of 10 years will be closest to:
A. $109,000
B. $143.200
C. $151,900
D. $165,600
Answer:
The problem gives the annuity amount A = $10,000, the interest rate r = 0.09, and time periods N = 10. Time units are all annual (compounded annually) so there is no need to convert the units on either r or N. However, the starting today introduces a wrinkle. The annuity being described is an annuity due, not an ordinary annuity, so to use the FV annuity factor, we will need to change our perspective to fit the definition of an ordinary annuity.
Drawing a timeline should help visualize what needs to be done:
An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9% interest compounded annually. The amount of money in the account of the end of 10 years will be closest to:
A. $109,000
B. $143.200
C. $151,900
D. $165,600
Answer:
The problem gives the annuity amount A = $10,000, the interest rate r = 0.09, and time periods N = 10. Time units are all annual (compounded annually) so there is no need to convert the units on either r or N. However, the starting today introduces a wrinkle. The annuity being described is an annuity due, not an ordinary annuity, so to use the FV annuity factor, we will need to change our perspective to fit the definition of an ordinary annuity.
Drawing a timeline should help visualize what needs to be done:
 |
| Figure 2.1: Cashflow Timeline |
The definition of an ordinary annuity is a cash flow stream beginning in one period, so the annuity being described in the problem is an ordinary annuity starting last year, with 10 cash flows from t0 to t9. Using the FV annuity factor formula, we have the following:
FV annuity factor = ((1 + r)N - 1)/r = (1.09)10 - 1)/0.09 = (1.3673636)/0.09 = 15.19293
Multiplying this amount by the annuity amount of $10,000, we have the future value at time period 9. FV = ($10,000)*(15.19293) = $151,929. To finish the problem, we need the value at t10. To calculate, we use the future value of a lump sum, FV = PV*(1 + r)N, with N = 1, PV = the annuity value after 9 periods, r = 9.
FV = PV*(1 + r)N = ($151,929)*(1.09) = $165,603.
The correct answer is "D".
Notice that choice "C" in the problem ($151,900) agrees with the preliminary result of the value of the annuity at t = 9. It's also the result if we were to forget the distinction between ordinary annuity and annuity due, and go forth and solve the problem with the ordinary annuity formula and the given parameters. On the CFA exam, problems like this one will get plenty of takers for choice "C" - mostly the people trying to go too fast!!
Usually I never comment on blogs but your article is so convincing that I never stop myself to say something about it. You’re doing a great job Man,Keep it up. ACCA Notes are available
ReplyDelete