Quantitative Methods - Time Value Of Money Calculations

Here we will discuss the effective annual rate, time value of money problems, PV of a perpetuity, an ordinary annuity, annuity due, a single cash flow and a series of uneven cash flows. For each, you should know how to both interpret the problem and solve the problems on your approved calculator. These concepts will cover LOS' 5.b and 5.c.

The Effective Annual Rate
CFA Institute's LOS 5.b is explained within this section. We'll start by defining the terms, and then presenting the formula.

The stated annual rate, or quoted rate, is the interest rate on an investment if an institution were to pay interest only once a year. In practice, institutions compound interest more frequently, either quarterly, monthly, daily and even continuously. However, stating a rate for those small periods would involve quoting in small fractions and wouldn't be meaningful or allow easy comparisons to other investment vehicles; as a result, there is a need for a standard convention for quoting rates on an annual basis.

The effective annual yield represents the actual rate of return, reflecting all of the compounding periods during the year. The effective annual yield (or EAR) can be computed given the stated rate and the frequency of compounding. We'll discuss how to make this computation next.

Formula 2.1
Effective annual rate (EAR) = (1 + Periodic interest rate)m - 1
Where: m = number of compounding periods in one year, and
periodic interest rate = (stated interest rate) / m

Example: Effective Annual Rate
Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR:

EAR = (1 + (0.09/12))12 - 1 = (1.0075) 12 - 1 = (1.093807) - 1 = 0.093807 or 9.38%

Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR.

Solving Time Value of Money Problems
Approach these problems by first converting both the rate r and the time period N to the same units as the compounding frequency. In other words, if the problem specifies quarterly compounding (i.e. four compounding periods in a year), with time given in years and interest rate is an annual figure, start by dividing the rate by 4, and multiplying the time N by 4. Then, use the resulting r and N in the standard PV and FV formulas.

Example: Compounding Periods

Assume that the future value of $10,000 five years from now is at 8%, but assuming quarterly compounding, we have quarterly r = 8%/4 = 0.02, and periods N = 4*5 = 20 quarters.

FV = PV * (1 + r)N = ($10,000)*(1.02)20 = ($10,000)*(1.485947) = $14,859.47

Assuming monthly compounding, where r = 8%/12 = 0.0066667, and N = 12*5 = 60.

FV = PV * (1 + r)N = ($10,000)*(1.0066667)60 = ($10,000)*(1.489846) = $14,898.46

Compare these results to the figure we calculated earlier with annual compounding ($14,693.28) to see the benefits of additional compounding periods.

Exam Tips and Tricks
On PV and FV problems, switching the time units - either by calling for quarterly or monthly compounding or by expressing time in months and the interest rate in years - is an often-used tactic to trip up test takers who are trying to go too fast. Remember to make sure the units agree for r and N, and are consistent with the frequency of compounding, prior to solving.

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

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

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.

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.

Formula 2.4
Future Value Annuity Factor = ((1 + r)n - 1)/r

Formula 2.5
Present Value Annuity Factor = (1 - (1 + r)-n

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

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!!

PV and FV of Uneven Cash Flows
The FV and PV annuity formulas assume level and sequential cash flows, but if a problem breaks this assumption, the annuity formulas no longer apply. To solve problems with uneven cash flows, each cash flow must be discounted back to the present (for PV problems) or compounded to a future date (for FV problems); then the sum of the present (or future) values of all cash flows is taken. In practice, particularly if there are many cash flows, this exercise is usually completed by using a spreadsheet. On the CFA exam, the ability to handle this concept may be tested with just a few future cash flows, given the time constraints.

It helps to set up this problem as if it were on a spreadsheet, to keep track of the cash flows and to make sure that the proper inputs are used to either discount or compound each cash flow. For example, assume that we are to receive a sequence of uneven cash flows from an annuity and we're asked for the present value of the annuity at a discount rate of 8%. Scratch out a table similar to the one below, with periods in the first column, cash flows in the second, formulas in the third column and computations in the fourth.

Time Period Cash Flow Present Value Formula Result of Computation
1 $1,000 ($1,000)/(1.08)1 $925.93
2 $1,500 ($1,500)/(1.08)2 $1,286.01
3 $2,000 ($2,000)/(1.08)3 $1,587.66
4 $500 ($500)/(1.08)4 $367.51
5 $3,000 ($3,000)/(1.08)5 $2,041.75

Taking the sum of the results in column 4, we have a PV = $6,208.86.

Suppose we are required to find the future value of this same sequence of cash flows after period 5. Here's the same approach using a table with future value formulas rather than present value, as in the table above:

Time Period Cash Flow Future Value Formula Result of computation
1 $1,000 ($1,000)*(1.08)4 $1,360.49
2 $1,500 ($1,500)*(1.08)3 $1,889.57
3 $2,000 ($2,000)*(1.08)2 $2,332.80
4 $500 ($500)*(1.08)1 $540.00
5 $3,000 ($3,000)*(1.08)0 $3,000.00

Taking the sum of the results in column 4, we have FV (period 5) = $9,122.86.

Check the present value of $9,122.86, discounted at the 8% rate for five years:

PV = ($9,122.86)/(1.08)5 = $6,208.86. In other words, the principle of equivalence applies even in examples where the cash flows are unequal.

Time Value Of Money Applications
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