Quantitative Methods - Time Value Of Money Applications

Most of the problems from the time value material are likely to ask for either PV or FV and will provide the other variables. However, on a test with hundreds of problems, the CFA exam will look for unique and creative methods to test command of the material. A problem might provide both FV and PV and then ask you to solve for an unknown variable, either the interest rate (r), the number of periods (N) or the amount of the annuity (A). In most of these cases, a quick use of freshmen-level algebra is all that's required. We'll cover two real-world applications - each was the subject of an example in the resource textbook, so either one may have a reasonable chance of ending up on an exam problem.

Annualized Growth Rates
The first application is annualized growth rates. Taking the formula for FV of a single sum of money and solving for r produces a formula that can also be viewed as the growth rate, or the rate at which that sum of money grew from PV to FV in N periods.

Formula 2.6
Growth rate (g) = (FV/PV)1/N - 1

For example, if a company's earnings were $100 million five years ago, and are $200 million today, the annualized five-year growth rate could be found by:

growth rate (g) = (FV/PV)1/N - 1 = (200,000,000/100,000,000) 1/5 - 1 = (2) 1/5 - 1 = (1.1486984) - 1 = 14.87%

Monthly Mortgage Payments
The second application involves calculating monthly mortgage payments. Periodic mortgage payments fit the definition of an annuity payment (A), where PV of the annuity is equal to amount borrowed. (Note that if the loan is needed for a $300,000 home and they tell you that the down payment is $50,000, make sure to reduce the amount borrowed, or PV, to $250,000! Plenty of folks will just grab the $300,000 number and plug it into the financial calculator.) Because mortgage payments are typically made monthly with interest compounded monthly, expect to adjust the annual interest rate (r) by dividing by 12, and to multiply the time periods by 12 if the mortgage loan period is expressed in years.

Since PV of an annuity = (annuity payment)*(PV annuity factor), we solve for annuity payment (A), which will be the monthly payment:

Formula 2.7
Monthly mortgage payment = (Amount of the loan)/(PV annuity factor)

Example: Monthly Mortgage Payments
Assuming a 30-year loan with monthly compounding (so N = 30*12 = 360 months), and a rate of 6% (so r = .06/12 = 0.005), we first calculate the PV annuity factor:

PV annuity factor = (1 - (1/(1 + r)N)/r = (1 - (1/(1.005)360)/0.005 = 166.7916

With a loan of $250,000, the monthly payment in this example would be $250,000/166.7916, or $1,498.88 a month.

Exam Tips and Tricks
Higher-level math functions usually don\'t end up on the test, partly because they give an unfair advantage to those with higher-function calculators and because questions must be solved in an average of one to two minutes each at Level I. Don\'t get bogged down with understanding natural logs or transcendental numbers.

Savings and retirement planning are sometimes more complicated, as there are various life-cycles stages that result in assumptions for uneven cash inflows and outflows. Problems of this nature often involve more than one computation of the basic time value formulas; thus the emphasis on drawing a timeline is sound advice, and a worthwhile habit to adopt even when solving problems that appear to be relatively simple.

Example: Retirement Savings
To illustrate, we take a hypothetical example of a client, 35 years old, who would like to retire at age 65 (30 years from today). Her goal is to have enough in her retirement account to provide an income of $75,000 a year, starting a year after retirement or year 31, for 25 years thereafter. She had a late start on saving for retirement, with a current balance of $10,000. To catch up, she is now committed to saving $5,000 a year, with the first contribution a year from now. A single parent with two children, both of which will be attending college starting in five years, she won't be able to increase the annual $5,000 commitment until after the kids have graduated. Once the children are finished with college, she will have extra disposable income, but is worried about just how much of an increase it will take to meet her ultimate retirement goals. To help her meet this goal, estimate how much she will need to save every year, starting 10 years from now, when the kids are out of college. Assume an average annual 8% return in the retirement account.

To organize and summarize this information, we will need her three cash inflows to be the equivalent of her one cash outflow.

1.The money already in the account is the first inflow.
2. The money to be saved during the next 10 years is the second inflow.
3. The money to be saved between years 11 and 30 is the third inflow.
4.The money to be taken as income from years 31 to 50 is the one outflow.

All amounts are given to calculate inflows 1 and 2 and the outflow. The third inflow has an unknown annuity amount that will need to be determined using the other amounts. We start by drawing a timeline and specifying that all amounts be indexed at t = 30, or her retirement day.

Next, calculate the three amounts for which we have all the necessary information, and index to t = 30.

(inflow 1) FV (single sum) = PV *(1 + r)N = ($10,000)*(1.08)30 = $100,627

(inflow 2) FV annuity factor = ((1 + r)N - 1)/r = ((1.08)10 - 1)/.08 = 14.48656

With a $5000 payment, FV (annuity) = ($5000)*(14.48656) = $72,433

This amount is what is accumulated at t = 10; we need to index it to t = 30.

FV (single sum) = PV *(1 + r)N = ($72,433)*(1.08)20 = $337,606

(cash PV annuity factor = (1 - (1/(1 + r)N)/r = (1 - (1/(1.08)25/0.08 = 10.674776.outflow)

With payment of $75,000, PV (annuity) = ($75,000)*(10.674776) = $800,608.

Since the three cash inflows = cash outflow, we have ($100,627) + ($337,606) + X = $800,608, or X = $362,375 at t = 30. In other words, the money she saves from years 11 through 30 will need to be equal to $362,375 in order for her to meet retirement goals.

FV annuity factor = ((1 + r)N - 1)/r = ((1.08)20 - 1)/.08 = 45.76196

A = FV/FV annuity factor = (362,375)/45.76196 = $7919

We find that by increasing the annual savings from $5,000 to $7,919 starting in year 11 and continuing to year 30, she will be successful in accumulating enough income for retirement.

How are Present Values, Future Value and Cash Flows connected?
The cash flow additivity principle allows us to add amounts of money together, provided they are indexed to the same period. The last example on retirement savings illustrates cash flow additivity: we were planning to accumulate a sum of money from three separate sources and we needed to determine what the total amount would be so that the accumulated sum could be compared with the client's retirement cash outflow requirement. Our example involved uneven cash flows from two separate annuity streams and one single lump sum that has already accumulated. Comparing these inputs requires each amount to be indexed first, prior to adding them together. In the last example, the annuity we were planning to accumulate in years 11 to 30 was projected to reach $362,375 by year 30. The current savings initiative of $5,000 a year projects to $72,433 by year 10. Right now, time 0, we have $10,000. In other words, we have three amounts at three different points in time.

According to the cash flow additivity principle, these amounts could not be added together until they were either discounted back to a common date, or compounded ahead to a common date. We chose t = 30 in the example because it made the calculations the simplest, but any point in time could have been chosen. The most common date chosen to apply cash flow additivity is t = 0 (i.e. discount all expected inflows and outflows to the present time). This principle is frequently tested on the CFA exam, which is why the technique of drawing timelines and choosing an appropriate time to index has been emphasized here.

Net Present Value and the Internal Rate of Return
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