Time Value Concepts Chapter 32 Tools & Techniques of Investment Planning Copyright 2007, The...

Time Value Concepts Chapter 32Tools & Techniques of

Investment Planning

Copyright 2007, The National Underwriter Company 1

What is it?

• Whenever people “invest” or “save” money, they are deferring the pleasure of currently consuming what that money could buy now until some later time.


Investment Planning


What is it?

• Investors and savers never have any absolute assurance that money which is saved or invested today will be able to purchase consumption goods of comparable value and appeal to the investors in the future as the consumption goods they could have bought today.– Investors or savers bear some risk that they will get less

“value” later than the value they have forgone today.– Therefore, they will demand not only a return of their money

but a return on their money with the level of return they demand on their money increasing with the risk that it might be worth less tomorrow than it is today.


Investment Planning


What is it?

• The longer is the time that investors postpone their consumption, the greater is the chance something adverse will occur to erode or lessen the value of their investment.– The longer is the period of time their money is tied up in an

investment, the greater is the return they will demand on their money as compensation for deferring current consumption.

• Investors or their advisers must fully understand time value concepts.– In order to evaluate alternative investment opportunities or

differing financial planning strategies.


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How does it work?

• Time value analysis is founded on the basic economic premise that value or price today depends on the future cash flows or other tangible benefits that flow from an investment.

• From this basic premise, all other basic time value concepts follow, including that the analysis must:– Account for all anticipated cash flows, benefits, and costs,– Properly account for differences in the timing of cash flows,

benefits, and costs by applying a properly risk-adjusted discount rate,

– Thereby evaluate all cash flows, benefits, and costs as of the same point in time.


Investment Planning


How does it work?

• The first concept basically states that cash is king.– In general, the only thing one can spend is cash or other

property that is readily convertible to cash.

• Time value analysis is so-named because differences in the timing of cash flows matter.

• The proper risk-adjusted discount rate (the hurdle rate) should be the investor’s “opportunity cost rate” which is equal to the rate of return the investor could earn on the next best available investment or project of comparable risk.– The purpose of the discount rate is to adjust all the cash flows

to the same time period.


Investment Planning


When is the use of this technique indicated?

• Time value analyses are required whenever a person is evaluating investment alternatives or financial planning strategies or tactics that involve differences in the timing of cash flows.– Various vehicles and strategies differ not only with respect to

the timing of cash flows and tax payments, but also with respect to the character of return for tax purposes and the tax rates applicable to the components of the return.

– Adjustments for inflation must be considered as well.

• Time value analysis is required in virtually every investment and financial planning situation.


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Advantages

• Properly applied time value analyses have the advantage of being the theoretically correct and complete means to quantitatively evaluate investment alternatives and other financial planning strategies and tactics according to economic principles.


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Disadvantages

• Accurate assessments of future cash flows, discount factors, and risks are difficult.– Investors or planners implicitly have to make assumptions about

future interest rates, inflation rates, tax rates, risks of default, business and economic conditions, and the like.

– Investors or advisers must always keep in mind the level of confidence that they can place in their estimates and temper the reliance and weight they place in the quantitative analyses accordingly.


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Disadvantages

• Time value analyses can become quite complex.– In many cases, analyses require several conceptual and

computational steps and must incorporate a number of other complicated adjustments, such as for growth factors for payment streams, inflation, and tax rules.

– Great care must be taken to assure that time value principles are properly applied and properly computed.


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How is it implemented?

• Terminology:– PV = Present Value - A value today– FV = Future Value - A value at some future date.– Pmt = Level periodic Payment - Constant dollar amount paid

or received at regularly spaced time intervals• Such as an annuity payment

– r = Interest rate or rate of return per period– n = Number of periods or the term of the investment or

valuation– ln = Natural logarithm operator


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How is it implemented?

• An annuity is a stream of level periodic payments.– An ordinary annuity is an annuity paid at the end of each

period.– An annuity due is an annuity paid at the beginning of each

period.– For positive interest rates with more than one cash flow, an

annuity due is always worth more than an ordinary annuity.


Investment Planning


Present Value Calculations: Overview

• Present value is the value of future cash flows “discounted” at an assumed interest rate or rate of return back from the future to their value today.

• The discount rate that is used to compute present values and other time adjusted dollar values is extremely important in most cases.– It should generally represent the best estimate or guess of the

true “opportunity” cost of money for investment opportunities of comparable risk over the time period cash flows are being valued.

– It should represent a rate that is comparable to the rate they could earn on the best alternative use of their money, hence the term “opportunity cost rate.”

• “Hurdle rate”


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Present Value of Future Lump Sum Payment

• An investment of a single dollar today at a periodic rate of return of r per period will provide that dollar plus the return, or (1+r) dollars, at the end of the first period.– At the end of n periods, investors would have (1+r)n dollars for

each dollar invested at the beginning of the first period.

• If investors want to know how much they would have to invest at the beginning of the first period to accumulate just one dollar at the end of n periods, divide $1 by (1+r)n.– To find the present value of FV dollars in n periods, divide FV by

(1+r)n

PV = FV / (1+r)n = FV x (1+r)-n


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Present Value of Perpetual Annuity

• An investment paying a fixed periodic payment forever • If an annuity is paying a fixed amount forever, the

annuity payment must just exactly equal the rate of return earned on the investment.– The principal amount neither increases nor decreases over

time, permitting the principal amount to generate a fixed level periodic payment forever.

• If the original principal amount is PV and the periodic rate of return is r, the periodic payment, Pmt, must be equal to PV * r.

PV = Pmt / r


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Present Value of Ordinary Annuity

• The valuation of an annuity can be viewed as simply the sum of a whole series of present values of future lump sum values where each future value is equal to the same level amount, Pmt.

• If the payments occur at the end of each period, the sum of the present values for n periods would be represented by the following equation:

n

PV = Pmt x 1/(1+r)j

j=1


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Present Value of Ordinary Annuity

• However, for simplicity, the present value of an n-period annuity paying Pmt per period can be written as the value of a perpetual annuity today, less the value of a perpetual annuity in n periods discounted back to today:

PV = Pmt/r - {Pmt/r * (1+r)-n }, if r

• By factoring out Pmt/r and rearranging terms, one can derive the most commonly used present value formula for an ordinary annuity:

PV = Pmt x {(1- (1+r)-n) / r}, if r 0.

• If the discount rate is zero for an n-period annuity:PV = Pmt x n, if r = 0


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Present Value of Annuity Due

• The present value of a level payment annuity with payments at the beginning of each period for n periods can be viewed as an ordinary annuity for (n-1) periods plus one payment immediately, which results in the same as the formula for the present value of an ordinary annuity multiplied by (1+r).

PV = Pmt x ((1- (1+r)-n) / r) x (1+r), if r 0.

If r = 0, PV = Pmt x n.


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Future Value of Present Lump Sum

• The future value of a lump sum is the inverse of the present value.

• Specifically, if one multiplies both sides of the present value formula by (1+r)n the result is the future value formula.

FV = PV x (1+r)n


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Future Value of Ordinary Annuity

• The future value calculation of an ordinary annuity can be derived directly from the formula for the present value of an ordinary annuity by substituting the present value of a lump sum formula for PV into the present value of the formula for an ordinary annuity and solving for FV.

FV = Pmt x {[(1+r)n - 1]/ r}, if r 0

if r = 0, FV = Pmt x n


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Future Value of Annuity Due

• Analogous to the derivation of the present value of an annuity due formula, the future value of an annuity due formula is just the future value of an ordinary annuity formula multiplied by (1+r).

FV = Pmt x {[(1+r)n - 1]/r} x (1+r), if r 0

if r = 0, FV = Pmt x n.


Investment Planning


Annuity Payment Calculations

• The calculation of the level periodic payment for an annuity depends on:– Whether it is computed in reference to a present value.– Whether it is computed in reference to a future value.– Whether the payments are to be made at the beginning or at

the end of each period.

• Each of the level periodic payment formulas can be derived from the corresponding present value and future value annuity formulas previously described by making simple arithmetic adjustments to isolate payment value on the left-hand side of the equation.


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• Ordinary Annuity Payment for a Given Present Value:

Pmt = PV x {r / [1 - (1+r)-n]}, if r 0

Pmt = PV / n, if r = 0

• Annuity Due Payment for a Given Present Value:

Pmt = PV x [r / {[1 - (1+r)-n] x (1+r)}], if r 0

Pmt = PV / n, if r = 0

• Periodic Payment for a Perpetual Annuity:

Pmt = PV x r


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• Ordinary Annuity Payment for a Given Future Value:

Pmt = FV x {r / [(1+r)n - 1]}, if r 0

Pmt = FV / n, if r = 0

• Annuity Due Payment for a Given Future Value:

Pmt = FV x {r / {[(1+r)n - 1] x (1+r)}}, if r 0.

Pmt = FV / n, if r = 0.


Investment Planning


Term Calculations

• The calculations to solve for the number of periods (n) for annuities depends on:– Whether n is calculated in reference to a present value or a

future value– Whether the annuities are ordinary annuities or annuities due.

• These calculations require the use of the natural logarithm operator.– This is built into most spreadsheet programs and financial

calculators.


Investment Planning


Term Calculations

• Number of Periods for Present Lump Sum Value to Reach a Future Value

n = {ln(FV/PV)} / {ln(1+r)}, for r>-100%; (FV/PV)>0

• Number of Periods Present Value Will Provide Ordinary Annuity Payments

n = {ln[1-(PV x r)/Pmt]} / {ln(1+r)}, for r>-100%; (PVxr)/Pmt<1

• Number of Periods Present Value Will Provide Annuity Due Payments.

n = [ln{[1-(PV x r)]/[Pmt x (1+r)]}] / [ln(1+r)],

for r>-100%; (PV x r)/(Pmt x (1+r))<1


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Term Calculations

• Number of Periods of Ordinary Annuity Payments to Reach Future Value

n = ln{[(FV x r) / Pmt] + 1} / ln(1+r),

for r>-100%; (FV x r) / Pmt > -1

• Number of Periods of Annuity Due Payments to Reach Future Value.

n = ln({(FV x r) / [Pmt x (1+r)]} +1) / ln(1+r),

for r>-100%; (FV x r) /[Pmt x(1+r)] > -1.


Investment Planning


• There is a mathematical anomaly attributable to the fact that the formulas solve for the number of periods in what is really a discreet sequence of payments.– If a result of the calculation includes a partial interval, then the

formula effectively assumes the investor makes one final additional partial payment related to the partial period.

• This non-integer value may overstate or understate the actual period slightly, resulting in a need to adjust by using appropriate formulas.

Final Payment Adjustment When n is a Non-Integer Value


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• These adjustments may result in two possible outcomes different from the initial results:– In all cases where the investment earnings in the last period

are not sufficient to reach the goal before the end of the last period, the actual number of periods, “n”, necessary to reach the target are really integer values, but the investor is required to make one final partial payment.

– In all cases where the investment earnings in the last period are more than sufficient to reach the accumulation target before the end of the next full period, the actual “n” will be a non-integer value, but the investor will not have to make a final partial payment.

Final Payment Adjustment When n is a Non-Integer Value


Investment Planning


Rate-of-Return Calculations

• Rate of Return Required for Lump Sum Present Value to Reach Future Value

R = (FV / PV)1/n -1

• Rate of Return of Annuities– There is no reduced-form analytical equation to solve for the

rate of return of an annuity. – Numerical methods, trial-and-error methods, or built-in

spreadsheet or calculator algorithms must be employed to compute this value.


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Rate Conversions, Compounding Periods, and Effective Interest Rates

• Frequently, financial planners and investors need to – Convert a rate quoted for one interval into an equivalent rate

for another interval, or– Convert a rate determined for one compounding period into an

equivalent rate for another compounding period

• For example, most mortgages are generally paid monthly and the compounding period is also generally monthly, while interest rates are almost always quoted in annual terms.


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• To understand interest-rate conversions, a few terms must be defined:

• c = The number of compounding periods per year.

• rc = The rate per compounding period.

• p = The number of payment periods per year.

• rp = The rate per payment period.

• rsa = The stated annual rate.

• rea = The effective annual rate.

– The values of the number of compounding periods, c, and for the number of payment periods, p, are often the same, but they need not be.



Investment Planning


Rate Conversions, Compounding Periods, and Effective Interest Rates• Stated Annual Rate

– The “stated annual rate” is usually defined as the compounding rate per period times the number of compounding periods per year.

rsa = c x rc

– Generally, the calculation actually goes the other way where one knows the stated annual rate and wants to find the compounding period rate:

rc = rsa/c


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• Effective Annual Rate– The “effective annual rate” is the rate one is really effectively

paying over a year’s time on a loan with more than one compounding period per year.

– This rate will virtually always be higher than the stated annual rate, unless there is only one compounding period per year.

rea = (1 + rc)c-1

– If c and p are different, apply this equation to the prior result:rp = (1 + rea)1/p -1


Time Value Concepts Chapter 32 Tools & Techniques of Investment Planning Copyright 2007, The...

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