Time Value of Money

Time Value of Money

The time value of money is money's potential to grow in value over time and this growth is associated with the expected return form ‘investment’ of money. Because of this potential, money that's available in the present is considered more valuable than the same amount in the future. For example, if you were given $100 today and invested it at an annual rate of only 1%, it could be worth $101 at the end of one year, which is more than you'd have if you received $100 at that point. After all, you should receive some compensation for foregoing spending. If you invest 1 dollar for one year at a 6% annual interest rate you can say that the future value of the dollar is $1.06 given a 6% interest rate and a one-year period. It follows that the present value of the $1.06 you expect to receive in one year is only $1.

The time value of money can be used to calculate how much you need to invest now to meet a certain future goal.

Inflation has the reverse effect on the time value of money. Because of the constant decline in the purchasing power of money, an uninvested dollar is worth more in the present than the same uninvested dollar will be in the future.

Time Value of Money (TVM) is an important concept in financial management. It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities.

A key concept of TVM is that a single sum of money or a series of equal, evenly-spaced payments or receipts promised in the future can be converted to an equivalent value today.  Conversely, you can determine the value to which a single sum or a series of future payments will grow to at some future date.

Calculations

You can calculate the fifth value if you are given any four of: Interest Rate, Number of Periods, Payments, Present Value, and Future Value.  Each of these factors is very briefly defined in the right-hand column below.  The left column has references to more detailed explanations, formulas, and examples.

Interest ·   Simple ·   Compound	Interest is a charge for borrowing money, usually stated as a percentage of the amount borrowed over a specific period of time.   Simple interest is computed only on the original amount borrowed. It is the return on that principal for one time period.  In contrast, compound interest is calculated each period on the original amount borrowed plus all unpaid interest accumulated to date.  Compound interest is always assumed in TVM problems.
Number of Periods	Periods are evenly-spaced intervals of time. They are intentionally not stated in years since each interval must correspond to a compounding period for a single amount or a payment period for an annuity.
Payments	Payments are a series of equal, evenly-spaced cash flows.  In TVM applications, payments must represent all outflows (negative amount) or all inflows (positive amount).
Present Value ·   Single Amount ·   Annuity	Present Value is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate.  The future amount can be a single sum that will be received at the end of the last period, as a series of equally-spaced payments (an annuity), or both.  Since money has time value, the present value of a promised future amount is worth less the longer you have to wait to receive it.
Future Value ·   Single Amount ·   Annuity	Future Value is the amount of money that an investment with a fixed, compounded interest rate will grow to by some future date. The investment can be a single sum deposited at the beginning of the first period, a series of equally-spaced payments (an annuity), or both.  Since money has time value, we naturally expect the future value to be greater than the present value. The difference between the two depends on the number of compounding periods involved and the going interest rate.
Loan Amortization	A method for repaying a loan in equal installments. Part of each payment goes toward interest and any remainder is used to reduce the principal. As the balance of the loan is gradually reduced, a progressively larger portion of each payment goes toward reducing principal.
Cash Flow Diagram	A cash flow diagram is a picture of a financial problem that shows all cash inflows and outflows along a time line.  It can help you to visualize a problem and to determine if it can be solved by TVM methods.

Investing For a Single Period:

Suppose you invest $100 in a savings account that pays 10 percent interest per year. How much will you have in one year? You will have $110. This $110 is equal to your original principal of $100 plus $10 in interest. We say that $110 is the future value of $100 invested for one year at 10 percent, meaning that $100 today is worth $110 in one year, given that the interest rate is 10 percent.

In general, if you invest for one period at an interest rate r, your investment will grow to (1 + r) per dollar invested. In our example, r is 10 percent, so your investment grows to 1 + .10 = 1.10 dollars per dollar invested. You invested $100 in this case, so you ended up with $100 x 1.10 = $110.

Investing For More Than One Period:

Consider your $100 investment that has now grown to $110. If you keep that money in the bank, what will you have after two years, assuming the interest rate remains the same?

You will earn $110 x .10 = $11 in interest after the second year, making a total of $100 + $11 = $121. This $121 is the future value of $100 in two years at 10 percent.

Another way of looking at it is that one year from now, you are effectively investing $110 at 10 percent for a year. This is a single-period problem, so you will end up with $1.10 for every dollar invested, or $110 x 1.1 = $121 total.

The process of leaving the initial investment plus any accumulated interest in a bank for more than one period is reinvesting the interest. This process is called compounding. Compounding the interest means earning interest on interest so we call the result compound interest. With simple interest, the interest is not reinvested, so interest is earned each period is on the original principal only.

Solve the problems:

1. Suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of two years? How much of this is simple interest? How much is compound interest?
2. How many years a sum of money would take to double itself if the interest rate applied is 10% p.a compounded annually?

All of the standard calculations for time value money derive from the most basic algebraic expression for the present value of a future sum. The same rate of interest r used to get future value from a present value can be the ‘discount rate’ can be used to convert the future value into present value. For example,

A sum of FV to be received in one year is FV = PV (1 + r)ⁿ , where r is the rate of interest and n is the number of interest periods (if interest is calculated quarterly, there will be 4 periods in a year and if annually, there is 1 period in a year).

And if we have the FV, the present value can be derived as PV = ^FV/_{(1 + r)}ⁿ , where (1 + r) is the discount factor and we get PV by discounting the FV. Also, PV is the value at time = 0 and FV is the value at time = n

The cumulative present value of future cash flows can be calculated by summing the contributions of FV_t, the value of cash flow at time=t

Note that this series can be summed for a given value of n, or when n is ∞. This is a very general formula, which leads to several important special cases given below.

Present value of an annuity (immediate) for n payment periods

In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for:

1. PV(A) is the value of the annuity at time=0
2. A is the value of the individual payments in each compounding period
3. i equals the interest rate that would be compounded for each period of time
4. n is the number of payment periods.

To get the PV of an annuity due, multiply the above equation by (1 + i).

Present value of a growing annuity

In this case each cash flow grows by a factor of (1+g). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of g as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.

Where i ≠ g :

To get the PV of a growing annuity due, multiply the above equation by (1 + i).

Where i = g :

Present value of a perpetuity

When n → ∞, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.

When this is an increasing perpetuity, this i becomes i’ 1+i’=(1+i)/(1+g) i’=(i-g)/(1+g)

so A/i’ = A x (1+g)/(i-g) not (A/(i-g))

Present value of a growing perpetuity

When the perpetual annuity payment grows at a fixed rate (g) the value is theoretically determined according to the following formula. In practice, there are few securities with precisely these characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.

This is the well known Gordon Growth model used for stock valuation.

Future value of a present sum

The future value (FV) formula is similar and uses the same variables.

Future value of an annuity (immediate)

The future value of an annuity (FVA) formula has four variables, each of which can be solved for:

1. FV(A) is the value of the annuity at time = n
2. A is the value of the individual payments in each compounding period
3. i is the interest rate that would be compounded for each period of time
4. n is the number of payment periods

To get the FV of an annuity due, multiply the above equation by (1 + i).

Future value of a growing annuity

The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for:

Where i ≠ g :

Where i = g :

1. FV(A) is the value of the annuity at time = n
2. A is the value of initial payment at time 0
3. i is the interest rate that would be compounded for each period of time
4. g is the growing rate that would be compounded for each period of time
5. n is the number of payment periods

Derivations

Annuity derivation

The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.

A single payment C at future time m has the following future value at future time n:

Summing over all payments from time 1 to time n, then reversing the order of terms and substituting k = n − m:

Note that this is a geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series, we get

The present value of the annuity (PVA) is obtained by simply dividing by (1 + i)ⁿ:

---

Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:

Principal = C/i + goal

Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:

FV = PV(1 + i)ⁿ

Initially, before any payments, the present value of the system is just the endowment principal (PV = C / i). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (FV = C / i + FVA). Plugging this back into the equation:

Perpetuity derivation

Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:

can be seen to approach the value of 1 as n grows larger. At infinity, it is equal to 1,

 leaving    as the only term remaining.

Examples

Example 1: Present value

One hundred euros to be paid 1 year from now, where the expected rate of return is 5% per year, is worth in today's money:

So the present value of €100 one year from now at 5% is €95.23.

Example 2: Present value of an annuity — solving for the payment amount

Consider a 10 year mortgage where the principal amount P is $200,000 and the annual interest rate is 6%.

The number of monthly payments is

and the monthly interest rate is

 

The annuity formula for (A/P) calculates the monthly payment:

Example 3: Solving for the period needed to double money

Consider a deposit of $100 placed at 10% (annual). How many years are needed for the value of the deposit to double to $200?

Using the algrebraic identity that if:

Then  

The present value formula can be rearranged such that:

(years)

This same method can be used to determine the length of time needed to increase a deposit to any particular sum, as long as the interest rate is known. For the period of time needed to double an investment, the Rule of 72 is a useful shortcut that gives a reasonable approximation of the period needed.

Example 4: What return is needed to double money?

Similarly, the present value formula can be rearranged to determine what rate of return is needed to accumulate a given amount from an investment. For example, $100 is invested today and $200 return is expected in five years; what rate of return (interest rate) does this represent?

The present value formula restated in terms of the interest rate is:

see also Rule of 72

Example 5: Calculate the value of a regular savings deposit in the future.

To calculate the future value of a stream of savings deposit in the future requires two steps, or, alternatively, combining the two steps into one large formula. First, calculate the present value of a stream of deposits of $1,000 every year for 20 years earning 7% interest:

This does not sound like very much, but remember - this is future money discounted back to its value today; it is understandably lower. To calculate the future value (at the end of the twenty-year period):

These steps can be combined into a single formula:

Example 6: Price/earnings (P/E) ratio

It is often mentioned that perpetuities, or securities with an indefinitely long maturity, are rare or unrealistic, and particularly those with a growing payment. In fact, many types of assets have characteristics that are similar to perpetuities. Examples might include income-oriented real estate, preferred shares, and even most forms of publicly-traded stocks. Frequently, the terminology may be slightly different, but are based on the fundamentals of time value of money calculations. The application of this methodology is subject to various qualifications or modifications, such as the Gordon growth model.

For example, stocks are commonly noted as trading at a certain P/E ratio. The P/E ratio is easily recognized as a variation on the perpetuity or growing perpetuity formulae - save that the P/E ratio is usually cited as the inverse of the "rate" in the perpetuity formula.

If we substitute for the time being: the price of the stock for the present value; the earnings per share of the stock for the cash annuity; and, the discount rate of the stock for the interest rate, we can see that:

And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).

Of course, stocks may have increasing earnings. The formulation above does not allow for growth in earnings, but to incorporate growth, the formula can be restated as follows:

If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:

Time value of money formulas with continuous compounding

Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:

See below for formulaic equivalents of the time value of money formulæ with continuous compounding.

Present value of an annuity

Present value of a perpetuity

Present value of a growing annuity

Present value of a growing perpetuity

Present value of an annuity with continuous payments

Option Time value

In finance, the value of an option consists of two components, its intrinsic value and its time value. Time value is simply the difference between option value and intrinsic value. Time value is also known as theta, extrinsic value, or instrumental value.

Intrinsic value

Option Value

Intrinsic value is the greater of zero and the difference between the exercise price of the option (strike price, K) and the current value of the underlying instrument (spot price, S); see formulae below. If the option does not have positive monetary value, it is referred to as out-the-money. If an option is out-the-money at expiration, its holder will simply "abandon the option" and it will expire worthless. Because the option owner will never choose to lose money by exercising, an option will never have a value less than zero.

For a call option: value = Max [ (S – K), 0 ]

For a put option: value = Max [ (K – S), 0 ]

As seen on the graph, the call option's intrinsic value begins when the underlying asset's spot price exceeds the option's strike price.

Option value

Option value (i.e. price) is found via a formula such as Black-Scholes or using a numerical method such as the Binomial model. This price will reflect the "likelihood" of the option finishing "in-the-money". For an out-the-money option, the further in the future the expiration date - i.e. the longer the time to exercise - the higher the chance of this occurring, and thus the higher the option price; for an in-the-money option the chance in the money decreases; however the fact that the option cannot have negative value also works in the owner's favor. The sensitivity of the option value to the amount of time to expiry is known as the option's "theta"; see The Greeks. The option value will never be lower than its intrinsic value.

As seen on the graph, the full call option value (intrinsic and time value) is the red line.

Time value

Time value is, as above, the difference between option value and intrinsic value, i.e.

Time Value = Option Value - Intrinsic Value.

More specifically, an option's time value captures the possibility, however remote, that the option may increase in value due to volatility in the underlying asset. Numerically, this value depends on the time until the expiration date and the volatility of the underlying instrument's price. The time value of an option is always positive and declines exponentially with time, reaching zero at the expiration date. At expiration, where the option value is simply its intrinsic value, time value is zero. Prior to expiration, the change in time value with time is non-linear, being a function of the option price.

Discounting

Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The discount, or charge, is simply the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

The discount is usually associated with a discount rate, which is also called the discount yield. The discount yield is simply the proportional share of the initial amount owed (initial liability) that must be paid to delay payment for 1 year.

Discount Yield  =  "Charge" to Delay Payment for 1 year  /  Debt Liability

It is also the rate at which the amount owed must rise to delay payment for 1 year.

Since a person can earn a return on money invested over some period of time, most economic and financial models assume the "Discount Yield" is the same as the Rate of Return the person could receive by investing this money elsewhere (in assets of similar risk) over the given period of time covered by the delay in payment. The Concept is associated with the Opportunity Cost of not having use of the money for the period of time covered by the delay in payment. The relationship between the "Discount Yield" and the Rate of Return on other financial assets is usually discussed in such economic and financial theories involving the inter-relation between various Market Prices, and the achievement of Pareto Optimality through the operations in the Capitalistic Price Mechanism, as well as in the discussion of the "Efficient (Financial) Market Hypothesis". The person delaying the payment of the current Liability is essentially compensating the person to whom he/she owes money for the lost revenue that could be earned from an investment during the time period covered by the delay in payment. Accordingly, it is the relevant "Discount Yield" that determines the "Discount", and not the other way around.

As indicated, the Rate of Return is usually calculated in accordance to an annual return on investment. Since an investor earns a return on the original principle amount of the investment as well as on any prior period Investment income, investment earnings are "compounded" as time advances. Therefore, considering the fact that the "Discount" must match the benefits obtained from a similar Investment Asset, the "Discount Yield" must be used within the same compounding mechanism to negotiate an increase in the size of the "Discount" whenever the time period the payment is delayed is extended. The “Discount Rate” is the rate at which the “Discount” must grow as the delay in payment is extended. This fact is directly tied into the "Time Value of Money" and its calculations.

The "Time Value of Money" indicates there is a difference between the "Future Value" of a payment and the "Present Value" of the same payment. The Rate of Return on investment should be the dominant factor in evaluating the market's assessment of the difference between the "Future Value" and the "Present Value" of a payment; and it is the Market's assessment that counts the most. Therefore, the "Discount Yield", which is predetermined by a related Return on Investment that is found in the financial markets, is what is used within the "Time Value of Money" calculations to determine the "Discount" required to delay payment of a financial liability for a given period of time.

BASIC CALCULATION

If we consider the value of the original payment presently due to be $P, and the debtor wants to delay the payment for t years, then an r% Market Rate of Return on a similar Investment Assets means the "Future Value" of $P is $P * (1 + r%)^t , and the "Discount" would be calculated as

Discount = $P * (1+r%)^t - $P  

where r% is also the "Discount Yield".

If $F is a payment that will be made t years in the future, then the "Present Value" of this Payment, also called the "Discounted Value" of the payment, is

$P = $F / (1+r%)^t   

Example

To calculate the present value of a single cash flow, it is divided by one plus the interest rate for each period of time that will pass. This is expressed mathematically as raising the divisor to the power of the number of units of time.

Consider the task to find the present value PV of $100 that will be received in five years. Or equivalently, which amount of money today will grow to $100 in five years when subject to a constant discount rate?

Assuming a 12% per year interest rate it follows

Discount rate

The discount rate which is used in financial calculations is usually chosen to be equal to the Cost of Capital. The Cost of Capital, in a financial market equilibrium, will be the same as the Market Rate of Return on the financial asset mixture the firm uses to finance capital investment. Some adjustment may be made to the discount rate to take account of risks associated with uncertain cash flows, with other developments.

The discount rates typically applied to different types of companies show significant differences:

• Startups seeking money: 50 – 100 %
• Early Startups: 40 – 60 %
• Late Startups: 30 – 50%
• Mature Companies: 10 – 25%

Reason for high discount rates for startups:

• Reduced marketability of ownerships because stocks are not traded publicly.
• Limited number of investors willing to invest.
• Startups face high risks.
• Over optimistic forecasts by enthusiastic founders.

One method that looks into a correct discount rate is the capital asset pricing model. This model takes in account three variables that make up the discount rate:

1.      Risk Free Rate: The percentage of return generated by investing in risk free securities such as government bonds.

2.      Beta: The measurement of how a company’s stock price reacts to a change in the market. A beta higher than 1 means that a change in share price is exaggerated compared to the rest of shares in the same market. A beta less than 1 means that the share is stable and not very responsive to changes in the market. Less than 0 means that a share is moving in the opposite of the market change.

3.      Equity Market Risk Premium: The return on investment that investors require above the risk free rate.

Discount rate= risk free rate + beta*(equity market risk premium)

Discount factor

The discount factor, P(T), is the number which a future cash flow, to be received at time T, must be multiplied by in order to obtain the current present value. Thus, a fixed annually compounded discount rate is

For fixed continuously compounded discount rate we have