Time Value of Money

Time Value of Money Definition

Time Value of Money (TVM) means that money received in present is of higher worth than money to be received in the future as money received now can be invested and it can generate cash flows to enterprise in future in the way of interest or from investment appreciation in the future and from reinvestment.

The Time Value of Money is also referred to as the Present Discounted value. Money deposited in a savings bank account earns a certain interest rate to compensate for keeping the money away from them at the current point of time. Hence, if a bank holder deposits $100 in the account, the expectation will be to receive more than $100 after one year.

Understanding Time Value of Money

Time Value of Money is a concept that recognizes the relevant worth of future cash flows arising as a result of financial decisions by considering the opportunity cost of the funds. Since money tends to lose value over time, there is inflation, which reduces the buying power of money. However, the cost of receiving money in the future rather than now shall be greater than just the loss in its real value on account of inflation. The opportunity cost of not having the money right now also includes the loss of additional income, which could be earned by simply having possession of cash earlier.

Moreover, receiving money in the future rather than now may involve some risk and uncertainty regarding its recovery. For these reasons, future cash flows are worth less than the present cash flows.

Time Value of Money Formula

The formula to calculate time value of money (TVMTVMThe Time Value of Money (TVM) principle states that money received in the present is of higher worth than money received in the future because money received now can be invested and used to generate cash flows to the enterprise in the future in the form of interest or from future investment appreciation and reinvestment.read more) either discounts the future value of money to present value or compounds the present value of money to future value. FV = PV * (1 + i/n )^n*t  or    PV = FV / (1 + i/n )^n*t

• FV = Future value of money,
• PV = Present value of money,
• i = Rate of interest or current yieldCurrent YieldThe current yield formula essentially calculates the yield on a bond based on the market price instead of face value. The current yield of bond= Annual coupon payment/current market priceread more on similar investment,
• t = Number of years and
• n = Number of compounding periods of interest per year

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Time Value of Money Calculations

Top 6 Time Value of Money Concepts

#1 – Future Value of A Single Amount

The first one in the time value of money concept that we discuss is to calculate the future value of a single amount.

Suppose one invests $1,000 for 3 years in a Savings account, which pays 10% interest per year. If one allows the interest incomeInterest IncomeInterest Income is the amount of revenue generated by interest-yielding investments like certificates of deposit, savings accounts, or other investments & it is reported in the Company’s income statement. read more to be reinvested, the investment shall grow as follows:

Future Value at the End of First Year

• Principal at the beginning of the year                                      $1,000
• Interest for the year ($1,000 * 0.10)                                         $100
• Principal at the end                                                                     $1,100

Future Value at the End of Second Year

• Principal at the beginning of the year                                      $1,100
• Interest for the year ($1,100 * 0.10)                                          $110
• Principal at the end                                                                     $1,210

The process of investing money and reinvestingReinvestingReinvestment is the process of investing the returns received from investment in dividends, interests, or cash rewards to purchase additional shares and reinvesting the gains. Investors do not opt for cash benefits as they are reinvesting their profits in their portfolio.read more the interest earned is called Compounding. The future value or compounded value of an investment after “n” year when the interest rate is “r” % is:

Time value of Money Formula = FV = PV (1+r) ⁿ

As per the above equation, (1+r) ⁿ is called the future value factor. There are pre-defined tables that specify the rate of interest and its value after ‘n’ number of years. It can also be utilized with the help of a calculator or an excel spreadsheet as well. The below snapshot is an instance of how the rate is calculated for different interest rates and at different time intervals.

Hence, taking the above instance, the FV of $1,000 can be used as:

FV = 1000 (1.210) = $1210

#2 – Time Value of Money: Doubling Period

The first important aspect of the time value of money (TVM) concept is the doubling period.

Investors are generally keen to know by when their investment can double up at a given Interest. Though a little crude, an established rule is the “Rule of 72,” which states that the doubling period can be obtained by dividing 72 by the interest rate.

For e.g., if the interest is 8%, the doubling period is 9 years [72/8=9 years].

A slightly more calculative rule is the “Rule of 69Rule Of 69The Rule of 69 is a common rule for estimating the time it will take to double an investment with a continuous compounding interest rate. It does not provide an exact time, but it does provide a near approximation without relying on a mathematical formula.read more” which states the doubling period as 0.35 + 69/Interest

#3 – Present Value of A Single Amount

The third important point in the time value of money (TVM) concept is to find the present value of a single amount.

This scenario states the Present Value of a sum of money, which is expected to be received after a given time period. The process of discounting used for computation of the present value is simply the inverse of compounding. The PV formula can be readily obtained by using the below formula:

Time Value of Money Formula = PV = FV _n [1 / (1+r) ⁿ]

For instance, if a client is expected to receive $1,000 after 3 years @ 8% ROI, its value at the Present time can be calculated as:

PV = 1000 [1/1.08]³

PV = 1000*0.794 = $794

#4 – Future Value of An Annuity

The fourth important concept in the time value of money (TVM) concept is to calculate the future value of an annuity.

An annuity is a stream of constant cash flows (receipts or payments) occurring at regular time intervals. The premium payments of a life insurance policy, for instance, are an annuity. When the cash flows occur at the end of each period, the annuity is called an Ordinary annuityOrdinary AnnuityAn ordinary annuity refers to recurring payments of equal value made at regular intervals for a fixed period. The frequency of these consecutive payments can be weekly, monthly, quarterly, half-yearly or yearly.read more or deferred annuity. When this flow occurs at the beginning of each period, it is called Annuity due. The formula for an annuity dueAnnuity DueAnnuity Due can be defined as those payments which are required to be made at the start of each annuity period instead of the end of the period. The payments are generally fixed and there are two values for an annuity, one would be future value, and another would be present value.read more is simply (1+r) times the formula for corresponding ordinary annuity. Our focus will be more on the deferred annuity.

Let’s take an example whereby one deposits $1,000 annually in a bank for 5 years, and the deposit is earning compound interestCompound InterestCompound Interest is the interest earned from the initial Principal & the previously accumulated Interest amount. This is also known as “Interest on Interest” & it is always higher than the Simple Interest. read more at 10% ROI, the value of the series of deposits at the end of 5 years:

Future Value = $1,000(1+1.10)⁴ + $1,000(1+1.10)³ + $1,000(1+1.10)² + $1,000(1.10) + $1,000 = $6,105

In general terms, the future value of the annuity is given by the following formula:

• FVA _n = A [(1+r) ⁿ – 1] / r
• FVA _n is the FV of annuity having duration of ‘n’ periods, ‘A’ is the constant periodic flow, and ‘r’ is the ROI per period. The term [(1+r) ⁿ – 1] / r is referred to as the future value interest factor for an annuity.

#5 – Present Value of Annuity

The fifth important concept in the time value of money concept is to calculate the present value of an annuity.

This concept is a reversal of the future value of annuityFuture Value Of AnnuityThe future value of annuity due is the amount to be received in future where each payment is made at the beginning of each period. FVA due = P * [(1 + r)n – 1] * (1 + r)/rread more just instead of FV; the focus will be on PV. Suppose one is expecting to receive $1,000 annually for 3 years with each receipt occurring at the end of the year, the PV of this stream of benefits at the discount rate of 10% would be calculated as below:

$1,000[1/1.10] + 1,000 [1/1.10]² + 1,000 [1/1.10]³ = $2,486.80

In general terms, the present value of an annuity can be expressed as follows:

• A = [{1 – (1/1 + r) ⁿ} / r]

#6 – Present Value of Perpetuity

The sixth concept in the time value of money (TVM) is to find the present value of a perpetuity.

Perpetuity is an annuity of indefinite duration. For instance, the British government has issued bonds called ‘consols,’ which pay yearly interest throughout its existence. Although the total face value of the perpetuity is infinite and undeterminable, its Present value is not. According to the Time Value of Money (TVM) principle, the Present Value of the perpetuity is the sum of the discounted value of each periodic payment of the perpetuity. The formula for computing the present value of the perpetuityPresent Value Of The PerpetuityPerpetuity can be defined as the income stream that the individual gets for an infinite time. Its present value is derived by discounting the identical cash flows with the discounting rate. Here the cash flows are endless, but its current value amounts to a limited value.read more is:

Fixed periodic payment / ROI or the discount rate per compounding period

For e.g., calculating the PV on Jan 1, 2015, of a perpetuity paying $1,000 at the end of each month starting from January 2015 with a monthly discount rate of 0.*8% can be shown as :

• PV = $1,000 / 0.8% = $125,000

Growing Perpetuity

This is a scenario in which the perpetuity will keep on changing, like Rental payments. For e.g., an office complex is expected to generate a net rental of $3 million for the forthcoming year, which is expected to increase by 5% every year. If we assume that the increase will continue indefinitely, the rental system will be termed as growing perpetuity. If the discount rate is 10%, the PV of the rental stream will be:

In an algebraic formula, it can be displayed as follows,

• PV = C / r-g, where ‘C’ is the rental to be received during the year, ‘r’ is the ROI and ‘g’ is the growth rate.

Time Value of Money – Intra-Year Compounding & Discounting

In this case, we consider the case where compounding is done on a frequent basis. Assuming a client deposit $1,000 with a finance company that pays 12% interest on a semi-annual basis, which indicates that the interest amount is paid every 6 months. The deposit amount will grow as follows:

• First six months: Principal at the beginning = $1,000
• Interest for 6 months        = $60   ($1,000 * 12%) /2
• Principal at the end            = $1,000 + $60 = $1,060

Next six months: Principal at the beginning = $1,060

• Interest for 6 months        = $63.6   ($1,060 * 12%) /2
• Principal at the end            = $1,060 + $63.6 = $1,123.6

It is to be noted that if the compounding is done annually, the principal at the end of one year would be $1,000 * 1.12 = $1,120. The difference of $3.6 (between $1,123.6 under semi-annual compounding and $1,120 under annual compounding) represents interest on interest for the second half-year.

Time Value of Money Examples

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Example #1 – Dividend Discount Model

This is a Time value of money real-life example of its usage in valuations using the Dividend Discount Model.

The dividend discount model prices a stock by adding its future cash flowsCash FlowsCash Flow is the amount of cash or cash equivalent generated & consumed by a Company over a given period. It proves to be a prerequisite for analyzing the business’s strength, profitability, & scope for betterment. read more discounted by the required rate of return that an investor demands for the risk of owning the stock.

Here the CF = Dividends.

However, this situation is a bit theoretical, as investors normally invest in stocks for dividends as well as capital appreciationCapital AppreciationCapital appreciation refers to an increase in the market value of assets relative to their purchase price over a specified time period. Stocks, land, buildings, fixed assets, and other types of owned property are examples of assets.read more. Capital appreciation is when you sell the stock at a higher price then you buy for. In such a case, there are two cash flows –

1. Future Dividend Payments
2. Future Selling Price

Intrinsic Value = Sum of Present Value of Dividends + Present Value of Stock Sale Price

This DDM price is the intrinsic value of the stock.

Let us take an example of a Dividend Discount Model DDM here.

Assume that you are considering the purchase of a stock which will pay dividends of $20 (Div 1) next year and $21.6 (Div 2) the following year.  After receiving the second dividend, you plan on selling the stock for $333.3. What is the intrinsic value of this stock if your required return is 15%? 

This problem can be solved in 3 steps –

Step 1 – Find the present value of Dividends for Year 1 and Year 2.

• PV (year 1) = $20/((1.15)^1)
• PV(year 2) = $20/((1.15)^2)
• In this example, they come out to be $17.4 and $16.3, respectively, for 1st and 2nd-year dividends.

Step 2 – Find the Present value of future selling price after two years.

• PV(Selling Price) = $333.3 / (1.15^2)

Step 3 – Add the Present Value of Dividends and the present value of Selling Price

• $17.4 + $16.3 + $252.0 = $285.8

Example #2 – Loan EMI Calculator

A loan is issued at the beginning of year 1.  The principal is $15,000,000, the interest rate is 10%, and the term is 60 months. Repayments are to be made at the end of each month. The loan must be fully repaid by the end of the term.

• Principal –  $15,000,000
• Interest Rate (monthly) – 1%
• Term = 60 months

To find the Equal Monthly Installment or EMI, we can use the PMT function in ExcelPMT Function In ExcelPMT function is an advanced financial function to calculate the monthly payment against the simple loan amount. You have to provide basic information, including loan amount, interest rate, and duration of payment, and the function will calculate the payment as a result.read more. It requires Principal, Interest, and term as inputs.

EMI = $33,367 per month

Example #3 – Alibaba Valuation

Let us see how the Time Value of Money (TVM) concept was applied for valuing Alibaba IPO. For Alibaba’s valuation, I had done the financial statement analysis and forecast financial statements and then calculated the Free Cash Flow to the Firm. You can download Alibaba Financial Model here

Presented below is the Free Cash Flow to the Firm of Alibaba. The Free Cash flow is divided into two parts – a) Historical FCFF and b) Forecast FCFF

• Historical FCFF is arrived at from the Income StatementIncome StatementThe income statement is one of the company's financial reports that summarizes all of the company's revenues and expenses over time in order to determine the company's profit or loss and measure its business activity over time based on user requirements.read more, Balance Sheet Balance Sheet A balance sheet is one of the financial statements of a company that presents the shareholders' equity, liabilities, and assets of the company at a specific point in time. It is based on the accounting equation that states that the sum of the total liabilities and the owner's capital equals the total assets of the company.read moreand Cash Flows of the company from its Annual Reports
• Forecast FCFF is calculated only after forecasting the Financial StatementsFinancial StatementsFinancial statements are written reports prepared by a company's management to present the company's financial affairs over a given period (quarter, six monthly or yearly). These statements, which include the Balance Sheet, Income Statement, Cash Flows, and Shareholders Equity Statement, must be prepared in accordance with prescribed and standardized accounting standards to ensure uniformity in reporting at all levels.read more (we call this as preparing the Financial Model in excel). Core Financial ModelingCore Financial ModelingFinancial modeling refers to the use of excel-based models to reflect a company's projected financial performance. Such models represent the financial situation by taking into account risks and future assumptions, which are critical for making significant decisions in the future, such as raising capital or valuing a business, and interpreting their impact.read more is slightly tricky, and I will not discuss the details and types of Financial Models in this article.
• In order to find the valuation of Alibaba, we must find the present value of all the future financial years (till perpetuity – Terminal valueTerminal ValueTerminal Value is the value of a project at a stage beyond which it's present value cannot be calculated. This value is the permanent value from there onwards. read more)
• For a complete analysis, you can refer to this detailed note – Alibaba Valuation ModelAlibaba Valuation ModelAlibaba is the most profitable Chinese e-commerce company and its IPO is a big deal due to its size. With its huge size and network, Alibaba IPO may look at international expansion beyond China and may lead to price wars and intensive competition in the US.read more.

Conclusion

The Time Value of Money concept attempts to incorporate the above considerations into financial decisions by facilitating an objective evaluation of cash flows from different time periods by converting them into present value or future value equivalents. This will only attempt to neutralize the present and future value of money and arrive at smooth financial decisions.

Time Value of Money Video

Recommended Articles

This has been a guide to what is Time Value of Money and its meaning. Here we discuss how to calculate the time value of money using its formula, concept and examples. You may learn more about financing from the following articles –

• Key Differences – Time vs Money
• TimeValue in VBA
• Present Value Example
• Examples of Bond Yield Formula