Financial Modeling Tutorials

- Excel Modeling
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- Time Value of Money
- Future Value Formula
- Present Value Factor
- Perpetuity Formula
- Present Value vs Future Value
- Annuity vs Pension
- Present Value of an Annuity
- Doubling Time Formula
- Annuity Formula
- Present Value of an Annuity Formula
- Future Value of Annuity Due Formula
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- Annuity vs Perpetuity
- Annuity vs Lump Sum
- Deferred Annuity Formula
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- Discount Rate vs Interest Rate
- Discounting Formula
- Rule of 72
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- Mean Examples
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- APR vs APY
- Mortgage APR vs Interest Rate
- Normal Distribution Formula
- Standard Normal Distribution Formula
- Normalization Formula
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- F-Test Formula
- Quantitative Research
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The **Time Value of Money** concept indicates that money earned today will be more than its intrinsic value in the near future. This is due to the potential earning capacity of the given amount of money. Time Value of Money (TVM) is also referred to as 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.

In this article, we discuss Time Value Concepts in Detail –

- #1 – Future Value of A Single Amount
- #2 – Time Value of Money: Doubling Period
- #3 – Present Value of A Single Amount
- #4 – Future Value of An Annuity
- #5 – Present Value of Annuity
- #6 – Present Value of Perpetuity
- Time Value of Money – Intra-Year Compounding & Discounting
- Time Value of Money Example #1 – Dividend Discount Model
- Time Value of Money Example #2 – Loan EMI Calculator
- Time Value of Money Example #3 – Alibaba Valuation
- Conclusion – Time Value of Money

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## #1 – Future Value of A Single Amount

The first one in time value of money concept that we discuss is to calculate 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 income 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 reinvesting the interest earned is called as Compounding. The future value or compounded value of an investment after *“n”* year when the interest rate is *“r” *% is:

FV = PV (1+r) ^{n}

As per the above equation, (1+r) ^{n }is called the future value factor. There are pre-defined tables which 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 spread sheet as well. The below snapshot is an instance of how the rate is computed 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 in 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 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 69” 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:

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 computed 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.

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An annuity is a stream of constant cash flows (receipts or payments) occurring at regular time interval. 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 annuity or deferred annuity. When this flow occurs at the beginning of each period, it is called as Annuity due. The formula for an annuity due 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 a compound interest 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)*is referred as future value interest factor for an annuity.^{n}– 1] / r

## #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 annuity 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 computed 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 an 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 perpetuity 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 a 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 deposits $1,000 with a finance company which pays 12% interest on a semi-annual basis which indicates that 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 Example #1 – Dividend Discount Model

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

Dividend discount model prices a stock by adding its future cash flows 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 appreciation. 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 –

- Future Dividend Payments
- 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 Dividend Discount Model 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 dividend.

**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 present value of Selling Price**

- $17.4 + $16.3 + $252.0 = $285.8

## Time Value of Money 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 Excel. It requires Principal, Interest and term as inputs.

EMI = $33,367 per month

## Time Value of Money Example #3 – Alibaba Valuation

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

Presented below is the Free Cash Flow to 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 Statement, Balance Sheet and Cash Flows of the company from its Annual Reports
- Forecast FCFF is calculated only after forecasting the Financial Statements (we call this as preparing the Financial Model) Core Financial Modeling 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 value)
- For complete analysis, you can refer to this detailed note – Alibaba Valuation Model

## Conclusion – Time Value of Money

Time Value of Money (TVM) is a concept which 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 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 neutralise the present and future value of money and arrive at smooth financial decisions.

### Time Value of Money Video

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