<p>This reading provides an introduction to the concepts and techniques of time value of money. These techniques are a fundamental tool for financial analysis, corporate finance, valuation, and other investment decisions.</p>

The Time Value of Money

Let’s say that you are given a choice to receive $100 today or $100 one year from now. Which choice will you prefer? The more likely answer is that you will want to receive $100 today. You could purchase something with that $100 today or you could deposit it in a savings account. If you deposit it in a savings account or any other form of investment, what you will get after one year is likely to be more than the $100 that you started with. This means that money is more valuable today than it is tomorrow or after one year.

Another important principle inherent in the time value of money concept is that your investments earn compound interest. The investments that you make will not only earn interest on the original principal but will also earn interest on the interest that has been accumulated over the period.

Your objective in this reading should be to be able to solve the time value problems as quickly as possible using the financial calculator prescribed by CFA Institute. You may be asked to calculate the present value or future value of the cash flows arising from different types of investments.

<strong>Use of Calculator</strong>

<em>For the CFA exam, there are only <a href="https://financetrain.com/cfa-calculator-financial-calculator-for-cfa-exam/" target="_blank" rel="noopener noreferrer">two calculators allowed by the CFA Institute</a>. The candidates can buy either of these and must carry them to the Exam Centre on exam day. The two calculators are:</em>
<ul>
 	<li><em>Texas Instruments BA II Plus Financial Calculator (including BA II Plus Professional)</em></li>
 	<li><em>HP 12C Financial Calculator (including the HP 12C Platinum, 12C Platinum 25th anniversary edition, 12C 30th anniversary edition, and HP 12C Prestige)</em></li>
</ul>
<em>In this reading we will demonstrate the use of BA II Plus calculator for solving time value of money problems.</em>

Introduction - Time Value of Money

<p>Interest rates are how we measure the time value of money. While making an investment, an investor will need to know the interest rate that the investment will earn. The interest rates can be interpreted in many ways.</p>



<h3><strong>Required Rate of Return</strong></h3>



<p>Required rate of return is the minimum return that an investor demands for a specific asset based on its riskiness. This is the minimum interest rate at which investors will be willing to invest or lenders will be willing to lend their money.</p>



<h3><strong>Opportunity Cost &nbsp;</strong></h3>



<p>The required rate of return also reflects the opportunity cost of forgoing the next best investment. Opportunity cost is what a person sacrifices when he chooses one option over the other. Say you decided to spend the money (current consumption). If investing that money instead of consuming it earned you an interest rate of 6%, then 6% is the opportunity cost.</p>



<h3><strong>Discount Rate</strong></h3>



<p>The interest rates are also referred to as the discount rates and are used to calculate the present value of future cash flows. If you are expecting to receive $1,000 after one year, you will use the discount rate to calculate the present dollar equivalent of that future payment. Generally, a single discount rate is used for all future period cash flows.</p>



<p>When calculating the intrinsic value of a stock, the investor will apply a discount rate that is based on the risk-free rate of return plus some equity risk premium.</p>

Interest Rates

<p>The required interest rate that an investor earns from an investment is made up of various components. The general interest rate equation is expressed below:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/interest-rate-2.gif" alt="interest-rate-2" class="wp-image-12265"/></figure></div>



<p>The nominal risk-free rate itself is expressed as the sum of real-risk free rate and inflation premium.</p>



<p>It is important to understand the difference between the nominal and real risk-free interest rates. <a title="Nominal Interest Rate and Effective Yield" href="https://financetrain.com/nominal-interest-rate-and-effective-yield/">Nominal interest rates</a> are what we observe everyday as published by banks and other financial institutions. For example, when a T-bill pays 6% interest that is the nominal risk-free interest rate. It already includes the premium for expected inflation. On the other hand, real interest rates take purchasing power parity into consideration. That’s the rate, which tells how much more you will be able to buy with your grown investment after one year.</p>



<p>So, the minimum that you expect from a risk-free investment is the nominal risk-free rate. Apart from that you will also expect a risk premium for various types of additional risks that you take by investment in a particular asset or security.</p>



<p>These risks come in many forms and a premium for each should be added to the risk-free rate to arrive at the required rate of return from an investment. The important types of risk are:</p>



<ul><li><b>Default Risk:</b> The risk that the borrower will be unable to meet its payment obligations or keep up with the terms of payments.</li><li><b>Liquidity Risk:</b> The risk that the investment is less liquid and that the investor may have to sell the bond at a price lower than the expected price.</li><li><b>Maturity Risk:</b> The longer the maturity of a bond, the more volatile will be its price.</li></ul>



<p>A premium for each of these risks is added to the nominal risk-free rate (such as T-bill rate) to arrive at the required rate of return. We can rewrite the interest rate equation as follows:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/interest-rate-1.gif" alt="interest-rate-1"/></figure></div>

Interest Rate Equation

<p>When you go to a bank enquiring about the deposit rates, the rates specified by the bank can be expressed in two ways: nominal interest rate, and the effective annual yield.</p>



<p>The difference between the two is that the nominal rate does not take the compounding into consideration, while the effective annual yields take the effect of compounding into consideration.</p>



<p>Consider an investment of $100 at a nominal rate of 10% compounded monthly.</p>



<p>The future value of the investment will be:</p>



<div class="wp-block-image"><figure class="aligncenter"><a href="https://financetrain.com/wp-content/uploads/2010/06/p_175.gif"><img src="https://financetrain.com/wp-content/uploads/2010/06/p_175.gif" alt="" class="wp-image-176" title="p_175"/></a></figure></div>



<p>The effective yield will be the absolute increase as a percentage of the principal invested.</p>



<p>Therefore, the effective annual yield will be:</p>



<p>EY = 10.4713/100 or 10.4713%</p>



<p>Since the effective yield considers compounding effect, it will always be greater than nominal yield.</p>



<p>The effective yield can be calculated using the following formula:</p>



<div class="wp-block-image"><figure class="aligncenter"><a href="https://financetrain.com/wp-content/uploads/2010/06/p_175b.gif"><img src="https://financetrain.com/wp-content/uploads/2010/06/p_175b.gif" alt="" class="wp-image-177" title="p_175b"/></a></figure></div>



<h4>Why Calculate Effective Annual Yield?</h4>



<p>Effective yield is useful when you are considering various investment options where the interest rates are expressed at different compounding rates. In such a situation, you can convert all the rates into effective annual yields and then make an informed decision.</p>



<p>Form example, assume that you have a choice between investing in a bond A offering a nominal interest rate of 5% compounded semiannually, and another bond B offering a nominal interest rate of 4.9% compounded monthly. Since the compounding periods are different, a direct comparison is difficult. Therefore, will calculate the effective annual yields for both the bonds:</p>



<p>EY (Bond A) = (1+0.05/2)^2 – 1 = 5.0625%</p>



<p>EY (Bond B) = (1+0.049/12)^12 – 1 = 5.0848%</p>



<p>We can see that the effective yield for Bond B is higher, so, that’s a better investment.</p>



<p>The nominal interest is also know as Annual Percentage Rate (APR). The Effective Annual Yield is also known as Annual Percentage Yield (APY).</p>

Nominal Interest Rate and Effective Yield

<p>Let’s first review the time value money concept using a very simple example.</p>



<h3>Example 1</h3>



<p>Let’s say you have $2000 to invest. You decide to invest it for 3 years in an account that pays you an interest of 6% per annum. How much will your investment grow to in 3 years?</p>



<p>We are calculating the future value of an investment after 3 years. This will be calculated as follows:</p>



<p>FV = $2000*(1.06)<sup>3</sup> = $2382.03</p>



<h3>Example 2</h3>



<p>Your target is to have $10,000 saved in your account in 5 years. How much money should you invest now to reach your target in 5 years when your investment account earns you 8% per annum?</p>



<p>We are calculating the present value of a future cash flow. This will be calculated as follows:</p>



<p>PV = $10,000/(1.08)^5 = $6805.83</p>



<p>This means that if you invest 46805.83 now for 5 years at 8% interest rate per annum, you will receive $10,000 at the end of 5 years.</p>



<p>A common assumption in both the above problems was that the frequency of compounding was annual. That is, the interest is compounded only annually. However, this is not always the case. The frequency of compounding could be anything, most commonly being, monthly, quarterly, semi-annually, or annually. Let’s look at how our future value and present value will change if we use a different frequency of compounding.</p>



<h3>Example 1 (With Quarterly Compounding)</h3>



<p>In our first example, if the compounding frequency was quarterly, then how much will our investment grow to?</p>



<p><b>Step 1: Calculate the quarterly rate</b></p>



<p>Quarterly rate = 6%/4 = 1.5%</p>



<p><b>Step 2: Calculate number of compounding periods</b></p>



<p>Compounding periods = 3 years * 4 = 12 periods</p>



<p><b>Step 3: Calculate Future Value</b></p>



<p>FV = $2,000*(1.015)^12 = $2391.24</p>



<p>As you can see, the future value based quarterly compounding is more than future value based on annual compounding.</p>



<p>Note that we could also calculate the <a title="Nominal Interest Rate and Effective Yield" href="https://financetrain.com/nominal-interest-rate-and-effective-yield/">effective annual yield</a> and then calculate future value as shown below:</p>



<p>EAY = (1.015)^4 – 1 = 6.13635%</p>



<p>FV = $2,000(1.0613635)^3 -1 = $2,391.24</p>



<p>Note that both the methods produce same results.</p>



<h3>Example 2 (With Monthly Compounding)</h3>



<p>In our second example, if the compounding frequency was monthly, how much should we invest now to reach our target of $10,000 in 5 years with an annual interest rate of 8%?</p>



<p>The monthly rate is 8%/12 = 0.667% and the number of compounding periods is 5*12 = 60.</p>



<p>PV = $10,000/(1.00667)^60 = $6710.77</p>



<p>As you can see, with monthly compounding we need to invest less to reach our target.</p>

Time Value of Money for Different Compounding Frequencies

<p>Future value of a single cash flow refers to how much a single cash flow today would grow to over a period of time if put in an investment that pays compound interest.</p>



<p>The formula for calculating future value is:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/fv1.gif" alt="fv1" class="wp-image-12283"/></figure></div>



<p><b>Example</b></p>



<p>Calculate the future value (FV) of an investment of $500 for a period of 3 years that pays an interest rate of 6% compounded semi-annually.</p>



<p>FV = 500*(1+6%/2)^(2*3) = $597.03</p>



<p>We can also solve this problem using the calculator as follows:</p>



<p><b>Calculator Variables</b></p>



<p>The BA II Plus calculator has the following five variables for Time Value of Money (TVM) functions.</p>



<p>N = Number of Periods (mT in our formula)</p>



<p>I/Y = Interest Rate Per Year (r)</p>



<p>PV = Present Value</p>



<p>FV = Future Value</p>



<p>PMT = Payment</p>



<p>The calculations are simple; you input the values that you know, and calculate the unknown.</p>



<p>To assign a value to a TVM variable, key in the number and press a TVM key – N, I/Y, PV, PMT, and FV.</p>



<p>To change the number of payments (P/Y) press 2nd, key in the number and press ENTER.</p>



<p>To calculate the unknown value, press Compute (CPT) and then press the key for unknown variable.</p>



<p>In our above example, enter PV = 500, change P/Y = 2 (semi-annual compounding), I/Y = 6, N = 6.</p>



<p>Then press CPT &gt; FV. We get:</p>



<p>FV = 597.026</p>



<p>Similarly we can calculate the Future Value for any compounding frequency.</p>

Future Value of a Single Cash Flow

<p>Present value of a single cash flow refers to how much a single cash flow in the future will be worth today. The present value is calculated by discounting the future cash flow for the given time period at a specified discount rate.</p>



<p>The formula for calculating future value is:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/pv1.gif" alt="pv1" class="wp-image-12291"/></figure></div>



<p><b>Example</b></p>



<p>Calculate the present value (FV) of a payment of $500 to be received after 3 years assuming a discount rate of 6% compounded semi-annually.</p>



<p>FV = 500/((1+6%/2)^(2*3)) = $418.74</p>



<p>We can also solve this problem using the calculator as follows:</p>



<p>In our above example, enter FV = 500, change P/Y = 2 (semi-annual compounding), I/Y = 6, N = 6.</p>



<p>Then press CPT &gt; PV. We get:</p>



<p>FV = 418.74</p>

Present Value of a Single Cash Flow

<p>An annuity refers to a series of equal cash flows that occur periodically such as monthly, quarterly or annually. For example, an investment that gives you fixed monthly payments for a specified period. There are two types of annuities, namely, ordinary annuities and annuities due. .</p>



<p>In an ordinary annuity, the first cash flow occurs at the end of the first period, and in an annuity due, the first cash flow occurs at the beginning (at time 0).</p>



<p>The present value and future values of these annuities can be calculated using a simple formula or using the calculator.</p>



<h3><b>Future Value of an Ordinary Annuity</b></h3>



<p>Let’s say we have an ordinary annuity that pays $500 every year for the next 5 years. The expected rate of return is 8%. The future value of this annuity can be represented as follows:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/ordinary-annuity.gif" alt="ordinary-annuity" class="wp-image-12295"/></figure></div>



<p>This can be calculated using the following formula:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/fv-oa.gif" alt="fv-oa" class="wp-image-12296"/></figure></div>



<p>While you can use the above formula to calculate the future value of annuity, you can simply calculate the future value using the BAII Plus calculator. Note that in our example, m = 1, since the compounding frequency is 1.</p>



<p><b>Calculator usage</b></p>



<p>Enter PMT = $500, N = 5, I/Y = 8%.</p>



<p>Since compounding frequency is 1, set Number of Compounding Periods (C/Y) to 1by pressing [2nd][P/Y][Down Arrow].</p>



<p>Since it’s an ordinary annuity, we should set End-of-period payments [END]. This can be set by pressing the key [2nd][BGN]</p>



<p>To compute the future value, press the key CPT &gt; FV</p>



<p>FV = $2933.2</p>



<h3><b>Present Value of an Ordinary Annuity</b></h3>



<p>Calculate the present value of an ordinary annuity that pays $500 at the end of each year for the next 5 years. The discount rate is 8%.</p>



<p>This can be calculated using the TVM functions of BAII Plus calculator as follows:</p>



<p>PMT = 500</p>



<p>N = 5</p>



<p>I/Y = 8%</p>



<p>To compute present value, press the key CPT &gt; PV.</p>



<p>PV = 1996.355</p>



<p>Without the calculator, you would calculate this as follows:</p>



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/pv-oa.gif" alt="pv-oa" class="wp-image-12297"/></figure></div>



<p>PV = 1996.355</p>

Future Value and Present Value of Ordinary Annuity

In an annuity due, the first cash flow occurs at the beginning (at time 0). We can use our BA II Plus calculator to calculate the present value and future value of the annuity due using the same procedure as above, just by making one minor adjustment.

<em>By default the payment period in the calculator is set to END (End-of-period payments). However, for annuity due, the payment occurs at the beginning of the period. So, we need to change the payment period to the beginning BGN (Beginning-of-period payments). To make this change, follow the following steps:</em>

<em>On your calculator, press: [2ND][BGN][2ND][SET]</em>

<em>The mode is now changed to BGN and you will see BGN displayed on the upper right corner of the display.</em>

<em>To switch back to END mode, repeat the above steps. Once the mode is changed, BGN will disappear from the screen.</em>

We will use the same examples as we used for ordinary annuity and calculate the PV and FV of the annuity due.
<h3><strong>Future Value of an Annuity Due</strong></h3>
An annuity due pays $500 every year for the next 5 years. The expected rate of return is 8%. The future value of this annuity can be calculated as follows:
<table>
<tbody>
<tr>
<td width="616"><strong>&nbsp;</strong><strong>Calculator Usage</strong></td>
</tr>
<tr>
<td width="616">&nbsp;Since it’s an annuity due, we should set payment period to beginning-of-period payments [BGN].

Enter PMT = $500, N = 5, I/Y = 8%.

Since compounding frequency is 1, set Number of Compounding Periods (C/Y) to 1 by pressing [2nd][P/Y][Down Arrow].

To compute the future value, press the key CPT &gt; FV

FV = $3167.965</td>
</tr>
</tbody>
</table>
<h3><strong>Present Value of an Annuity Due</strong></h3>
Calculate the present value of an ordinary annuity that pays $500 at the end of each year for the next 5 years. The discount rate is 8%.
<table>
<tbody>
<tr>
<td width="616"><strong>&nbsp;</strong><strong>Calculator Usage</strong></td>
</tr>
<tr>
<td width="616">&nbsp;This can be calculated using the TVM functions of BA II Plus calculator as follows:

PMT = 500

N = 5

I/Y = 8%

To compute present value, press the key CPT &gt; PV.

PV = $2,156.063</td>
</tr>
</tbody>
</table>

Present Value and Future Value of Annuity Due

A perpetuity is a type of annuity that pays equal cash flows that occur periodically such as monthly, quarterly or annually for an infinite period of time.

The present value of an annuity is calculated using the following formula:

PV = A/r

Where, A is the annuity payment, and r is the interest rate.

Assume that an perpetuity pays $500 per year. The rate of return is 8%. The present value of this perpetuity is calculated as follows:

PV = 500/0.08 = $6,250

If the investor invests $6,250 in the perpetuity paying 8% rate of return, he will receive a payment of $500 per year for an infinite period.

Present Value of a Perpetuity

We have looked at the PV/FV calculations for single sums of money and for annuities in which all the cash flows are equal. However, there may be an investment where the cash flows are not equal. We will now look at how to calculate the PV and FV of such an uneven series of cash flows.

Look at the following cash flows:

<img class="aligncenter size-full wp-image-18826" src="https://financetrain.com/wp-content/uploads/uneven.gif" alt="uneven" width="473" height="96">

Assuming an interest rate of 8%, we will now calculate the present value and future value of this uneven series of cash flows.
<table>
<tbody>
<tr>
<td width="616"><strong>&nbsp;</strong><strong>Calculator Usage: Future value</strong></td>
</tr>
<tr>
<td width="616">&nbsp;To calculate the future value of this series of cash flows, we will need to treat each cash flow as an independent cash flow and calculate its future value. We will adopt the procedure that we used to calculate the future value of a single cash flow.

The following calculations are demonstrated using BA II Plus calculator.

FV1: PV = -500, N = 4, I/Y = 8. CPT &gt; FV = -$680.244

FV2: PV = -600, N = 3, I/Y = 8. CPT &gt; FV = -$755.827

FV3: PV = +1,000, N = 2, I/Y = 8. CPT &gt; FV = +$1,166.400

FV4: PV = +1,500, N = 1, I/Y = 8. CPT &gt; FV = +$1,620

FV5: PV = +2,000, N = 0, I/Y = 8. CPT &gt; FV = +$2,000

Future Value of cash flows = Sum of all Future Values = $3350.328</td>
</tr>
</tbody>
</table>
&nbsp;
<table>
<tbody>
<tr>
<td width="616"><strong>&nbsp;</strong><strong>Calculator Usage: Present value</strong></td>
</tr>
<tr>
<td width="616">To calculate the future value of this series of cash flows, we will need to treat each cash flow as an independent cash flow and calculate its future value. We will adopt the procedure that we used to calculate the present value of a single cash flow.

PV1: FV = -500, N = 1, I/Y = 8. CPT &gt; PV = -$462.963

PV2: FV = -600, N = 2, I/Y = 8. CPT &gt; PV = -$514.403

PV3: FV = +1,000, N = 3, I/Y = 8. CPT &gt; PV = +$793.832

PV4: FV = +1,500, N = 4, I/Y = 8. CPT &gt; PV = +$1,102.545

PV5: FV = +2,000, N = 5, I/Y = 8. CPT &gt; PV = +$1,361.166

Future Value of cash flows = Sum of all Future Values = $2280.177</td>
</tr>
</tbody>
</table>
The present value of the uneven series of cash flows can also be calculated using the Cash Flow (CF) key and NPV function.

Present Value and Future Value of Uneven Cash Flows

In all the above examples for annuities, we assumed that the compounding frequency is annual. However, this may not always be the case and an annuity may have monthly, quarterly, or even semi-annual compounding. We can solve the time value of money problems for any of these compounding frequencies using the BA II Plus calculator.

Let’s take an example. An ordinary annuity pays $250 every quarter for the next 5 years. The expected rate of return is 8% per annum. Calculate the present value of this annuity.

We can solve this problem using two methods.

<strong>Method 1:</strong>

Since payments are made quarterly, change the number of payments per year to 4. Press [2ND][P/Y]. Input 4 and then press ENTER.

Now enter the following values for variables.

N=20 (20 quarters in 5 years)

I/Y = 8 (Interest rate per annum)

PMT = $250

Then compute the Present Value [CPT][PV].

PV = $4087.85

<strong>Method 2</strong>

Keep the number of periods per as 1 (Annual Compounding).

Enter the following variables:

N=20 (20 quarters in 5 years)

I/Y = 2% (Annual interest rate /4)

PMT = $250

Then compute the Present Value [CPT][PV].

PV = $4087.85

Both the methods are accurate and produce the same result. Try both methods and follow the one you are more comfortable with. One problem in the first method is that you will have to change the compounding frequency (P/Y) and then reset it after you have solved the problem.

Annuities with Different Compounding Frequencies

When solving a time value of money problem, it is sometimes easy to draw a timeline to present the cash flows on it. Once we have the timeline, we can easily understand the variables and visualize the present value or future value calculations.

In the previous pages, we demonstrated the time line for an ordinary annuity and for uneven cash flows.

Let’s take one more example to demonstrate the use of a time line.

<strong>Example: Loan Payments</strong>

You have taken a loan of $10,000 at an annual interest rate of 12% for a period of 2 years. Calculate the monthly payments you will make on this loan.

The payments (PMT) or Equated Monthly Instalments will be paid monthly for the next 24 months.

The above problem can be demonstrated on a timeline as follows:

<img class="aligncenter size-full wp-image-18845" src="https://financetrain.com/wp-content/uploads/timeline.gif" alt="timeline" width="589" height="98">
<table>
<tbody>
<tr>
<td width="616"><strong>Calculator Usage</strong></td>
</tr>
<tr>
<td width="616">To calculate the monthly payment:

Set compounding frequency to 12 (P/Y)

PV = 10,000

I/Y = 12

N = 24

PMT = 470.735</td>
</tr>
</tbody>
</table>

Time Value of Money

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