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Athens University of Economics and Business Yiannis Dendramis

Principles of Finance with Excel [email protected]

Part I

Time Value of money

1. Future valueThe future value tells you the value in the future of money deposited in a bank

account today and left in the account to draw interest. The future value uses the

concept of compound interest. That is, you earn interest on interest.

The value of x deposited today in an account paying r% annually and left in the

account for n years is its future value

Figure 1

In the spreadsheet above you can see the FV of 100 for 2 interest rates: 6% and 8%.

Notice that interest is earned at the end of the year, so at time 0, we just have our

initial investment.

Consider the case that you intend to make 4 annual deposits of 100, with the first

deposit made at time 0 (today). The next spreadsheet answers the question of how

much you will have accumulated at the end of period 3.


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Figure 2

Figure 2 analyses in a step by step manner how money accumulates in a typical

savings plan. The excel function FV simplifies all these calculations. It computes the

future value of any series of constant payments. It requires as inputs the Rate of

interest, the number of periods Nper, the annual payment Pmt and the Type which

tells excel whether payments are made at the beginning of the period (type=1) or at

the end of the period (type=0). To see the difference, see the calculations in the next

figure.

Figure 3

Here, each deposit is in the account one year less and earns one years less interest.

2. Present valueThe present value is the value today of a payment that will be made in the future.

The present value of x to be received in n years when the interest rate is r% is

The interest ris also called the discount rate.


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Figure 4

Figure 4 presents a simple example. If you anticipate getting 100 in 3 years and the

bank pays 6% interest on savings accounts, the future payment worths 83.96

today. That is, if you put 83.96 today in a bank account, in 3 years you would have

100.

We next discuss the PV of an annuity1. The PV of an annuity tells you the value today

of all the future payments on the annuity.

Figure 5

The PV of an annuity that gives you 100 at the end of each year, for the next 3 years

is 267.30. You can calculate this in two ways: find the present value of each value

and then sum the individual discounted values, or use the excels function PV. This,

calculates the PV of an annuity and it looks like the FV preciously discussed. Again,

the type denotes whether the payments are made at the beginning or the end of the

year (default=0).

1Annuity is a series of equal periodic payments


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3. Net present value

The NPV of a series of future cash flows is their present value minus the initialinvestment required to obtain the future cash flows. It is the increase in wealth

which you get if you make the investment.

Figure 6

Figure 6 presents an example. We pay 800 today to get the series of future cash

flows F63:F65. So, the increase in wealth that we obtain from this investment is

258.8.

4. Internal rate of returnThe IIR of a series of cash flows is the discount rate the sets the net present value of

the cash flows equal to zero. We can calculate this in two ways: using the NPV excel

function, or using the IRR excel function. Figure 7 presents an example. If the initial

investment is 800 and the future cash flows are in cells E74:E76, the IRR of this

project is 2.8%. In cells E79:E85 of figure 7 we also calculate the NPV for various

discount rates. As you can see, somewhere between 2% and 3% the NPV becomes

zero. This means that if a bank gives you an r>2.8% you prefer the bank account

than this investment plan, and if the bank gives you an r


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Figure 7

Figure 8


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Part II

Portfolio management

1. The problemMarkowitz reduced the optimization problem to that of finding mean-variance

efficient portfolios, with efficient points having the highest expected return for a

given level of risk (variance or std deviation of portfolio returns).

Given a set of N risky assets and a set of weights that describe howthe portfolio investment is split, the general formulas for expected return and

variance are:

Or in matrix format:

Where is the Nx1 vector of weights, is the Nx1 vector of expected returns, isthe variance covariance matrix (symmetric and positive semidefinite).

Then the single investor faces the problem:

s.t. , where is an Nx1 vector of ones.


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The solution to the above problem is the minimum variance portfolio withexpected return. We say that is mean variance efficientif there exists no othersuch that

If there are no short selling constraints the unique solution can be obtainedby minimizing the Lagrangian of the problem. Otherwise we can use excel build in

functions to numerically optimize with respect to the portfolio weights.

2. Excel implementationExcel has several individual functions for quickly summarizing the features of a

dataset. These include AVERAGE(array) which returns the mean, STDEV(array) for

the standard deviation, MAX(array) and MIN(array), COVAR(array1,array2) &

CORREL(array1,array2) for the correlation/covariance between two arrays. The next

figure illustrates basic calculations for three asset classes (Bond, Stock, Exchange

rate).


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Figure 1

A covariance matrix is a symmetric matrix whose element in the position isthe covariance between the th and th elements of a random vector. There aretwo alternatives to calculate the covariance matrix in excel. The first alternative uses

the COVAR(array) excel function to calculate the element of the covariancematrix (see J52:J54 cells in figure 1).

Another way to calculate the covariance matrix is to use the excel Data Analysis

ToolPak. This is a Microsoft Office Excel add-in program which must be activated

before you use it2. To calculate the variance covariance matrix go to Tools -> Data

analysis and choose the choice Covariance.

2To activate it go to tools->Add-Ins->Analysis Toolpak
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Figure 2

Then choose the Input Range (the returns in our case), the Grouped Byset it to Rows

if your dataset is in row vectors, check the Labels in first row if the Input Range

contains labels of the data in the first row and finally choose the Output Options,

that is, the area that excel displays the output.

Figure 3

As we have already seen in section one, the portfolio is characterized by the weight

vector . Assume that the percentage composition of our portfolio is : 20% stocks,40% bond, 40% exchange rate. The daily return of our portfolio is the weighted sum

of the individual securities. To compute the portfolio return and variance one can

use the excel functions: AVERAGE, VAR, STDEV. Notice the C78 cell in figure 4. This

displays the portfolio return formula. The $ sign before and after each portfolio

weight ($B$76, $C$76, and $D$76) tell Excel that in pasting the formula, it should not


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modify the cell reference relative to the cell where the formula is being pasted. The

$B$76 ($C$76, and $D$76) is called an absolute cell reference, which is not modified

when formulas are copied and pasted.

Figure 4

Figure 5

We should emphasize that the portfolio variance and return could also be computed

using the analytic formulas (). An interesting result from our calculations is that our

portfolio has lower risk than any individual asset. This can be seen intuitively

because different types of assets often change in value in opposite ways. For

example, as prices in the stock market are negatively correlated with prices in

the bond market, a collection of both types of assets can have lower overall risk. This

is why we want the Diversification of the portfolio.

3. Using the Excel SolverGiven the same asset data set (Bond, Stock, Exchange rate), suppose that we want to

construct an efficient portfolio producing target return 7%. The problem is to find
http://en.wikipedia.org/wiki/Stock_markethttp://en.wikipedia.org/wiki/Bond_markethttp://en.wikipedia.org/wiki/Bond_markethttp://en.wikipedia.org/wiki/Stock_market


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the split across assets that achieve the target return whilst minimizing the variance

return. This is a standard optimization problem amenable to excels solver, which

contains a range of iterative methods for optimization. The solver requires changing

cells, a target cellfor minimization and the specification of constraints. To calculatethe objective function we need the excel functions TRANSPOSE and MMULT. The

Transpose function returns a transposed range of cells. For example, a vertical range

of cells is returned if a horizontal range of cells is entered as a parameter.

TRANSPOSE must be entered as an array formula (use brackets {} or

CTRL+SHIFT+ENTER) in a specified range.

Figure 6

The changing cells for optimization are cells C141:C137. The target cell to be

minimized is the portfolio variance, C155. There are two explicit constraints, namely

the expected return (C146) must equal the target level (C158), and the weights must

sum to 1 (C159).
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Figure 7

The next figure plots the frontier portfolio.

Figure 8

Notice that there are no constraints on the weights assigned to individual assets,

that is, short selling is possible.

We next extend our analysis to include a riskless asset. That is, an assets with zero

variance. Now, the investor faces the problem:


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s.t.

Again is the target level and is the risk free rate. Notice that the constraint isno longer there. To see why this constraint is no longer needed, note that the

optimization is written only in terms of the N risky assets, so once the weights are chosen, we can always choose the weight of the riskless asset to be .The excel file is modified as follows:

Figure 9

Now, the efficient frontier is linear in volatility:

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