Basic Finance: Interest Rates, Discounting, Investments, Loans

Abstract

Interest rates allow estimating the value today of future cash flows and defining and calculating the return on investment or the cost of financing. Some fundamental concepts such as financial leverage or the distinction between real and nominal cash flows and interest rates are also discussed. We first explain how to represent lending and borrowing and, more generally, investments and financing through cash flow sequences. We then present the various interest rates in use in the market and the different conventions they give rise to. We then discuss the methods of estimating present values and their applications to the analysis of investments, before studying the mechanics of long-term credits.

Appendices

Appendix 1: Geometric Series and Discounting

Financial calculations with cash flows over many periods often employ the properties of geometric sequences, also called geometric progressions.

In a geometric progression, each element is derived from the previous one through multiplication by a factor called the ratio of the sequence. Thus a geometric sequence with first term a and ratio q is written: a, aq, aq²,…, aq^n-1, aqⁿ,…the (n + 1)st term (aqⁿ) being obtained from nth (aq^n-1) by multiplication of it by the constant ratio q.

The sum S_n of the first n terms of a geometric sequence, whose initial term is a and whose ratio is q, is given by:

$$ {S}_n\equiv \underset{n\kern1em terms}{\underbrace{a+ aq+{aq}^2\dots +{aq}^{n-1}=}}a\frac{1-{q}^n}{1-q}. $$

(2.16)

Indeed

$$ {S}_n=a+ aq+\dots +{aq}^{n-1}; $$

therefore

$$ {qS}_n= aq+{aq}^{2+}+\dots +{aq}^n. $$

Subtracting the second equality from the first we get:

$$ {S}_n\left(1-q\right)=a-{aq}^n;\mathrm{from}\ \mathrm{which}:{S}_n=a\frac{1-{q}^n}{1-q}. $$

Let us consider, in particular, the present value of a cash flow sequence of $1 for n years: $\frac{1}{\left(1+r\right)}+\frac{1}{{\left(1+r\right)}^2}+\dots +\frac{1}{{\left(1+r\right)}^n}$.

We have a sum of the first n terms of a geometric progression whose initial term and ratio are both equal to $\frac{1}{\left(1+r\right)}$. (2.16) thus gives us:

$$ \sum \limits_{i=1}^n\frac{1}{{\left(1+r\right)}^i}=\frac{1-{\left(1+r\right)}^{-n}}{r}. $$

(2.17)

Example 20

$$ 1000+1000\times 1.01+1000\times {(1.01)}^2\kern0.5em +{}^{\dots }+1000\times {(1.01)}^{11}=\kern0.5em 1000\left[1-{(1.01)}^{12}\right]/\left[1-(1.01)\right]=12,683. $$

The capital resulting at the end of 1 year (earned value) in the case of a savings plan with monthly interest of 1% or where the payments are $1000 at the end of each month is therefore $12,683.

Equation (2.16) giving the sum of the first n terms of a geometric progression allows us, by taking the limit, to calculate the sum when the number of terms is infinite, if the ratio is smaller than one; indeed if ∣q ∣  < 1, qⁿ tends to zero as n tends to infinity, and the sum ${S}_n=a\frac{1-{q}^n}{1-q}$ tends to $\frac{a}{1-q}$; i.e.:

$$ {\mathrm{Lim}}_{n\to \infty}\left({S}_n\right)\equiv {S}_{\infty }=\frac{a}{1-q}. $$

(2.18)

Such a computation comes up when one considers a perpetual annuity that provides a constant annual coupon (interest) indefinitely. For a discount rate r, its value is:

$$ V(r)=\frac{a}{1+r}+\frac{a}{{\left(1+r\right)}^2}+\dots +\frac{a}{{\left(1+r\right)}^n}+\dots =\frac{\frac{a}{1+r}}{1-\frac{1}{1+r}}=\frac{a}{r} $$

In particular:

$$ \sum \limits_{i=1}^{\infty}\frac{1}{{\left(1+r\right)}^i}=\frac{1}{r}. $$

(2.19)

Example 21

$$ {S}_{\infty }=\frac{1000}{1,05}+\frac{1000}{{\left(1,05\right)}^2}+\frac{1000}{{\left(1,05\right)}^3}+\dots +\frac{1000}{{\left(1,05\right)}^n}+\dots =\frac{1000}{0,05}=20,000. $$

If the interest rate is 5%, the value of a perpetual annuity giving a coupon of $1000 each year thus equals S_∞ = $20,000.

Appendix 2: Using Financial Tables and Spreadsheets for Discount Computations

1.1 1. Financial Tables

• (a) Computation of present value .

Tables A1 and A2, at the end of the book, give present value factors for different values of r and different maturities n. In Table A1, each line corresponds to a period n, and each column to an interest rate r. The value of $\frac{1}{{\left(1+r\right)}^n}$ is entered at the intersection of a line and a column. The terms of the sum of formula (2.6) can then be calculated one by one.

Financial Table A1 Present value of 1 currency unit: 1/(1 + r)ⁿ

Financial Table A2 Present value of a sequence of 1 currency unit per period during n periods 1/(1 + r) + 1/(1 + r)² + … + 1/(1 + r)ⁿ

For example, to determine the present value of {−50, 6, 6, 6, 56} discounted at a rate of 6%, we read, from the intersection of the sixth column (rate = 6%) and the lines 1 to 4, respectively, the coefficients 1/(1.06)¹, 1/(1.06)², 1/(1.06)³ and 1/(1.06)⁴. Thus we obtain a net present value of:

$$ -50+6\times 0.9434+6\times 0.89+6\times 0.8396+56\times 0.7921=10.396. $$

For constant cash flows, it is preferable to use Tables A2 which give, at the intersection of line n and column r, the value of $\sum \limits_{i=1}^n\frac{1}{{\left(1+r\right)}^i}=\frac{1-{\left(1+r\right)}^{-n}}{r}$, see Eq. (2.17), in the preceding appendix.

For example, Tables A2 indicate that $\sum \limits_{i=1}^{10}\frac{1}{{\left(1,1\right)}^i}=6.1446$. As a result, the present value of the cash flow sequence: {−1200; 360; …; 360} discounted at a rate of 10%, equals:

$$ -1200+360\times 6.1446=1012.06. $$

• (b) Computation of a YTM by interpolation, using tables.

If one has neither a calculator nor a spreadsheet , computing the YTM may be done using financial tables, by successive approximation: one brackets the IRR between two near values, one too large (NPV <0) and the other too small (NPV >0), then one makes a linear interpolation.

Example 22

Let us consider the cash flow sequence: $\left\{-1000;\underset{9\kern0.5em \mathrm{times}}{\underbrace{40;\dots; 40}};1090\right\}$

NPV $(r)=-1000+40\sum \limits_{i=1}^9\frac{1}{{\left(1+r\right)}^i}+1090\frac{1}{{\left(1+r\right)}^{10}}$; the curve of this function of r is drawn in Fig. 2.13.

Fig. 2.13

Computing a yield-to-maturity

We wish to determine r* so that NPV (r*) = 0 (abscissa of the point a in the figure, intersection of the curve of NPV (r) with the horizontal axis). We proceed by increments, finishing with a linear interpolation.

For r = 4%, we find NPV = + 33.78, which means the r* sought is > 4%. For r = 5%, we obtain NPV = − 46.52, which means the rate r* sought is < 5%.

By linear interpolation, we find NPV = 0 for r = 4.42%:

($4\%+\frac{33,78}{33,78+46,52}\times 1\%=4,42\%$; abscissa of point b).

The actual IRR (4.409%, abscissa of the point a) is in fact slightly smaller than the result of linear interpolation, due to the convexity of the function NPV(r) (see the figure).

Remark that Table A2 may allow us to find, relatively simply, the YTM of a sequence of constant cash flows F following an initial disbursement of F₀. As a matter of fact, in this case,

r* is simply the solution of the equation: $\sum \limits_{i=1}^9\frac{1}{{\left(1+r\ast \right)}^i}=-{F}_0/F$.

To find the value of r*, it suffices to consider the line corresponding to n periods, and to run along this line until one finds the closest value to– F₀/F; the corresponding column gives the approximate value of the period YTM.¹³

• (c) Using spreadsheets

Spreadsheets allow calculating the present value or the IRR very easily. However, reading the manual is absolutely necessary to check the mathematical formula used and the assumptions that are made for its use (particularly the reference time) which differ from one piece of software to another.

In Excel, for example, the NPV of a sequence is computed using the function NPV(discount rate; sequence). However, this function discounts the first flow of cash. Let us consider for example the sequence {−1000; 40; …; 40; 1090} whose terms are written in the cell range A1:K1 (see the Excel table below):

NPV $\left(0.04;\mathrm{A}1:\mathrm{K}1\right)=-1000\frac{1}{1,04}+40\sum \limits_{i=1}^9\frac{1}{{\left(1,04\right)}^{i+1}}+1090\frac{1}{{\left(1,04\right)}^{11}}=32.48$.

But the usual NPV is: $-1\kern0.5em 000+40\sum \limits_{i=1}^9\frac{1}{{\left(1,04\right)}^i}+1090\frac{1}{{\left(1,04\right)}^{10}}=33.78$; to obtain this we have to write: NPV (0.04;B1:K1) + A1 (or 1.04*NPV (0.04;A1:K1)).

Continuing with Excel, the financial function which calculates the IRR is IRR(sequence). For example, IRR(A1:K1) will give the IRR of the cash flow sequence entered in the cell range A1:K1, which is 4.409% in our example.