ch01 by Salim Kasap

Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte

Chapter 1 Interest rate

1. Measuring time  



In finance the most common unit of time is the year There exists various conventions to measure fractions of the year We will use the 30/360 rule to measure time between two dates: initial and final Note that the initial date starts at noon and the final date ends at noon. Thus there is only one whole day between e.g. 2 February 2007 and 3 February 2007

Chapter 1 Interest rate

Exhibit p.1: 30/360 rule Rule

Result

Example: from 15 January 2006 to 13 March 2009

1. Count the number of whole Y years

3 (from 15 January 2006 to 15 January 2009)

2. Count the number of M/12 remaining months and divide by 12

1/12 (from 15 January 15 2009 to 15 February 2009)

3. Count the number of D/360 remaining days (the last day of the month counting as the 30th unless it is the final date) and divide by 360

28/360 (under the 30/360 convention there are 16 days from 15 February 2009 at noon to 1 March 2009 at noon and 12 days from 1 March 1 2009 at noon to 13 March 2009 at noon)

TOTAL

Y + M/12 + D/360 3 + 1/12 + 28/360 = 3.161111…

Chapter 1 Interest rate

Exhibit p.2: 30/360 rule (2) Semester (Half year)

0.5 year

Quarter

0.25 year

Month

1/12 year

Week

7/360 year

Day

1/360 year

Chapter 1 Interest rate

2. Interest Rate 



In the economic sphere there are two types of agents whose interests are by definition opposed to each other: Investors, who have money and want that money to make them richer while they remain idle, Entrepreneurs, who don't have money but want to get rich actively using the money of others. Banks help to reconcile these two interests by serving as an intermediary

Chapter 1 Interest rate

Exhibit p.2: Economic agents Capital Investors

Loan Bank

Interest – Fee

Interest

Entrepreneurs

Chapter 1 Interest rate

2.1 Gross interest rate 

If I is the total interest paid on a capital K, the gross interest rate over the considered period is defined as:

I r= K 

An interest rate is meaningless if no time period is specified

Chapter 1 Interest rate

2.2 Compounding 

Starting with initial capital K one can build a compounding table of the capital at the end of each interest period Period

Capital

Example: r = 10%

$2,000

K(1 + r)

2, 000 × (1 +10%) = $2, 200

K(1 + r)²

2, 200 × (1 +10%) = $2, 420

…

...

K(1 + r)n

2, 000 × (1 +10%) n

Chapter 1 Interest rate

2.2 Compounding 

From the compounding table we obtain a formula for the amount of accumulated interest received after n periods:

I n = K (1 + r ) − K n



We may now define the compound interest rate over n periods, corresponding to the total accumulated interest:

[n]

In n = = (1 + r ) − 1 K

Chapter 1 Interest rate

2.3 Conversion Formula 

Two compound interest rates over periods τ1 and τ2 are said to be equivalent if they satisfy:

1 + r 

1 [τ1 ] τ 1

 = 1 + r

1 [τ 2 ] τ 2



τ1 and τ2 are two positive real numbers (for instance 1.5 represents a year and a half)

Chapter 1 Interest rate

2.4 Annualization 



Annualization is the process of converting a given compound interest rate into its annual equivalent. This allows one to rapidly compare the profitability of investments whose interests are paid out over different periods. If r[annual] is an annualized rate then the compound interest rate over T years is:

[T ]

= (1 + r

[ annual ]

) −1 T

Chapter 1 Interest rate

3 Discounting 



A dollar today is worth more than a dollar tomorrow. Two principal reasons can be put forward: Inflation: the increase in consumer prices implies that one dollar will buy less tomorrow than today. Interest: one dollar today produces interest between today and tomorrow. With this principle in mind the next step is to determine the value today of a dollar tomorrow – or generally the present value of an amount received or paid in the future.

Chapter 1 Interest rate

3.1 Present value 

The present value of an amount C paid or received in T years is the equivalent amount that, invested today at the compound rate r, will grow to C over T years:

PV × (1 + r ) = C T

or:

C PV = T (1 + r )

Chapter 1 Interest rate

3.2 Discounting 



Discounting is the process of computing the present value of various future cash flows. Similar to annualization, it is a key concept in finance as it makes amounts received or paid at different points in time comparable to what they are worth today. In practice, the choice of the discount rate r is crucial when calculating a present value and depends on the expected return of each investor. The minimum expected return for all investors is the interest rate offered by such ‘infallible’ institutions as central banks or government treasury departments.