Financial Mathematics Fundamentals

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Financial Mathematics Fundamentals

In this comprehensive guide, we delve into the fundamental principles of financial mathematics, which are essential for calculating various investment banking products. We will explore the intricacies of spot interest rates, day count conventions, and the differences between the money and capital markets. Additionally, we will examine the key interest rates such as Euribor, Libor, and EONIA, and discuss essential concepts such as annualizing, present value, and discount factors. By understanding these foundational concepts, professionals in the field of finance will be better equipped to navigate the complex world of investment banking and make informed decisions.

Spot Interest Rate

The spot rate (also known as the spot interest rate or cash interest rate) is an exchange rate in the cash market, which assumes an immediate capital intake (initial capital) with guaranteed interest up to a fixed point in the future. No inflows/outflows occur during the entire term. At maturity, the cash flow consists of the final capital (initial capital + interest).

The spot rate is the annualized interest rate for capital growth over the fixed term and is thus comparable to a zero-coupon bond.

Daycount Convention

The Daycount Convention determines the number of days between a start date d_0 and a maturity date d_E. Furthermore, the total number of days in the reference year is determined.

Method Description Use
(British Method)
Actual number of days divided by 365 days.
Money market interest for the British Pound
(Effective Interest Method)
Actual term in calendar days in relation to the actual days of the year.
Eurobonds and US Treasuries
(Euro Method)
Actual number of days divided by 360 days in the year.
EUR and USD money market
(German Method)
Assumes a month of 30 days and a year of 360 days.
Some corporate Bonds

Furthermore, the Business Day Convention determines the date on which a transaction is executed. The holidays of the involved currencies must be taken into account. If a relevant date is not a business day, the transaction is usually executed on the next business day (“Following”).

1 Money Market

1.1 Money Market Interest (Euro)

The money market comprises invested funds with a term of up to 12 months. Money market interest in the EURO area is therefore calculated on a sub-annual basis using act/360 and linear interest.

\(Capital_{d_E} = Capital_{d_0} * (1 + i * \frac{d_E – d_0}{360})\)

The factor by which the initial capital is to be multiplied is called the capital growth factor and takes into account the money market interest rate i and the number of days between the start date d0 and the maturity date dE.

1.1.1 Euribor & LIBOR and EONIA

Three common rates are used to determine interest rates.

EURIBOR (European InterBank Offered Rate)

The EURIBOR is an arithmetic average of the interest rates reported in interbank transactions. These rates are unsecured loans between banks. This rate is determined for 1 week, 2 weeks, 1/2/3/6/9/12 months and published daily.

LIBOR (London InterBank Offered Rate)

The LIBOR provides average interbank interest rates for different currencies. It is similar in nature to the EURIBOR.

EONIA (European Overnight Index Average)

The EONIA is a one-day interest rate levied on unsecured loans in interbank transactions “overnight”. The basis for calculation is the average overnight interest rates of the largest credit institutions.

1.2 Annualizing (Money Market)

To convert an interest rate achieved in the money market for a specific period into an annual interest rate, the money market formula is rearranged for an interest rate per annum.

\(i_{per\ annum} =   \left(\frac{Capital_{d_E}}{Capital_{d_0}}-1\right)*\frac{360}{d_E-d_0}\)

1.3 Present Value (Money Market)

The present value (also known as the fair value) is the fair price today for future cash flows, taking into account current market interest rates. The calculation is systematically carried out in the opposite direction to future interest. It is thus “discounted” or “discounted.”

In the money market with sub-annual interest, the calculation is done using the following formula:

\(PV = \frac{CF}{1 + i * \frac{d_E – d_0}{360}}\)

The future cash flow (CF) is thus discounted over the number of days (d_E-d_0).

1.4 Discount Factor (Money Market)

The discount factor is used as a standardized factor for calculating present values. It is determined in the money market from the number of days (d_E-d_0) and the term-congruent interest rate i.

\(DF = \frac{1}{1 + i * \frac{d_E – d_0}{360}}\)

It is now sufficient to simply multiply a cash flow by the term-congruent discount factor to determine the present value.

2 Capital Market

The capital market refers to transactions with terms exceeding 12 months. In this case, the counting method is used in years to calculate interest.

2.1 Spot Rate (Capital Market)

The spot rate or spot interest rate in the capital market is determined depending on the term. As with the money market, this is an annualized interest rate for a transaction with no additional inflows/outflows (cash flows) during the term.

The spot interest rate can be compared to the yield of a zero-coupon bond since, in this case, too, only the capital and interest are paid out on the maturity date.

2.2 Interest Calculation

In the capital market, due to the term of several years, exponential interest occurs. Thus, the final capital results from the initial capital, which is compounded over t years with the term-congruent spot interest rate. (Provided that no further cash flows occur during the term.)

\(FV = PV \times (1 + i_t)^t\)

2.3 Annualizing (Capital Market)

By annualizing, the annual interest rate can be determined if the initial capital and final capital are given. To obtain an annual interest rate (per annum), the interest formula for the capital market is rearranged:

\(i = \sqrt[t]{\frac{FV}{PV}} – 1\)

2.4 Present Value (Capital Market)

The present value (PV) describes the current value of a future cash flow. Taking into account the term in t years and the term-congruent spot interest rate i_t, the future cash flow is discounted.

\(PV = \frac{FV}{(1 + i_t)^t}\)

If there are multiple cash flows over n years, they are discounted with their respective term-congruent spot interest rates i_t and then added together.

\(PV = \sum_{t=1}^n \frac{CF_t}{(1 + i_t)^t}\)