Interest Rates

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Interest rates are pretty much the most important parameter in finance.

Terms

The time value of money is relevant here, too, just like it is explained in the corporate finance part.
Interest is the price of money. In exchange for the use of money, the debtor pays a fraction back to the creditor. However, there are different types of interest rates:

Interest Rate	Characteristic
Deposit rate	Received for deposits at a bank
Debt (borrowing) rate	Paid for borrowing capital from the bank
Prime rate	Interest rate at which banks lend to their most creditworthy clients
Key interest rates or federal funds rate	Interest rates at which central banks borrow funds from or lend funds to commercial banks (ECB: see next slide; FED: target federal funds rate)
Money market rates	(Risk-free) Interest rate on short-term (approx. up to 1 year) transactions
Interbank rates	Rates at which banks lend to each other
Bond yields	Yield that can be earned on mid-term and long-term(risk-free) financial instruments
Nominal rate	Interest rate fixed in financial contracts
Effective rate	Effective interest rate paid/earned in a financial contract
Real interest rate	Yield earned above inflation rate

Key Interest Rates

Within the central bank system, there are several important rates:

• Fixed Rate: The central bank assigns a fixed interest rate to commercial banks for borrowing funds.
• Minimum Bid Rate: The minimum interest rate at which commercial banks can borrow funds from the central bank in an auction, not currently used by the ECB.
• Marginal Lending Facility: After using up their assigned fixed rate credit, banks may be able to borrow additional funds from the central bank at a higher rate, known as the marginal lending facility rate.
• Deposit Facility: The interest rate at which commercial banks can deposit excess funds with the central bank.

LIBOR Rates

LIBOR (London Interbank Offered Rate) was a benchmark interest rate at which major global banks lent to one another in the international interbank market for short-term loans. It was widely used as a reference rate for various financial instruments, including derivatives, loans, and mortgages. However, due to manipulation scandals and declining liquidity in the interbank market, LIBOR was replaced by alternative reference rates controlled by the central banks (e.g. ESTER, euro short-term rate, in the Eurozone; SOFR in the US).

Money Market Rates

Money market rates are short-term interests rates, of which prime rates are reserved for the most creditworthy clients. Interbank rates are the rates at which banks lend to each other, and they are typically higher than money market rates due to the increased risk of lending to other banks.

Bond Market Rates

Bond yields represent the return that investors can expect to earn on mid-term and long-term financial instruments, such as government bonds.

Nominal, Effective and Real Interest Rates

The nominal interest rate is the interest rate that is stated in financial contracts, while the effective interest rate takes into account the compounding of interest over time. The real interest rate is the nominal interest rate adjusted for inflation, which reflects the true purchasing power of the money being lent or borrowed.

Types of Interest

Varying Time Periods

To determine the exact investment period of transactions/securities that do not last an integer number of years, three day count conventions can be used:

• 30/360: Assumes 30 days in a month and 360 days in a year, the German method used for e.g. savings accounts.
• Actual/360: Uses the actual number of days in the month and assumes 360 days in a year, the French method used for e.g. money market instruments.
• Actual/Actual: Uses the actual number of days in the month and the actual number of days in the year, also called ICMA rule and used for Euro/US bond markets.

Simple Interests

In simple interest, compounding effects are ignored. The interest amount is calculated once on the initial amount and simply applied every year. This is practically only relevant for within-year savings accounts. A payout $C_n$ for an initial amount $C_0$ after $n$ years with an interest rate $r$ is given by:

$$C_n=C_0\cdot (1+r\cdot n)|C_0=\frac{C_n}{1+r\cdot n}$$

The payout is therefore the future value of the initial amount, which is the present value. See time value of money (uses compound interest). To determine the present value of a future payout, use the formula for $C_0$.

When using varying time period conventions, the formula simply adjusts to:

$$C_n=C_0\cdot \left(1+r\cdot \frac{d}{D}\right)$$

where $d$ is the number of days between the start and end date, and $D$ is the number of days in the year according to the chosen convention.

Compound Interests

Compound interest refers to the interest calculated on the initial amount and also on the accumulated interest from previous periods. Compounding is the opposite of discounting. The calculations are:

$$\begin{array}{rl}{C}_n=C_0\cdot (1+r{)}^n=C_0\cdot q^n|& C_0=\frac{C_n}{(1+r{)}^n}=\frac{C_n}{q^n}\\ & \\ q^n=\frac{C_n}{C_0}|& n=\frac{\ln \left(\frac{C_n}{C_0}\right)}{\ln (q)}\\ & \\ q=\sqrt[n]{\frac{C_n}{C_0}}|& r=q-1\end{array}$$

For example, the time required to double an investment at 5% interest is $n=\frac{\ln (2)}{\ln (1.05)}\approx 14$ years.

Rule of 72

A quick way to estimate the time required to double an investment at a given interest rate is

$$n\approx \frac{72}{r\cdot 100}$$

Intra-Year Interests

Intra-year compounding occurs when interest is calculated and added to the principal more than once per year. The formula for calculating the future value with intra-year compounding is:

$$C_n=C_0\cdot {\left(1+\frac{r}{m}\right)}^{m\cdot n}|C_0=\frac{C_n}{{\left(1+\frac{r}{m}\right)}^{m\cdot n}}$$

where $m$ is the number of compounding periods per year.

Instead of adjusting the compounding period, the effective interest rate for $m$ periods per year can be calculated, which is the interest rate that would yield the same future value if compounding were only done once per year:

$$r_{eff}={\left(1+\frac{r}{m}\right)}^m-1$$

For consumer loan and mortgage contracts in the EU, the effective rate must be disclosed, following ICMA rules (actual/actual day convention and all associated payments accounted for).

Nominal and Effective Rates

The nominal interest rate is the stated interest rate in a financial contract, usually annualized and not accounting for compounding effects. The effective interest rate takes into account the compounding of interest over time.

Continuous Compounding

Contrary to intra-year compounding with set compounding periods, continuous compounding assumes that interest is compounded an infinite number of times per year. It is often used in theoretical finance and for certain types of financial instruments, The formula for continuous compounding is:

$$C_n=C_0\cdot e^{r\cdot n}|C_0=C_n\cdot e^{-r\cdot n}$$

To determine the effective interest rate for continuous compounding (because ${\lim }_{m\to \infty }{\left(1+\frac{r}{m}\right)}^m=e^r$):

$$r_{eff}=e^r-1$$