Basic Concept

Basic Concept

Example 1-1 introduce the concept of future value of money invested today being the principal amount and the interest due. By definition, a future value is the value of today’s RM1 at a future date. Using Actual/365 day count convention, MYR1,002,465.75 to be received in one month is the future value of MYR1,000,000. Alternatively, the present value of MYR 1,002,465.75 is MYR 1,000,000.

Diagram 1-2

By definition, present value is the current value of a cash flow to be received in the future. This is represented generally by Equation 1-3 . Adjustment to the formula is neccessary for different convention as described in Wikipedia article. Equation 1-6 is an example of such adjustment for 'Actual/365 Fixed' and 'Actual/360'. When \(FV=1\), Equation 1-6 is simplified into:

\(P=\dfrac{1}{\left(1+\frac{rt}{365}\right)}\)

Discount factor is a special case of present value wen the future value is 1. The equation above is now modified to reflect the basic formula for a discount factor:

\( \begin{align} DF &=\dfrac{1}{\left(1+\frac{rt}{365}\right)} \\ \textrm{ where } DF &= \textrm{discount factor} \end{align} \)

Equation 1-9

where \(DF\) is discount factor. The equation is a general equation and will differ depending upon the day count convention.

Borrowing and Lending

One of the core incomes of banking business is net interest income – interest revenues less interest expenses. Revenues are generated from various form of lending to both retails and corporations. Funds lent are raised from retails and corporations in the form of deposits. Interests paid on these deposits are considered interest expenses. On daily basis financial institutions will receive interest and principal repayments from various borrowers and will make payments of interest and principal repayments to various lenders. Net of these cash flows rarely balances and financial institutions will have to borrow to cover the shortfall or lend any excesses.

The avenue to do so is through interbank money market in which financial institutions lend to and borrow from each other. For Malaysian Ringgit, the standard tenors for the activities are in Table 1-1 and typical quotations for interbank money market rates are as in Table 1-2.

Table 1-1

Table 1-2

Any borrowing and lending activities in the Malaysian interbank market is for value on the same day as the deal date. If a bank were to borrow MYR from another bank for 1 month, the lending bank will deliver the money on the same day and will expect a repayment from the borrower 1 month later.

We will be using interbank market for lending and borrowing without interim/periodic interest payment to illustrate the application of present value. Unlike retail and/or corporate loan products, there are plenty of interbank financial product fitting the requirement such as bankers acceptances, treasury bills, commercial papers, interbank lending/borrowing and commercial papers.

Quotations of the money market rates are provided by banks mostly through the money market brokers and published in Bloomberg and/or Reuters (or any other data vendors). The 1-month rate is quoted as 3.00%/3.10% means that the price provider is willing to borrow at 3.00% and lend at 3.10%. The price at which the price provider is wiling to borrow is called a bid while the price at which the price provider is willing to lend is called an offer.

Table 1-3

Different markets, however have different conventions in quoting the money market rates. For instance, USD money market rates are quoted as the bid having a higher value than the offer e.g. 3.45/3.35. This follows from the securities market which quoted prices in terms of prices i.e dollar and cents with lower price as the bid and higher price as the offer. Once these prices are converted to yields(rates) the bid will have a higher number than the offer due to the inverse relationship between prices and rates.

If another bank wishes to lend for one month, the bank has to give at the bid of 3.00%. On the other hand, if a bank wishes to borrow for one month, the bank has to take the offer at 3.10%.Assuming the fixed deposit received by the bank in Example 1-1 has to be placed out via lending. The bank can lend the money for one month at the interbank rate of 3.00% p.a. and breakevens.

Other currencies may have other value date convention. USD and EUR for instance are transacted for value spot or two working days from transaction date. The exceptions to this are in Table 1-3.

The following diagram illustrates the timing of the value dates and maturity dates.

Diagram 1-3

Termination

Early termination of a fixed deposit is not a norm. Nevertheless there are occurrences whereby a depositor would request termination or early refunds. Pre-maturing a deposit often results in the deposit earning zero or minimal interest.

Example 1-2

After twenty three (23) days, the depositor in Example 1-1 requested the fixed deposit to be terminated and the fund be returned to his account. How much will the depositor receive? Or how much interest is the bank willing to pay without incurring any losses?

Let's assume that the bank lent the money received form the deposit to another bank at 3.00% p.a. for one month. If the termination request is to be met, there will be cash flow mismatch and the bank has to borrow the fund on the twenty third (23rd) day to repay the depositor. The maturity of the new borrowing must match the interbank lending maturity to avoid further cash flow mismatch.

Diagram 1-4

Assuming that there are no changes in market rates, the bank will be able to borrow for 7 days (the remaining tenor of the interbank lending) at 3.05% (see Table 1-2). To match the cash flow on the 30th day, the maturity amount will be its present value using Equation 1-8.

\( \begin{align} P &= \dfrac{1,002,465.75} {\left(1+\frac{0.305\textrm{x}7}{365}\right)}\\ &=\textrm{1,001,879.72} \end{align} \)

CALCULATOR: TERMINATION

The calculator on the right can help reader to simulate the present value with different number of days remaining, rate, maturity or day count convention. 'Days Remaining' should be less than the number of days in a year.

The amount borrowed can then be paid to the depositor if penalty charges are not imposed. Diagram 1-5 illustrates that the timing and amount of the cash flow are matched and netted to zero. The bank can pay to the depositor a smaller amount than calculated to cover for other costs (such as operational cost) and profitability. By doing so, the bank will need to borrow a smaller amount resulting in smaller payment on maturity of the borrowing. The difference between the maurity amounts of the borrowing and original lending will be positive to cover costs and profits.

Diagram 1-5

Example 1-2 has a major significance in our discussion in the following chapter. At this stage, it is best to note that the maturity of 23-day borrowing from depositors can be rolled-over for the next 7 days to obtain the same maturity amount as if it was a single 30-day borrowing. Mathematically;

\( \begin{align} FV_{30}&=FV_{23}\left(1+\dfrac{r_{7}t_{7}}{365}\right) \\ \textrm{where } FV_{30} &= \textrm{maturity amount on the 30}^{\textrm{th}} \\ FV_{23} &= \textrm{maturity amount on the 23}^{\textrm{rd}} \textrm{day}\\ r_{7} &= \textrm{interest rate for the 7 days period}\\ t_{7} &= \textrm{tenor or number of days which is 7 days, in this case} \end{align} \)

Applying Equation 1-2 on \(FV_{30}\) and \(FV_{23}\) we obtain;

\( \begin{align} P\left(1+\dfrac{r_{30}t_{30}}{365}\right) &= P\left(1+\dfrac{r_{23}t_{23}}{365}\right) \left(1+\dfrac{r_{7}t_{7}}{365}\right) \\ \textrm{where } P &= \textrm{original principal or deposit amount}\\ r_{7} &= \textrm{interest rate for the 7-day period}\\ r_{23} &= \textrm{interest rate for the 23-day period}\\ r_{30} &= \textrm{interest rate for the 30-day period}\\ t_{7},t_{23},t_{30} &= \textrm{tenor or number of days for the respective interest rate} \end{align} \)

When \(P = 1\), the equation is reduced into;

\(\left(1+\dfrac{r_{30}t_{30}}{365}\right)=\left(1+\dfrac{r_{23}t_{23}}{365}\right)\left(1+\dfrac{r_{7}t_{7}}{365}\right)\)

This can be generalised further into:

\( \begin{align} \left( 1 + r_{l}dcf_{l} \right) &= \left( 1 + r_{s}dcf_{s} \right) \left( 1 + r_{f}dcf_{f} \right)\\ \textrm {where } dcf_{l} &= \textrm{day count factor for the full period}\\ dcf_{s} &= \textrm{day count factor for the deposit period}\\ dcf_{l} &= \textrm{day count factor for the remaining period}\\ \end{align} \)

Equation 1-10

Equation 1-10 is the underlying equation behind forward rate agreements and futures which will be discussed in Chapter 2. It also highlight the importance of compounding concept i.e. reinvestment of principal and interest which will be relevant in bonds discussed in Chapter 4.