## Single Period Random Cash-Flow [Basic Concepts]

An investment instrument that can be bought and sold is frequently called an asset. Suppose you purchase an asset at time zero, and one year later you sell the asset. The total on your investment is defined to be

Very often, the term return is used for total return.

The rate of return is

These two terms are related by

Or it can be written as

This shows that a rate of return acts much like an interest rate.

dividend is a distribution of a portion of a company’s earnings, decided by the board of directors, to a class of its shareholders. Dividends can be issued as cash payments, as shares of stock, or other property.

Portfolio is a bundle or a combination of individual assets or securities. Portfolio theory provides a normative approach to investors to make decisions to invest their wealth in assets or securities under risk. It is based on the assumption that investors are risk averse. This implies that the investors hold the diversified portfolios instead of investing their entire wealth in a single or a few assets. The second assumption of the theory is that the returns of assets are normally distributed. This means that the mean and variance analysis is the foundation of the portfolio decisions.

Suppose that this is done by apportioning an amount X0 among the n assets. We then select amounts X0i, i= 1,2,3……..,n , such that

Where X0i represents the amount invested in the ith asset. If we are allowed to sell short then some of the Xoi’s can be negative; otherwise we restrict the X0i’s can be non-negative.

The amount invested can be expressed as fractions of the total investment. Thus we write

i= 1,2,3,…….n

Where Wi is the weight or fraction of asset i in the portfolio. Clearly,

Some of the Wi’s may be negative if short selling is allowed.

Let Ri denote the total return of asset i. then the amount of money is generated at the end of the period by the ith asset is  . The total amount received by the portfolio at the end of the period is therefore   Hence we find that the overall total return of the portfolio is

Equivalently, since   , we have

We can conclude that ;

Portfolio return both the total return and the rate of return of a portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight (in purchase cost) in the portfolio , that is,

It can also be explained with the help of an example

 Security No of Shares Price Total Cost Weight in Portfolio A 100 40 4000 .25 B 400 20 8000 .50 C 200 20 4000 .25 Portfolio Total Value 16000 1.00

 Security Weight in Portfolio Rate of Return Weighted Rate A 25 17% 4.25% B 50 13% 6.50% C 25 23% 5.75% Portfolio rate of Return 16.50%

Note: I am assuming that you have covered the concepts Random Variables, Expected values, Variances and Covariance’s in the Statistics Paper .

# Fixed-Income Securities

### Basic Concepts

An interest rate is a price, or rent, for the most popular of all traded commodities—money. The one-year interest rate, for example, is just the price that must be paid for borrowing money for one year. Markets for money are well developed, and the corresponding basic market price—interest—is monitored by everyone who has a serious concern about the financial activity.

The overall market associated with interest rates is more complex than the simple bank accounts discussed before. There are varieties of bills, notes, bonds, annuities, futures contracts, and mortgages are part of the well-developed markets for money. These market items are not real goods in the sense have intrinsic value—such as potatoes or gold—but instead are traded only as pieces of paper, or as entries in a computer database. These items, in general, are referred to as financial instruments. Their values are derived from the promises they represent. If there is a well-developed market for an instrument, so that it can be traded freely and easily, then that instrument is termed a security. There are many financial instruments and securities that are directly related to interest rates and, therefore, provide access to income—at a price defined by the appropriate interest rate or rates.

Fixed income securities are financial instruments that are traded in well-developed markets and promise a fixed income to the holder over a span of time. They are Important to an investor because they define the market for money, and most investors participate in this market.

The classification of a security as being a fixed-income security is actually a bit vague. The only uncertainties about the promised stream were associated with weather the issuer of the security might default (by, say, going bankrupt), in which case the income would be discontinued or delayed.

There are many different kinds of fixed income securities, and we cannot provide a comprehensive survey of them here. However, we shall mention some of the principal types of fixed-income securities in order to indicate the general scope of such securities.

Savings Deposits

The most familiar fixed-income instrument is an interest-bearing bank deposit. These are offered by commercial banks, savings and loan institutions, and credit unions. The simplest demand deposit pays a rate of interest that varies with market conditions. Over an extended period of time, such a deposit is not strictly of fixed-income type. The interest rate is guaranteed in a time deposit account, where the deposit must be maintained for a given length of time (such as 6 months), or else a penalty for early withdrawal is assessed.

Money market instruments

The term money market refers to the market for short-term (1 year or less) loans by corporations and financial intermediaries like a bank. It is a well-organized market which is designed for large amounts of money, but it is not of great importance to long-term investors because of its short-term and specialized nature.

Mortgages

This is a legal agreement by which a bank, building society, etc lends money at interest in exchange for taking the title of the debtor’s property, with the conveyance of title becomes void upon the payment of the debt. The standard mortgage is structured so that equal monthly payments are made throughout its term. Most standard mortgages allow for early repayment of the balance. Hence from the mortgage holder’s viewpoint, the income stream generated is not completely fixed, since it may be terminated with an appropriate lump-sum payment at the discretion of the homeowner.

There are many variations on the standard mortgage. There may be modest-sized periodic payments for several years followed by a final balloon payment that completes the contract. Adjustable-rate mortgages adjust the effective interest rate periodically according to an interest rate index, and hence these mortgages do not really generate fixed income in the strict sense.

Mortgages are not usually thought of as securities since they are written as contracts between two parties, a homeowner, and a bank.

Annuities

An annuity is a contract that pays the holder money periodically, according to predetermined schedule or formula, over a period of time. Pensions are the best examples of annuities. Sometimes annuities are structured to provide a fixed payment every year for as long as the annuitant is alive, in which case the price of the annuity is based on the age of the annuitant when the annuity is purchased and on the number of years until payments are initiated.

Annuities are not really securities since they are not traded. However, these are considered to be investment opportunities that are available at standardized rates. Hence from an investor’s viewpoint, they serve the same role as other fixed-income instruments.

Perpetual annuities

As a step toward the development of the formula, we consider an interesting and conceptually useful fixed income instrument termed as perpetual annuity or perpetuity which pays a fixed amount periodically forever.

## Internal Rate of Return

Internal rate of return is another important concept of cash flow analysis. It pertains specifically to the entire cash flow stream associated with an investment, not to a partial stream such as a cash flow at a single period. The streams to which this concept is applied typically have both negative and positive elements: the negative flows correspond to the payments that must be made; the positive flows to payments received. A simple example is a process of investing in a certificate of deposit for a fixed period of 1 year. Here there are two cash flow elements: the initial deposit or payment (a negative flow) and the final redemption (a positive flow).

Given a cash flow stream (x0, x1,………, xn) associated with an investment, we write the present value formula as

If the investment that corresponds to this stream is constructed from a series of deposits and withdrawals from a constant ideal bank at interest rate r, then from the main theorem on the present value in the previous section, PV would be zero. The idea behind the internal rate of return is to turn the procedure around. Given a cash flow stream, we write the expression for present value and the value of r that renders this present value equal to zero.

That value is called the internal rate of return because it is the interest rate implied by the internal structure of the cash flow stream. The idea can be applied to any series of cash flows.

The preliminary formal definition of the internal rate of return (IRR) is as follows:

Internal rate of return : Let (x0, x1, …….xn)  be a cash flow stream. Then the internal rate of return of this section is a number r satisfying the equation

0 = x0 + x1/(1+r) +x2/(1+r)2 +……………….xn/(1+r)n

Equivalently, it is a number r satisfying  1/(1+r) =c [i.e., r= (1/c) – 1] , where c satisfies the polynomial equation

0 = x0 + x1c +x2c2 +……………….xncn

We call this a preliminary definition because there may be ambiguity in the solution of the polynomial equation of degree n.

IRR can also be defined as a discount rate at which net present value is equal to zero. In other words, t is a rate of return which equates investment output or cash outflow to present value or cash inflow from an investment.

IRR is determined entirely by the cash flows of the stream. This is the reason why it is called the internal rate of return; it is defined internally without reference to the external financial world. It is the rate that an ideal bank would have to apply to generate the given stream from an initial balance of zero.

As we have seen that the internal rate of return is a polynomial equation in c of degree n, which does not, in general, have an analytic solution. However, it is almost always easy to solve the equation with a computer. Form algebraic theory it is known that such an equation always has at least one root, and may have as many as n roots, but some or all of these roots may be complex numbers. Fortunately, the most common form of investment, where there is an initial cash outlay followed by several positive flows, leads to a unique positive solution. Hence the internal rate of return is then well defined and relatively easy to calculate. The formal statement of the existence of the positive root embodies the main result concerning the internal rate of return.

• Main theorem of internal rate of return Suppose the cash flow stream (x0, x1, …….xn) has x0 < 0 and  xk ≤ 0 for all k, k = 1, 2, ……n. with at least one term being strictly positive. Then there is a unique positive root to the equation

0 = x0 + x1c +x2c2 +……………….xncn

Furthermore, if   > 0 (meaning that the total amount returned exceeds the initial investment), then the corresponding internal rate of return r = (1/c) – 1 is positive.

If some or all solutions to the equations for internal rate of return are complex, the interpretation of these values is not simple. In general, it is reasonable to select the solution that has the largest real part and use that real part to determine the internal rate of return. In practice, however, this is not often a serious issue, since suitable real roots typically exist.

## Present and future value of streams

Previously we discussed the impact of interest on a single cash deposit or loan; that is, in a single cash flow. We now extend that discussion to the case where cash flows occur at several time periods, and hence constitute a cash flow stream or sequence.

#### The Ideal Bank

An ideal bank applies the same rate of interest to both deposits and loans, and it has no service charge or transaction fees. Its interest rate applies equally to any size of principal, from 1 cent (or fraction thereof) to \$1 million (or even more). Furthermore, separate transactions in an account are completely additive in their effect on future balances.

Note that the distinction of an ideal bank doesn’t imply that interest rates for all the transactions are identical. For example, a 2-year certificate of deposit (CD) might offer a higher rate than 1-year CD. However, the 2-year (CD) must offer the same rate as a loan that is payable in 2 years.

If an ideal bank has an interest rate that is independent of the length of time for which it applies, and that interest is compounded according to normal rules, it is said to be a constant ideal bank. For simplicity we will assume that interest rates are indeed constant.

#### Future Value

Now we return to the study of cash flow streams. Let us decide on a fixed time cycle of compounding ( for example, yearly ) and let a period be the length of this cycle. We assume that cash flows occur at the end of each period (although some flows might be zero). We shall take each cash flow and deposit it in a constant ideal bank as it arrives. (If the flow is negative, we cover it by taking out a loan. ) Under the terms of a constant ideal bank, the final balance in our account can be found by combining the results of the individual flows. Explicitly, consider the cash flow stream (x0, x1, …….xn). at the end of n periods, the initial cash flow x0 will have grown to x0(1+r)n, where r is the interest rate per period (which is the yearly rate divided by the number of periods per year). The next cash flow, x1, received after the first period, will at the final time have been in the account for only n-1 periods, and have a value of x1(1+r)n-1. Likewise, the next flow x2 will collect interest during n-2 periods and have value x2(1+r)n-2. The final flow xn will not collect any interest, so will remain xn. The total value at the end of n periods is therefore FV= x0(1+r)n + x1(1+r)n-1 +……………+ xn.

• Future Value of a stream: Given a cash flow stream (x0, x1, …….xn) and interest rate r each period, the future value of the stream is

FV= x0(1+r)n + x1(1+r)n-1 +……………+ xn

#### Present Value

The present value of a general cash flow stream- like the future value- can also be calculated by considering each flow element separately. Again consider the stream (x0, x1, …….xn). the present value of the first element x0 is just that value itself since no discounting is necessary. The present value of the flow x1 is x1/(1+r), because that flow must be discounted by one period (again the interest rate r is the per-period rate.). continuing this way, we find that the present value of the entire stream is PV= x0 + x1/(1+r) +x2/(1+r)2 +……………….xn/(1+r)n. We can summarize this important result as follows:

• Present Value of a stream: Given a cash flow stream (x0, x1, …….xn) and an interest rate r per period, the present value of this cash flow stream is

PV =  x0 + x1/(1+r) +x2/(1+r)2 +……………….xn/(1+r)n

The present value of a cash flow stream can be regarded as the present payment amount that is equivalent to the entire stream. Thus we can think of the entire stream as being replaced by a single flow at the initial time.

There is another way to interpret the formula for the present value that is based on transforming the formula for future value. Future value is the amount of future payment that is equivalent to the entire stream. We can think of the stream as being transformed into that single cash flow at period n. The present value of this single equivalent flow is obtained by discounting it by (1+r)n. that is, the present value and the future value are related by

PV = FV/(1+r)n

#### Present Value and an Ideal Bank

In general, if an ideal bank can transform the stream (x0, x1, …….xn) into the stream (y0, y1, …….yn), it can also transform in the reverse direction. Two streams that can be transformed into each other are said to be equivalent streams.

How can we tell whether two given streams are equivalent? The answer is the main theorem on present value.

• Main theorem on present value The cash flow streams x = (x0, x1, …….xn) and y = (y0, y1, …….yn) are equivalent for a constant ideal bank with interest rate r if and only if the present values of the two stream, evaluated at the bank’s interest rate, are equal.

This is important because it implies that present value is the only number needed to characterize a cash flow stream when an ideal bank is available. The stream can be transformed in a variety of ways by the bank, but the present value remains the same. So if someone offers you a cash flow stream, you only need to evaluate its corresponding present value, because you can then use the bank to tailor the stream with that present value to any shape you desire.

# Present Value and Discounting

#### Basic Concepts

Interest is often termed as the time value of money. The basic idea of interest is quite familiar.  If you invest ₹1000 in a bank account that pays you 8% interest per year, then at the end of one year you will have in your account the principal or original amount plus interest at 8% for the total of ₹1080. If interest rate is r, expressed as a decimal, then your initial investment would be multiplied by (1+r) after one year.

Under a simple interest rule, money invested for a period different from one year accumulates interest proportional to the total time of the investment. So after 2 years, the total interest due is 2r times the original investment and so forth. In other words, the investment produces interest equal to r times the original investment every year. Usually, partial years are treated in a proportional manner; that is, a fraction f of 1 year, interest of r.f times the original investment is earned. The general rule for simple interest is that of an amount ‘x’ is left in an account at simple interest r, the value after n years is

V = (1+rn) x

If the proportional rule holds for fractional years, then after any time t (measured in years), the account value is

V = (1+rt) x

The account grows linearly with time as shown above; the account value at any time is just the sum of principal amount and the accumulated interest, which is proportional to time.

If we talk about Compound Interest most banks and loans employ some form of compounding- producing compounding interest. Again consider an account that pays interest at a rate of r per year. If interest is compounded yearly, then after one year, the first year’s interest is added to the original principal to define a larger principal base for the second year. Thus during the second year, the account earns interest on interest. This is the compounding effect, which is continued year after year.

Under yearly compounding, money left in an account is multiplied by (1+r) after one year. After the second year, it grows by another factor of (1+r) to (1+r)2. After n years, such an account will grow to (1+r)n times its original value, and this is the analytic expression for the account growth under compound interest. This expression is said to exhibit geometric growth because of the nth -power form.

Figure 1: Simple Interest leads to linear growth over time whereas compound interest leads to an accelerated increase defined by geometric growth. The figure shows both cases for an interest rate of 10%.

Figure 1 shows a graph of a \$100 investment over time when it earns 10% interest under simple and compound interest rules. The figure shows the characteristic shapes of linear growth tor simple interest and of accelerated upward growth for compound interest. Note that under compounding, the value doubles in about 7 years. There is a cute little rule that can be used to estimate the effect of interest Compounding (More exactly, at 7% and 10 years, account increases by a factor of 1.97, whereas at 10% and 7 years it increases by a factor of 1.95)

• The Seven Ten rule- Money invested at 7% per year doubles in approximately 10 years. Also, money invested at 10% per year doubles in approximately 7 years.

#### PRESENT VALUE

As discussed above that money invested today leads to increased value in the future as a result of interest. The formulas above show how to determine this future value.

That whole set of concepts and formulas can be reversed in time to calculate the value that should be assigned now, in the present, to money that is to be received at a later time. This reversal is the essence of the extremely important concept of present value.

To introduce this concept, consider two situations:

(1) You will receive \$110 in 1 year; (2) you receive \$100 now and deposit it in a bank account for 1 year at 10% interest.

Clearly, these situations are identical after 1 year you will receive \$110. We can restate this equivalence by saying that \$110 received in I year is equivalent to the receipt of \$100 now when the interest rate is 10% or we say that the \$110 to be received in 1 year has a present value of \$100 in general, \$1 to be received a year in the future has a present value of \$1/(1+ r), where r is the interest rate.

A similar transformation applies to future obligations such as the repayment of debt. Suppose that, for some reason, you have an obligation to pay someone \$100 in exactly 1 year. This obligation can be regarded as a negative cash flow that occurs at the end of the year. To calculate the present value of this obligation, you determine how much money you would need now in order to cover the obligation. This is easy to determine. If the current yearly interest rate is r, you need \$100/(1+r). If that amount of money is deposited in the bank now, it will grow to \$100 at the end of the year. You can then fully meet the obligation. The present value of the obligation is therefore \$100/(1+r).

The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting. The present value of a future monetary amount is less than the face value of that amount, so the future value must be discounted to obtain the present value. The factor by which the future value must be discounted is called the discount factor. The 1 year discount factor is dt=1/(1+r) where r is the 1-year interest rate. So if an amount A is to be received in 1 year, the present value is the discounted amount dtA.

The formula for present value depends on the interest rate that is available from a bank or other source. If that source quotes rates with compounding, then such a compound interest rate should be used in the calculation of present value. As an example, suppose that the annual interest rate, is compounded at the end of each of m equal periods each year; and suppose that a cash payment of amount A will be received at the end of the kth period, Then the appropriate discount factor is

dk = 1/[1+(r/m)]k

The present value of a payment of A to be received k periods in the future is dkA.