Tarheel ConsultancyServices Manipal, Karnataka
Course on Fixed Income Securities For XIM -Bhubaneshwar
For PGP-II 2003-2005 Batch Term-V: September-December 2004
Module-I Part-II Time Value of Money: Concepts & Illustrations
Introduction • If we were to be given a choice between receiving one rupee today, and receiving it one year later, would we be indifferent? • The answer is no, because if we were to receive it today, we can invest it at a rate of interest, so as to have more than one rupee tomorrow.
Introduction (Cont…) • Thus, in order for us to be indifferent, we would have to be offered more than a rupee next year. • Consequently, a rupee today is worth more than a rupee to be offered in the future. • Reversing the argument, a rupee to be received in the future, is worth less than a rupee today.
Introduction (Cont…) • This is an illustration of the concept of Time Value of Money. • What matters is not only how much you receive in monetary terms, but also when you receive the cash flow.
Interest • What is interest? • It is the compensation paid by the borrower of capital to the lender for permitting him to use it. • Technically, it is the economic rent paid by the borrower to compensate the lender for the loss of opportunity to use the amount, when it is on loan.
Rationale for Interest • One of the main reasons for charging interest is Time Preference. • Everyone would rather have a given amount of money today, than the same amount at a future point in time. • This is because, money received today can be used to meet current needs, whereas if the same amount is received later, it can only be used for deferred needs, and that too in an uncertain future.
Rationale (Cont…) • Thus interest can be defined as the price that is adequate to cause individuals to overcome their time preferences to retain money, and lend it instead. • Thus far we have looked at the supply side of funds. • On the demand side, all firms need access to capital, most of which is borrowed.
Rationale (Cont…) • For a firm to be successful, the return on capital employed should be greater than the cost of funds. • The prospect of high returns on capital employed, causes firms to compete for scarce capital, by offering the market rate of interest.
Determinants of Interest • In a free market economy the interest rate is determined by the demand for and the supply of capital. • The factors which influence the interest rate are the following. 1. There is what is called a Pure or base rate of interest. It is the rate of return that would prevail on a risk-less investment, in the absence of inflation.
Determinants (Cont…) • It can also be defined as the Real risk-less rate of return, where the word `real’ connotes that there is no loss in purchasing power. • For instance assume that a banana is currently worth 2 rupees. • Assume that an investment of Rs 2 in a risk-less asset will yield Rs 2.20 next period.
Determinants (Cont…) • If Rs 2.20 is adequate to purchase 1.10 bananas next period, then we can say that the real rate of return is 10%. 2. Inflation Premium • In the real world, the prices of goods do not remain constant. • For most people, however, wealth is in the form of financial and not physical assets.
Determinants (Cont…) • So what is observed in practice is not the `real’ but the `nominal’ or `money’ rate of interest. • In our example above the nominal rate is 10%. • In this case the real rate is equal to the nominal rate because there is no inflation.
Determinants (Cont…) • However, in practice, to calculate the real rate of return from an investment, we would have to estimate and factor in the rate of inflation.
Real versus Nominal Rates • Let us consider a simple economy where there is only one physical good. • Assume that the current price of the good is P0. • So Rs 1 can buy 1/P0 units of the good today. • Assume that the price of the good next period is P1, which is known with certainty today. • If so, a Rupee can buy 1/P1units of the good after one period.
Real & Nominal Rates (Cont…) • Assume that there is financial bond that will pay 1+R rupees next period per rupee invested now. • Also assume that there is a `goods bond’ which will pay 1+r units of the good in the next period, per unit of the good invested now.
Real & Nominal Rates (Cont…) • If you invest one rupee in the financial asset you will receive 1+R rupees next period, which will be adequate to buy (1+R)/P1 units of the good. • Similarly if you invest Rs 1 in the goods bond, which amounts to an investment of 1/P0 in goods terms, you will get (1+r)/P0 units of the good next period.
Real & Nominal Rates (Cont…) • In order for the market as a whole to be in equilibrium, an investor must be indifferent between the two bonds, which means that the two bonds should yield an identical rate of return. • So we require that: 1+R = 1+r P1(1+r) = (1+R) ----- ----- --------- P1 P0 P0
Real & Nominal Rates (Cont…) • Let us define (P1 – P0)/P0 as , which represents the rate of inflation. • So (1+)(1+r) = (1+R) • This relationship between the nominal or money rate of return, the real or the `goods’ rate of return, and the rate of inflation is called the Fisher Hypothesis.
Real & Nominal Rates (Cont…) • If r and are very small, we can write the approximate relationship as R = r + • In other words, the nominal rate of return is equal to the real rate of return plus the rate of inflation.
Uncertain Inflation • In order to derive the Fisher relationship, we assumed that the inflation rate was known with certainty. • In real life inflation is a random variable. • Hence we can only have an expectation of it. • Thus in real life, even risk-less securities do not provide a guaranteed real rate of return.
Uncertain Inflation (Cont…) • We will first assume that investors do not demand a risk premium for bearing inflation risk, even though the rate of inflation is uncertain. • If the real rate of return required by them is r, then the nominal rate demanded by them will be: R = r + E()
Uncertain Inflation (Cont…) • Thus the required nominal rate of return in this case will equal the required real rate of return plus the rate of inflation. • Once the nominal rate is specified, it will not change. • However the actual real rate of return could differ from the expected real rate, because the actual rate of inflation can be different from the expected rate of inflation.
Uncertain Inflation (Cont…) • Technically we say that the ex-ante inflation measure need not equal the ex-post inflation measure. • The term ex-ante refers to a forecasted measure, whereas the term ex-post refers to a measure based on actual results. • Consequently the realized or ex-post real rate will differ from the expected real rate.
Illustration • Assume that the current inflation is 8%. • Assume that the probability distribution for inflation is as follows: • Inflation Probability 4% .25 6% .25 8% .25 10% .25
Illustration (Cont…) • The expected rate of inflation is: .04x.25 + .06x.25 + .08x.25 + .10x.25 = .07 • If the required real rate of return is 3%, then the nominal rate demanded will be 10%. • However if the actual rate of inflation is 8%, then the realized real rate will be: .10 - .08 = .02 2%
Risk Premia • One of the fundamental assumptions made in Finance, is that investors are risk averse. • Since inflation is uncertain, a higher than expected rate of inflation, could result in a lower than expected real rate of return. • Hence in order to induce investors to purchase financial assets, they must be offered a nominal rate that factors in this uncertainty, besides the expected inflation rate.
Risk Premia (Cont…) • If we denote the premium for bearing this risk as R.P, then the Fisher hypothesis can be rewritten as: R = r + E() + R.P. • Since investors are assumed to be risk averse, the premium will be positive.
Illustration • We will assume that investors expect to be compensated by the difference between the actual rate of inflation and the expected rate of inflation, whenever the actual rate is higher. So: R.P = (.08-.07)x.25 + (.10-.07)x.25 = .01
Illustration (Cont…) • According to our assumption the risk premium demanded is 1%. • So the required nominal rate will be: R = .03 + .07 + .01 = .11 11% • Thus when investors are not indifferent to uncertainty, what matters is not just the expected inflation, but the possibility of deviations from the expected value.
Other Factors Influencing the Rate of Interest 3. Risk and Uncertainty • Inflation risk is obviously one source of risk. • And risk averse investors will demand a risk premium whenever confronted with a source of risk.
Other Factors (Cont…) • In practice, most investments, other than those in government securities, carry a risk of default. • Default Risk, also known as Credit Risk, is the possibility that the borrower may not make principal and/or interest payments as promised at the outset.
Other Factors (Cont…) • Hence, in real life, whenever an investor is contemplating an investment, he will factor in a default risk premium. • The premium demanded will be a function of the perceived creditworthiness of the borrower.
Other Factors (Cont…) 4. Length of Investment • The rate of return demanded by an investor will depend on the time to maturity of the financial claim that he receives in return. • Lenders in general like to lend short-term, whereas borrowers like to borrow long-term.
Other Factors (Cont…) • Thus in order to raise long-term funds, borrowers have to tempt lenders with a higher nominal rate of return. 5. Quality of Information • In practice lenders and borrowers may not have full information about each other. If so, the rate of return required/offered may not be consistent with what is suggested by economic factors alone.
Other Factors (Cont…) 6. Legal Restrictions • In reality many rates of interest are subject to government control. • This has tended to be true of command economies. • But even a country like the U.S. has had legislations like Regulation Q in the past.
Other Factors (Cont…) • To give an example from India, although the general level of interest rates has come down, the government has not reduced the rate of return on contributions to the Employees Provident Fund (EPF). • Thus the EPF rate does not reflect market realities, and is artificially maintained at a high level.
Other Factors (Cont…) 7. Fiscal and Monetary Policies • These have an impact on interest rates. • The RBI controls the money supply in India. • If it wants to increase the supply of money it can buy T-bills from the market. • It can also reduce the SLR/CRR which will increase the funds availability.
Other Factors (Cont…) • An increase in money supply is designed to stimulate the economy, but will invariably have an impact on the inflation rate. • On the contrary if the economy is perceived to be over-heated, the RBI can increase the reserve requirements, or sell T-bills as a part of its Open-Market operations.
Other Factors (Cont…) • The budgetary deficits of the government also impact the system. • The higher the shortfalls in tax collections, the greater will be the need for the government to resort to borrowings. 8. Random Fluctuations • Finally, over and above all the objective causes, interest rates also exhibit random fluctuations.
Simple Interest • According to this principal, whenever we deposit money, the interest accrued per period is based only on the original principal deposited. • That is, interest earned at an intermediate stage, does not itself earn any further interest. • Thus the amount of interest accrued every period is a constant.
Simple Interest (Cont…) • Consider an investor who deposits $ P in an account that pays interest at the rate of r% per period, for a duration of N periods. • At the end of the first period the balance would be P(1+r). • At the end of two periods it would be P(1+2r). • In general, after N periods, it would be P(1+rN). • N need not be an integer. That is, interest may accrue for a fractional period.
Illustration-1 • Caroline deposits $ 10,000 with a bank for 3 years. • Assume that the bank pays simple interest at the rate of 10% per annum. • What will be her balance at the end of 3 years? • At the end of one year she would have 10000(1+.10) = 11000.
Illustration (Cont…) • During the second period, only the original principal of $ 10,000 will earn interest. • So the balance after two years will be 11000 + 1000 = 12000. • Finally, at the end of 3 years, the balance will be 12000 + 1000 = 13000 • 13000 = 10000(1+.10x3) = P(1+rN)
Illustration-2 • Gulati deposited Rs 10000 with ICICI Bank 5 years and 6 months ago. • If the bank pays 8% per annum on a simple interest basis, how much can he withdraw? • P(1+rN) = 10000(1+.08x5.5) = 14,400
Compound Interest • According to the principle of compound interest, when an amount is deposited, it is not only the original principal which earns interest every period, but the interest accrued at the end of an intermediate period continues to earn interest till the maturity of the deposit. • Thus each time interest is accrued, it is automatically reinvested at the same rate for the next compounding period.
Compound Interest (Cont…) • Thus if an investment of $ P is made for N periods, with a bank which pays interest on a compounded basis at the rate of r% per period, then the terminal balance will be: P(1+r)N • Once again, N need not be an integer.