In this article we will discuss about:- 1. Calculation of Interest Rates and Discounting of Cash-Flows 2. Term Structure and Interest Rates.

### Calculation of Interest Rates and Discounting of Cash-flows:

#### Time Value of Money:

The value of money received today is different from the value of money received after some time in the future. An important financial principle is that the value of money is time dependent.

This principle is based on the following four reasons:

Inflation:

Under inflationary conditions the value of money, expressed in terms of its purchasing power over goods and services, declines.

Risk:

Re. 1 now is certain, whereas Re. 1 receivable tomorrow is less certain. This ‘bird-in-the-hand’ principle is extremely important in investment appraisal.

Personal Consumption Preference:

Many individuals have a strong preference for immediate rather than delayed consumption. The promise of a bowl of rice next week counts for little to the starving man.

Investment Opportunities:

Money like any other desirable commodity, has a price, given the choice of Rs. 100 now or the same amount in one year’s time, it is always preferable to take the Rs. 100 now because it could be invested over the next year at (say) 18% interest rate to produce Rs. 118 at the end of one year.

If 18% is the best risk-free return available, then you would be indifferent to receiving Rs. 100 now or Rs. 118 in one year’s time. Expressed another way, the present value of Rs. 118 receivable one year hence is Rs. 100.

#### Simple Interest:

Simple interest is the interest calculated on the original principal only for the time during which the money lent is being used. Simple interest is paid or earned on the principal amount lent or borrowed.

Simple interest is ascertained with the help of the following formula:

Interest = Pnr

Amount = P(1 + nr)

Where, P = Principal

r = Rate of Interest per annum (r being in decimal)

n = Number of years

Illustration 1:

What is the simple interest and amount of Rs. 8,000 for 4 years at 12% p.a.

Solution:

Interest = Pnr

= 8,000 x 4 x 0.12 = Rs. 3,840

Amount (i.e. principal + Interest)

= P(1 + nr) = 8,000 [1+(4X0.12)]

= 8,000 (1 + 0.48)

= 8,000 x 1.48 = Rs. 11,840

Interest = Amount – Principal

= 11,840 – 8,000 = Rs. 3,840

Illustration 2:

At what rate per cent will Rs. 26,435 amount to Rs. 31.722 in 4 years?

Solution:

A = P(1 + nr)

31,722 = 26,435 (1+4x r/100)

31,722 = 26435 + 1,05,740r/100

1,057.40 r = 31,722 – 26,435

r = 5,287/1,057.40= 5

... Rate of interest = 5%

Illustration 3:

A sum deposited at a bank fetches Rs. 13,440 after 5 years at 12% simple rate of interest. Find the principal amount.

Solution:

A = P(1 +nr)

13,440 =P (1+5 x 0.12)

13,440 = P + P0.6

1.6 P = 13,440

P = 13,440/1.6 = 8,400

... Principal amount = Rs. 8,400

#### Compound Interest:

If interest for one period is added to the principal to get the principal for the next period, it is called ‘compounded interest’. The time period for compounding the interest may be annual, semiannual or any other regular period of time.

The period after which interest becomes due is called ‘interest period’ or ‘conversion period’. If conversion period is not mentioned, interest is to be compounded annually. The formula used for compounding of interest income over ‘n’ number of years.

A = P (1 + i)n

Where, A = Amount at the end of ‘n’ period

P = Principal amount at the beginning of the ‘n’ period

i = Rate of interest per payment period (in decimal)

n = Number of payment periods

When interest is payable half-yearly

A = P (1 + i/2 )2n

When interest is payable quarterly

A = P(1+i/4)4n

When interest is payable monthly

A = P (1+i/12)12n

When interest is payable daily

A = P(1+i/365)365n

Illustration 4:

Find out compounded interest on Rs. 6,000 for 3 years at 9% compounded annually.

Solution:

A = P (1 + i)n

= 6,000(1 + 0.09)3

= 6,000 (1.09)3

= 6,000 x 1.29503 = Rs. 7,770

Illustration 5:

What sum will amount to Rs. 5,000 in 6 years’ time at 8 ½ % per annum.

A = P (1 + i)n

= 5,000 (1 +0.085)6

= 5,000 (1.085)6

= 5,000 x 1.63147 = Rs. 8,157

Illustration 6:

Find the compound interest on Rs. 2,500 for 15 months at 8% compounded quarterly.

Solution:

A = P(1+ i/4 )4n

= 2,500 (1+ 0.08/4)4×125

= 2,500 (1+ 0.02)5

= 2,500 (1.02)5

Let x = (1.02)5

... log x = 5 log 1.02 = 5×0.0086 = 0.0430

... x = antilog 0.0430 = 1.104

... A = 2,500 x 1.04 = Rs. 2,760

Compound Interest = 2,760 – 2,500 = Rs. 260

Illustration 7:

Find the present value of Rs. 2,000 due in 6 years if money is worth compounded semiannually.

Solution:

A = P (1+i/2)2n

2,000 = P (1+0.05/2)2×6

2,000 = P (1.025)12

Log 2,000 = log P +12 log 1.025

3.30103 = log P +12 x 0.01072

Log p = 3.30103 – 0.12864

Log p = 3.17239 = 3.1724

P = antilog 3.1724

P = Rs. 1,487.30

... The required present value is Rs. 1,487.30

Compounded interest = 2,000- 1,487.30 = Rs. 512.70

#### Present Value:

It is a method of assessing the worth of an investment by inverting the compounding process to give present value of future cash flows. This process is called ‘discounting’.

The present value of ‘P’ of the amount ‘A’ due at the end of ‘n’ conversion periods at the rate ‘i’ per conversion period.

The value of ‘P’ is obtained by solving the following equation:

P = A (1 + i)n

Illustration 8:

Ascertain the present value of an amount of Rs. 8,000 deposited now in a commercial bank for a period of 6 years at 12% rate of interest.

Solution:

P = A/(1 + i)n

8,000 = A/(1 + i)n

8,000 = A/(1 + 0.12)6

8,000 = A/1.97382

A = 8,000 x 1.97382 = Rs. 15,791

Illustration 9:

Find out the present value of Rs. 10,000 to be required after 4 years if the interest rate is 6%.

Solution:

... An amount Rs. 7,921 to be deposited into bank to get Rs. 10,000 at the end of 4 years at interest rate of 6%.

Calculation of Discount Factors:

The exercise involved in calculating the present value is known as ‘discounting’ and the factors by which we have multiplied the cash flows are known as the ‘discount factors’.

The discount factor is given by the following expression:

Where ‘i’ is the rate of interest per annum and ‘n’ is the number of years over which we are discounting.

Discounted cash-flow is an evaluation of the future cash-flows generated by a capital project, by discounting them to their present day value. The discounting technique converts cash inflows and outflows for different years into their respective values at the same point of time, allows for the time value of money.

Illustration 10:

A firm can invest Rs. 10,000 in a project with a life of three years. The projected cash inflow are as follows:

The cost of capital is 10% p.a. Should the investment be made?

Solution:

Firstly the discount factors can be calculated based on Re. 1 received in with ‘i’ rate of interest in 3 years.

Since the net present value is positive, investment in the project can be made.

The present value of future cashflow can also be ascertained as follows:

Compounding Rate and Capitalising Rate -The compounding rate is used in project evaluation to determine the present value of past investment / cashflow, whereas the capitalising rate is applied in the reverse process of discriminating present value of future cash flows. Both considers the time value of money.

#### Annuity:

An annuity is a cashflow, either income or outgoings, involving the same sum in each period. An annuity is the payment or receipt of equal cashflows per period for a specified amount of time. For example, when a company set aside a fixed sum each year to meet a future obligation, it is using annuity.

The time period between two successive payments is called ‘payment period or ‘rent period. The word ‘annuity’ is broader in sense, which includes payments which can be annual, semiannual, quarterly or any other fixed length of time. Annuity does not necessarily mean payment taken to be one year.

Future Value of Ordinary Annuity – An ordinary annuity is one in which the payments or receipts occur at the end of each period. In a five year ordinary annuity, the last payment is made at the end of the fifth year.

Where,

A = Annual or future value which is the sum of the compound amounts of all payments

P = Amount of each installment

i = Interest rate per period

n = Number of periods

Illustration 11:

Mr. X is depositing Rs. 2,000 in a recurring bank deposit which pays 9% p.a. compounded interest. How much amount Mr. X will get at the end of 5th year.

Solution:

Illustration 12:

Find the future value of ordinary annuity Rs. 4,000 each six months for 15 years at 5% p.a. compounded semiannually.

Solution:

A = P/I [1+i)n-1]

Where, P = Rs. 4,000

i = 0.05/2 = 0.025

n = 15 x 2 =30

A = 4,000/0.025 [(1 + 0.025)30 – 1]

A = 4,000/ 0.025 [(1.025)30-1]

Let x = (1.025)30

Log x = 30 log 1.025

= 30 x 0.0107 = 0.321

x = antilog 0.321 = 2.094

A = 4,000 /0.025 (2.094 – 1)

= 1,60,000 x 1.094 = Rs. 1,75,040

Present Value of Ordinary Annuity:

The present value of an ordinary annuity is the sum of the present value of a series of equal periodic payments.

V = P/I [1-(1+i)-n]

Where, V = Present value of annuity

Illustration 13:

Mr. Y is depositing Rs. 8,000 annually for 4 years, in a post office savings bank account at an interest of 5% p.a. Find the present value of annuity.

Solution:

V = P/I [1- (1+i)-n]

P = Rs. 8,000 i = 0.05 n = 4

V = 8,000/0.05 [1-(1+0205)-4]

= 1,60,000 [1 – (1.05) -4]

Let x = (1.05) -4

Log x = – 4 log 1.05

= – 4 X 0.0212 = – 0.0848

= – 1 + 1 – 0.0848 = 1.9152

x = antilog (1.9152) = 0.8226

V = 1,60,000 x (1 – 0.8226)

= 1,60,000 x .1774 = Rs. 28,384

Present Value of Deferred Annuity – An annuity where the first payment is delayed beyond one year, the annuity is called a ‘deferred annuity’.

The present value ‘V’ of a deferred annuity ‘P’ to begin at the end of ‘m’ years and to continue for ‘n’ years is given by:

Calculation of present value by applying the above formula would be extremely tedious. The simple way of calculation is presented in the following illustration:

Illustration 14:

Z Ltd. intend to invest Rs. 15,000 per annum at the end of years 5, 6, 7 and i of 12%. Find out the present value of the deferred annuity payments.

Solution

Present Value of Perpetuity:

A perpetuity is a financial instrument that promises to pay an equal cash flow per period forever, that is, an infinite series of payments and principal amount never be repaid.

The present value of perpetuity is calculated with the following formula:

V = P/i

Illustration 15:

X Ltd. had taken a freehold land for an annual rent of Rs. 1,200. Find out the present value of freehold land which is enjoyable in perpetuity if the interest rate is 8% p.a.

Solution:

#### Amortisation:

Amortisation is the gradual and systematic writing off of an asset or an account over a period. The amount on which amortisation is provided is referred to as ‘amortizable amount. Depreciation accounting is form of amortisation applied to depreciable assets. Depletion is a form of amortisation in case of wasting assets.

The gradual repayment or redemption of loan or debentures is also referred to as amortisation. Sinking fund method and Insurance policy method are used for systematic writing-off of an asset or redemption of bonds and other long-term debt instruments. Present value of an annuity interest factors can be used to solve a loan amortisation problem, where the objective is to determine the payments necessary to pay off or amortise a loan.

Illustration 16:

Mr. Balu has borrowed a loan of Rs. 5,00,000 to construct his house which repayable in 12 equal annual instalments the first being paid at the end of first year. The rate of interest chargeable on this loan is (a 4% p.a. compounded. How much of equal annual installments payable to amortize the said loan.

V = P/I [1-(1+i)-n]

V = Rs. 5,00,000 I = 0.04 n = 12

5,00,000 = P/0.04 [1-(1+4)-12]

5,00,000 = P/0.04 [1-(1.04)-12]

Let x = (1.04)-12

Log x = -12 log 1.04

= -12 x 0.0170 = -0.204

= -1 + 1 – 0.204 = 1.796

X = antilog 1.796 = 0.6252

5,00,000 = p/0.04 (1-0.6252)

5,00,000 = P/0.04 x 0.3748

5,00,000 = P x 9.37

P = 5,00,000/9.37 = Rs. 53,362

#### Sinking Fund:

It is a kind of reserve by which a provision is made to reduce a liability, e.g., redemption of debentures or repayment of a loan. A sinking fund is a form of specific reserve set aside for the redemption of a long-term debt. The main purpose of creating a sinking fund is to have a certain sum of money accumulated for a future date by setting aside a certain sum of money every year.

It is a kind of specific reserve. Whatever the object or the method of creating such a reserve may be, every year a certain sum of money is invested in such a way that with compound interest, the exact amount to wipe off the liability or replace the wasting asset or to meet the loss, will be available. The amount to be invested every year can be known from the compound interest annuity tables.

Alternatively, an endowment policy may be taken out which matures on the date when the amount required will be paid by the insurance company.

The advantage of this method is that a definite amount will be available while in the case of investment of funds in securities then exact amount may not be available on account of fall in the value of securities. After the liability is redeemed, the sinking fund is no longer required and as it is the undistributed profit, it may be distributed to the shareholders or may be transferred to the General Reserve Account.

Illustration 17:

A machine costs Rs. 3,00,000 and its effective life is estimated to be 6 years. A sinking fund is created for replacing the machine at the end of its effective life time when its scrap realizes a sum of Rs. 20,000 only. Calculate to the nearest hundreds of rupees, the amount which should be provided, every year, for the sinking if it accumulates at 8% p.a. compounded annually.

Solution:

For accumulation in sinking fund at compound rate we have:

A =P/i [(1+i)n-1]

A = 3,00,000 – 20,000 = 2,80,000

i = 0.08

n =6

2,80,000 = P/0.08 [(1+0.08)6 -1]

2,80,000 = P/0.08 [(1.08)6-1]

2,80,000 = P/0.08 (1.586874 -1)

2,80,000 = P/0.08 x 0.586874

2,80,000 = P x 7.33593

P = 2,80,000/7.33593 = Rs. 38,168

### Term Structure and Interest Rates:

#### Interest Rates:

The interest rate is an important consideration for a modern finance manager in taking investment and finance decisions. Interest rates are the measure of cost of borrowing. The interest rates of a country will also influence the foreign exchange value of its own currency. Interest rates are taken as a guide in making investments into shares, debentures, deposits, real estates, loan lending etc.

The interest rates differ in different market segments due to the following reasons:

(a) Risk:

Borrowers carrying high risk will pay higher rates of interest than the borrowers with less risk.

(b) Size of Loan:

The higher amounts of deposits carry higher interest than small deposits.

(c) Profit on Re-Lending:

Financial intermediaries make their profits from re-lending at a higher rate of interest than the cost of their borrowing.

(d) Type of Financial Asset:

Different types of financial assets attract different types of interest. For example deposit in a public sector bank carries interest rate of 10%, but a deposit in a private sector company may attract an interest rate of 15%.

(e) International Interest Rates:

The rate of interest may vary from country to country due to differing rates of inflation, Government policies and regulations, foreign exchange rates etc.

Nominal and Real Rates of Interest:

The nominal rates of interest are the actual rates of interest paid. The real rates of interest are the rates of interest adjusted for the inflation. The real rate is, therefore, a measure of the increase in the real wealth, expressed in terms of buying power, of the investor or lender.

The real rate of interest is calculated as follows:

Real rate of interest = 1 + Nominal rate of interest/1 + Rate of inflation -1

Illustration 18:

The nominal rate of interest is 12% and the rate of inflation is 5%. What is the real rate of interest?

Solution:

Real rate of interest = 1+0.12/1+0.05 -1 = 1.12/1.052-1 = 1.067 -1 = 0.067

... Real rate of interest = 6.7%

Simply, the real rate of interest is calculated as follows:

Real rate of interest = Nominal rate of return – Rate of inflation = 12% -5% = 7%

The real rate of interest will usually be positive, although when the rate of inflation is very high, because the lenders will want to earn a real return and will therefore want nominal rates of interest to exceed the inflation rate. A positive real rate of interest adds to an investor’s real wealth from the income he earns from his investments.

Interest Rates, Capital Gains and Losses:

The increase or decrease in the value of stock is calculated as follows:

Real value of stock = Face value of stock x Nominal rate of stock/Market Nominal rate

Illustration 19:

The long-term guilts issued by the Government with a face value of Rs. 100 and the coupon rate is 10%.

Calculate the resale value of guilts in the following situations:

(a) If the market nominal rate rises to 15%:

Resale value of stock = Rs. 100 x 10%/15% = Rs. 66.67

If the investor sells his stock we will incur a capital loss of Rs. 33.33 (le. Rs. 100 – Rs. 66.67)

(b) If the Market nominal rate falls to 7%:

Resale value of stock = Rs. 100 x 10%/7% = Rs. 142.86

If the investor sells his stock he will get a capital gain of Rs. 42.86 (i.e. Rs. 100 – Rs. 142.86)

Interest Rates and Share Prices:

The shares and debt instruments are alternative ways of investment. If the interest rates on debt instruments fall, shares become more attractive to buy. As demand for shares increases, their prices rise too, and so the dividend return gained from them fall in percentage terms. If interest rates went up, the shareholder would probably want a higher return from his shares and share prices would fall.

Changes in Interest Rates and Financing Decisions:

The changes in interest rates will have strong impact on financing decisions taken by a Finance manager.

Financial strategy to be followed when interest rates are low:

(i) Borrow more moneys at fixed rate of interest to increase the company’s gearing and to maximize return on equity.

(ii) Borrow long-term funds rather than short-term funds.

(iii) Replace the high cost debt with low cost debt.

Financial strategy to be followed when interest rates are higher:

(a) Raise funds by issue of equity shares and to stay away from raising debt finances.

(b) Debt finance can be taken for short-term rather than long-term.

(c) Surplus liquid assets can profitably be invested by switching of investments from equity shares to interest bearing investments.

(d) Reduce the need to borrow funds by selling unwanted and inefficient assets, keep the stocks and debtors balances at lower levels etc.

(e) New projects need to be given careful consideration, which must be able to earn the increased cost of financing the projects.

#### Theories on Term Structure of Interest Rates:

The term structure of interest rates and the levels of interest rates are obviously of prime importance. We will consider first the nature of the different types of interest rates.

The most commonly quoted interest rates in the financial markets are:

(a) The bank’s base rate.

(b) The interbank lending rate.

(c) The treasury bill rate.

(d) The yield on long-dated gilt-edged securities.

The term structure of interest rates describes the relationship between interest rates and loan maturities. Three theories have been advanced to explain the term structure of interest rates:

Expectations Theory:

It asserts that in equilibrium the long-term rate is a geometric average of today’s short-term rate and expected short-term rates in the long run.

Liquidity Preference Theory:

The future is inherently uncertain, thus the pure expectations theory must be modified. In a world of uncertainty investors will in general prefer to hold short-term securities because they are more liquid in the sense that they can be converted to cash without danger of loss of principal. Investor will, therefore, accept lower yields on short-term securities.

Borrowers will react in exactly the opposite way from investors. Business borrowers generally prefer long-term debt because short-term subjects a firm to greater dangers of having to refund debt under adverse conditions. Accordingly firms are willing to pay a higher rate, other things held constant, for long-term funds than for short-term funds.

Market Segmentation Theory:

This theory admits the liquidity preference argument as a good description of the behaviour of investors of short-term. Certain investors with long-term liabilities might prefer to buy long-term bonds because, given the nature of their liabilities, they find certainty of income highly desirable.

Borrowers typically relate the maturity of their debt to the maturity of their assets. Thus the market segmentation theory characterizes market participants’ maturity preferences and interest rates are determined by supply and demand in each segmented market, with each maturity constituting a segment.

Each of these theories carries some validity, and each must be employed to help explain the term structure of interest rates.

#### Yield to Maturity:

Yield to maturity means the rate of return earned on security if it is held till maturity. This can be presented in a graph called ‘yield to maturity curve’ which represents the interest rates and the maturity of a security.

The term structure of interest rates refers to the way in which the yield on a security varies according to the term of borrowing that is the length of time until debt will be repaid as shown by the ‘yield curve’.

In figure 22.1 yield is measured on the vertical axis and term to maturity is on the horizontal axis. Often the yield curve is upward sloping i.e., short-term securities yield less than long-term securities (curve A). Sometimes it is rather flat, short-term yields equal long-term yields (curve B).

And sometimes the yield curve is even downward sloping, short-term interest rates are above long- term rates (curve C). Normally, the longer the term of an asset to maturity, the higher the rate of interest paid on the asset.

In the normal situation, yield to maturity curve is upward sloping for the following reasons:

(a) The risk is more in holding securities for a longer period than short period. This is due to conditions of business which cannot be predicted with accuracy and hence the investors holding long-term securities prefer to be compensated for the additional risk than on the shorter term securities.

(b) In the long-term securities the funds of the investors are tied up for long periods and for this the investors naturally expects for higher return than the short-term securities.

The basic expectations theory maintains that the shape of the yield curve is determined by investors’ expectations of future short-term interest rates. In particular, if short-term rates are expected to rise, the yield curve will be upward sloping, while if short-term rates are expected to fall, the yield curve will be downward sloping.

A specific result is that long-term rates are averages of current and expected future short-term rates.

The relationship between maturity and yields is also known as the ‘term structure of interest rates’ and yield curve is represented as follows:

(i) Yield curves have an upward slope when long maturity bonds have higher interest rates than short maturity bonds.

(ii) Yield curves have a downward slope when short maturity bonds have higher interest rates.

#### Factors Determining Yields:

The general level of yields on stocks is determined by a complex of factors:

(a) Level of Interest Rates:

A change in interest rates will affect security prices and yields as follows:

1. A fall in interest rates brings about a rise in the price of fixed interest securities and a fall in yields.

2. A rise in interest rates has the opposite effect.

(b) Borrower’s Financial Standing:

The Government offers absolute security for its debts so that yields on gilt-edged are the finest in the market and establish a standard for other yields. Since the standing of other borrowers is lower, investors expect higher yields. This difference is known as the ‘yield differential’ or ‘yield gap’.

(c) Duration of the Loan:

The element of risk increases with the duration of the loan so that, investors expect higher yields on long-term bonds than for short, by way of compensation.

(d) General Outlook of Economy:

A bullish outlook for industry or the prospect of inflation and high rates of interest will cause investors to switch to equities to depress the prices of gilts and to raise their yields.

(e) Political Events:

Prices and yields are also influenced by political events, e.g., changes of Government, industrial unrest, publication of trade figures, international crises or any events likely to affect business confidence etc.

#### Yield of Stocks:

The prospective purchaser of gilt-edged stocks will compare the return, or ‘yield, of the investment with its opportunity cost, i.e. what it could earn elsewhere at current interest rates.

Two yields may be calculated as follows:

Flat Yield:

This is the annual return on the investment. It is appropriate for irredeemable stocks. Suppose 14 per cent undated bonds of Rs. 100 are quoted at Rs. 80.

Then:

Coupon rate Rs. 14

Flat yield = Coupon rate/Market price x 100 = Rs.14/Rs. 80 x 100 = 17.5%

Redemption Yield:

The true return on dated stocks must include the rate of interest and any capital gain which results from differences between their cost and redemption price.

Suppose 12 per cent conversion bonds redeemable at par in 2010 were quoted at Rs. 84 in 2009 then:

Flat yield = Rs. 12/Rs.84 x 100 = 14.29%

In addition, the investor will make a capital gain of Rs. 16 over six years since the bond was redeemable for Rs. 100 in 2010.

Capital gain may be expressed as an annual yield for the period. In practice this is found in actuarial tables, which allow for compounding, but it may be estimated as follows:

Capital gain/Years to redemption = Rs. 16/6years = Rs. 2.266 p.a.i.e., 2.66% on bond value of Rs.100

Redemption yield = Flat yield p.a. + Capital gain p.a. = 14.29% + 2.66% = 16.95%.