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Lecture  Notes  Class  1  Saturday,  12th  September  2015   Cashflows Cashflows are simply money that is paid or received at particular times. For example, a company operating an amusement park will receive entrance fees and ride fees. The company will pay out money for maintenance, debt repayment and for other management expenses. From the company’s viewpoint the entrance and ride payments are positive cashflows (ie money received) while the maintenance, debt repayments and other expenses are negative cashflows (ie money paid out). Similar cashflows arise in all businesses.

Zero-coupon bond

A zero-coupon bond is a security that is simply a contract to provide a specified lump sum at some specified future date. For the investor there is a negative cashflow at the point of investment and a single known positive cashflow on the specified future date.

Example

An investor gives the issuer of a zero-coupon bond BDT 40,000, and in return the investor will receive BDT 60,000 from the issuer in exactly 2 years’ time. The issuer may be a government or a large company.

Fixed-interest security

The holder of a fixed-interest security will receive a lump sum, of specified amount at a specified future time, together with a series of regular level interest payments until the repayment of the lump sum.

An industrial company, a local authority, or the government of a country or any similar body may raise money by floating a loan on the stock exchange by for e.g. issuing bonds of a stated nominal amount. The holder of a bond will receive a lump sum of specified amount at some specified future time together with a series of regular fixed interest payments until the repayment (or “redemption”) of the lump sum. The regular fixed interest payments are referred to as coupons. Thus a zero-coupon bond has zero interest payments.

Example

An investor buys a 15-year fixed-interest security of nominal amount (or face value) BDT 10,000. The investor is unlikely to pay exactly BDT 10,000 for this security but will pay a price that is acceptable to both parties, which may be higher or lower than BDT 10,000. The investor will then receive a lump sum payment in 15 years’ time. This lump sum is most commonly equal to the nominal amount, i.e. BDT 10,000. The investor will also receive regular payments throughout the 15 years of, say, BDT 500 per annum.

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Index-linked security

Investors are sometimes interested in a security for which the interest payments and final capital repayment are linked to an “index” which incorporates effects of inflation. Therefore future interest payments and final payment are unknown. But it is known that the amounts of the future cashflows will relate to an inflation index, hence, these cashflows are said to be known in “real” terms.

Real terms means taking into account inflation, whereas nominal means ignoring inflation. For example if your wages are rising at 5% pa and inflation is 7% pa, your wages are falling in real terms (you will be able to buy less with your “higher wages”), even though your wages are rising in nominal terms.

Example

If inflation is 10% per time period and the 1st regular coupon payment is BDT 500, then the 2nd payment will be 500×1.10 = BDT 550, and the 3rd payment will be BDT 605 and so on.

Inflation is often measured by reference to an index. For example an inflation index might take values as set out in the table below.

Date 1/1/2001 1/1/2002 1/1/2003 1/1/2004

Index 100.00 105.00 108.00 113.00

The rate of inflation during 2002 is !"#!"#

− 1 = 2.86% per annum.

Note that in practice the operation of an index-linked security will be such that the cashflows do not relate to the inflation index at the time of payment, due to delays in calculating the index. Sometimes the need of the borrower (or investors) to know the amounts of the payments in advance may lead to the use of an index from the previous period.

Cash on deposit

Let’s consider your own bank account that pays interest and allows instant access. You can choose when to deposit money and when to withdraw money. The interest you receive will be based on the current interest rate and this may change at any time. This type of deposit is called a call deposit. There are also term deposits where money is deposited for a fixed term, usually for between one week and one year. The interest rate can be fixed for the term, or vary at specified intervals.

Equity

Equity shares are securities that are held by the owners of an organization. For example if a company issues 4,000 shares and an investor buys 1,000, the investor owns 25 per cent of the company. Shareholders are entitled to a share in the company’s profits in proportion to the number of shares owned. The distributions of

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profits to shareholders are known as dividends.

Example

The following table shows the projected future cashflows for a shareholder who has just purchased a block of shares for BDT 10,000 and expects dividends to increase by 5% every year. Dividends are paid twice yearly. This shareholder expects the two dividends in the 1st year to be BDT 300 each and intends to sell all the shares after 2 years.

Time (years) Purchase price (£) Dividends (£) Sale proceeds (£)

0 –10,000

½ +300

1 +300

1½ +315

2 +315 +11,025

In this example we have assumed that the price also grows at 5% pa.

Annuity certain

An annuity certain provides a series of regular payments in return for a single premium. The number of years for which the annuity is payable, and the frequency of payment, will be specified. The payment amounts may be level or may vary. Annuity policyholders are usually called annuitants. An example is a whole life annuity where a policyholder receives a fixed amount of money per month until he dies, just like a pension.

Interest-only loan

An interest-only loan is a loan that is repayable by a series of interest payments followed by a return of the initial loan amount.

Repayment loan (or mortgage)

A repayment loan is a loan that is repayable by a series of payments, each including a partial repayment of the loan capital in addition to interest payments. In its simplest form, the interest rate will be fixed and the payment amounts will be fixed, paid at regular intervals.

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Practice Questions Q1 Describe the cashflows for an organization that issues a zero-coupon bond. Q2 An investor purchased a three-year index-linked security on 1/1/2001. In return the investor received payments at the end of each year plus a final redemption amount, all of which were increased in line with the index given in the inflation table above. The payments would have been BDT 600 each year and BDT 11,000 on redemption if there had been no inflation. Calculate the payments actually received by the investor. Q3 Repeat Q2 for a two-year index-linked security issued on 1/1/2002. The payments increase in line with the index with a one-year indexation lag, ie the index value one year before each payment is used.

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Time-value of money

Simple Interest

If an amount P is deposited into an account that pays simple interest at the rate i per annum, and the account is closed after n years, then the amount paid to the investor when the account is closed is

𝑃(1+  𝑛𝑖)  

This payment consists of a return of the initial deposit P, plus an interest of niP.

Example

An investor deposits BDT 10,000 in a bank account that pays simple interest at a rate of 5% pa, then after 7 years there will be

𝐵𝐷𝑇  10,000  𝑥  (1  +  0.05𝑥7)  =  𝐵𝐷𝑇  13,500

in the account.

Compound Interest

The essential feature of compound interest is that interest itself earns further interest, hence, it is called compounding. Consider a savings account that pays compound interest at a rate of i per annum. An amount of P is initially deposited into this account. If the account is closed after one year, the investor will receive 𝑃 1+ 𝑖 .

Moreover, let A be the amount received by the investor if the account is closed after n years. Then

𝐴 = 𝑃(1+ 𝑖)!

This payment consists of a return of the initial deposit, P, together with accumulated interest of amount:

𝑃[(1+ 𝑖)! − 1]    

Example

If an investor deposits BDT 5,000 in a bank account that pays compound interest at a rate of 4% pa, then after 6 years there will be

5,000  𝑥  (1+ 0.04)! = 𝐵𝐷𝑇  6,326.60

in the account. The interest accumulated is

5,000  𝑥   1.04! − 1 = 𝐵𝐷𝑇  1,326.60

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Present Values

In the previous section we saw how much a single payment accumulates to at a later time. Actuarial work involves making future payments for e.g. making pension payments to a worker after retirement or making a life assurance payment when an insured individual dies. So actuaries are interested in calculating how much money is needed now (in the present) to provide specific future payments. This amount is called the present value (PV) or discounted value of the payments.

This is the amount of money that if invested at time 0, will give A at time t.

𝑃𝑉 =𝐴

(1+ 𝑖)!

Example

An investor must make a payment of BDT 7,000 in 3 years’ time. The investor wishes to make provision for this payment by investing an amount now in a deposit account that pays 10% per annum compound interest. How much should the initial investment be?

𝑃𝑉 =𝐵𝐷𝑇  7,000(1+ 0.10)! = 𝐵𝐷𝑇  5,259.20

Let’s define the present value factor denoted by v as:

𝑣 =  1

(1+ 𝑖)

Therefore the discounted present value of A due at time t is:

𝑃𝑉 =  𝐴𝑣!

Using this notation, answer to the last example is:

𝑣 =  1

(1+ 0.10) =    0.909091

𝑃𝑉 = 7,000  𝑣! = 7000  𝑥  0.909091! = 𝐵𝐷𝑇  5,259.21

Simple discount

Suppose an amount A is due after n years and a rate of simple discount applies of d per annum. Then required investment now to accumulate to A in n years is

𝐴(1− 𝑛𝑑)

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Example

An 8-month loan, repayable by a single repayment, is issued at a rate of commercial discount rate of 15% per annum. If the amount of the repayment is BDT 100,000, then the initial loan amount is:

𝐵𝐷𝑇  100,000  𝑥   1−812 𝑥0.15 =  𝐵𝐷𝑇  90,000

Note that since the 15% was quoted as a per annum rate, we converted 8 months to !!"

years.

Investing over a period

We begin by considering investments where capital and interest are paid at the end of a fixed term. It is essential in any compound interest problem to define the unit of time for e.g. a month or a year (year is more commonly used). In certain situations, a different period (eg six months) is used as the basic time unit. For example, if BDT 200 is invested in a savings account that pays compound interest at a rate of 2% per half-year then the amount in the account at the end of 18 months is

200  𝑥  (1+ 0.02)!"! = 𝐵𝐷𝑇  212.24

The interest rates we have been discussing so far refers to an “effective rate of interest” that is different from a “nominal” and “flat” rates of interest, which will be discussed later.

Example

An investor makes an initial investment of BDT 9,000 and is credited with BDT 900 interest at the end of 1 year. The effective rate of interest i here is:

𝐵𝐷𝑇  9,000  𝑥   1+ 𝑖 =  𝐵𝐷𝑇  9,000+ 𝐵𝐷𝑇  900

𝑖 =9,9009,000− 1 = 0.10  

or another way to calculate i is !""!"""

= 0.10 or 10%

Converting between different effective rates

The accumulated value of BDT 100 in one year at 10% pa effective is 100  𝑥  1.10 =𝐵𝐷𝑇  110. We could express the interest rate in terms of an effective half-yearly rate, j, such that

100  𝑥  (1+ 𝑗)! = 110

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Solving the above equation gives 𝑗 = 4.88%. Therefore 10% per annum effective is equivalent to 4.88% per half-yearly effective.

A simpler way to convert between effective rates is:

1+ 0.10 ! = 1+ 𝑗 !

𝑗 = 1.10!! − 1 = 4.88%

Similarly for a longer time period, i.e. if the effective quarterly rate is 2%, then the effective annual rate is

1+ 0.02 ! = 1+ 𝑖

𝑖 = 8.24%

Practice Questions

Q1 An investor puts BDT 5,000 in a savings account that pays 10% simple interest at the end of each year. Compare how much the investor would have after 6 years if the money was:

(i) invested for 6 years

(ii) invested for 3 years, then immediately reinvested for a further 3 years.

Q2 Repeat Q1 but with the savings account paying 10% compound interest at the end of each year.

Q3 Calculate v, assuming an effective annual rate of interest of 4%.

Q4 A company is due to receive a payment of BDT 500,000 from a customer in 6 months’ time. To smooth its cashflows, the company would prefer to receive the payment immediately, and has agreed to transfer its entitlement to this payment to a third party (a discount house) in return for an immediate payment calculated using a rate of commercial discount of 16% per annum. How much will the immediate payment made by the discount house be?

Q5 Find the accumulation of a payment of BDT 400 for 6½ years under compound interest at rate 12½ % per annum.

Q6 Calculate the equivalent effective monthly rate and biennial (every 2 years) rate for 10% pa effective