Notes on basic finance

7/29/2019 Notes on basic finance

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THE TIME VALUE OF MONEY

The phrase the time value of moneyrefers to the idea that a given amount of is worth much more in

hand presently than at a later period.

A recurring concept is that of future value, which refers to the amount of money an investment will

grow to over some time period at some given interest rate.

Future value of a specific amount will depend on the nature of the interest element. In the case of

simple interest, interest is only earned on the initial amount invested over any given period. So for

example, an amount of $100 invested at a rate of 5% per year, will at the end of the first year grow to

$105, while at the end of the second year, it will grow to $110. The amount of $110 obtained after the

two years consists of the initial amount invested and the constant interest earned over each of the two

years.

There is also compound interest, in which case, interest is also earned on all interest accumulated prior

to the current period. If the $100 had been invested at a rate of 5% compounded, the invested amount

will grow to $105 after the first year but in the second year, interest will also be earned on the

additional $5 that the initial sum of $100 has grown by. So in the second year, 5% interest is earned on

the initial amount of $100, and 5% is earned on the additional $5 that has accumulated. This gives a

total additional interest element of $5.25 ($5 on the $100, and $0.25 on the $5 gained after one year).

Adding this up gives a total amount of $110.25, after two years.

In our analysis we shall be resorting more often to compound interest. Under compound interest, the

future value of any amount Cinvested for time tat a rate ris given by:

Future Value = C(1 + r)t

Note that ris very often the rate per annum and that tis often given in years.

The expression (1 + r)tis referred to as thefuture value interest factoror justfuture value factor.

In the case of an investment of $100 for five years at a rate of 5% per annum, the future value after the

five years will be:

$100(1 + 0.05)5 = $127.63

Present Value: Assuming at a rate rwe wish to determine the amount that has to be invested presently

in order for us to have Cdollars in tyears, we use the formula:

PV = C/ (1 + r)t, where PV or the Present Value is the amount to be invested presently at rate rfor t

years in order for us to have C dollars by time t.

In this instance, ris known as the discount rate and the expression 1 / (1 + r)t

is known as the discount

factor of the present value factor.


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Calculating the present value of a future cash flow to determine its worth today is commonly called

Discounted Cash Flow Valuation.

For example, assuming we wanted to determine how much we have to invest today in order to have

$1000 in 3 years if we earn 15% per annum, on our investment.

Let us first determine the discount factor. In this instance, ris 0.15 and tis 3. The discount factor

therefore becomes:

1 / (1 + 0.15)3 = 0.6575

This means PV is $1000 X 0.6575 = $657.50

Present Value for Annuity Cash flows

Suppose we want to examine an asset that promises to pay $500 at the end of each of the next three

years, assuming we wanted to earn 10% on our money each year. Applying the formula PV = C/ (1 + r)t

for each of the three years, we end up with

PV = 500/1.1 + 500/1.12 + 500/1.13

= 454.55 + 413.22 + 375.66

= 1243.43

The general formula for the present value of an annuity ofCdollars per period for tperiods at a rate of

return or interest rate ofris given by:

PV of annuity: Cx [ 1 (1/(1 + r)

t

)]/r

Future Value for Annuity Cash flows

Suppose you plan to contribute $2000 every year to a retirement account paying 8%. How much will you

have in 30 years when you retire? The future value of an annuity ofCdollars per period for tperiods at a

rate of return or interest rate ofris given by:

FV of annuity: Cx [( 1 + r)t 1]/r

Annuity due

An important alteration of an annuity, is an annuity due, in which cash flows occur at the beginning of

each period. For a loan repayments, the initial installment is usually made a month after the loan is

granted; this is an example of an ordinary annuity. However in the case of a lease of an apartment, the

first payment is usually due immediately


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An ordinary annuity is converted to an annuity due simply by multiplying the ordinary annuity by the

discount factor (1 + r). Hence the present value or future value of an annuity due is obtained simply by

multiplying the present value or the future value (respectively) of the corresponding ordinary annuity by

(1 + r)

Perpetuities

An annuity in which the cash flows continue forever are referred to as a perpetuity. In Canada and the

United Kingdom, perpetuities are called consols.

The present value of a perpetuity ofCdollars per period at a rate ris given by C/r

Preferred Stock

Preferred stock or preference stock is an important example of a perpetuity. When a corporation

sells preferred stock, the buyer is promised a fixed cash dividend every period usually quarterly

forever. This dividend must be paid before any dividend can be paid to regular stockholders, hence the

term preferred.

Effective Annual Rates and Compounding

The stated interest rate is the interest rate expressed in terms of the interest payment made each

period. Effective annual rate is the interest rate expressed as if it were compounded once per year.

If a rate is quoted as 10% compounded semiannually, it means that the investment actually pays 5%every six months. But 5% every six months does not yield 10% (the stated interest rate); rather it yields

10.25% (the effective annual interest rate (EAR))

Given a quoted rate and m,the number of times the interest is compounded, the formula for the

effective annual interest is given by:

EAR = [1 + (Quoted rate / m)]m

1

Now there is no limit to the number of times that a particular amount may be compounded within a

year or a unit period. Generally, as the number of times the amount is compounded increases, the

effective annual rate also increases. Ifq is the number of times the interest is compounded, as q getsextremely large, the effective annual rate (EAR) approaches: eq 1


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Loan types and loan amortization

Whenever a lender grants a loan, some provision will be made for the repayment of the principal. A loan

might be repaid in equal installments for example, or it might be repaid in a single lump sum. There is

actually an unlimited number of possible repayment arrangements, as the repayment arrangement is

usually determined by the parties involved. There are however some forms of repayment that come upmore often than the rest; we have pure discount loans, interest-only loans and amortization loans.

Pure discount loan: The borrower receives money today and repays a single lump sum at some time in

the future. A one-year 10% pure discount loan requires the borrower to repay $1.10 for every dollar

borrowed today. Assuming a lender wants a 12% rate on a five year loan that requires the borrower to

repay $25,000 at the end of the loan term, how much would the borrower have to lend today? The

answer would be to find the present value of $25,000 paid five years from now at a rate of 12%.

PV = C /(1 + r)t, with r = 12% and t = 5 and C = $25,000

= $25,000 / (1.12)5

= $14,186

Interest-only loans: Require the borrower to pay interest each period and to repay the entire principal

at some point in the future. For example, with a three-year 10% interest-only loan of $1,000, the

borrower will pay $1000 x 0.1 = $100 at the end of each of the first two years and pay the $1000 in

addition to the interest of $100 for the last year. In the extreme, the borrower pays the interest every

period forever and never repays the principal. This would be an example of a perpetuity. Most corporate

bonds have the general form of an interest-only loan.

Amortized loans: The process of providing for a loan to be paid off by making regular principal

reductions is called amortizing the loan. A simple way of amortizing a loan is to have the borrower pay

interest each period plus some fixed amount. For example, suppose a business takes out a $5000, five-

year loan at 9%. The loan agreement calls for the borrower to pay the interest on the loan balance each

year and to reduce the loan balance each year by $1000. Because the loan amount declines by $1000

each year, it is fully paid in five years. The total payment will decline each year. The reason is that the

loan balance goes down, resulting in a lower interest charge each year, whereas the $1000 principal

reduction is constant. For example, the interest in the first year will be $5000 x 0.09 = $450. The total

payment will be $1000 +$450 = $1450. In the second year, the loan balance is $4000, so the interest is

$4000 x 0.09 = $360, and the total payment is $1,360.

Probably the most common way of amortizing a loan is to have the borrower make a single, fixed

payment every period. Almost all consumer loans such as car loans and mortgages work this way.Suppose the initial five-year, 9%, $5000 loan was amortized this way, how would the amortization

schedule look? We would first have to determine the payment. From earlier discussions, we know that

the cash flows are in the form of an ordinary annuity, solved as follows:

$5000 = C[1 (1/( 1 + 0.09)5)]/0.09


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Solving, we find Cto be $1285.46. So for fixed payments every period, the borrower will have to pay

$1285.46 every year.


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INTEREST RATES AND BOND VALUATION

Usually, when the government or a corporation wishes to borrow money from the public on a

long-term basis, it issues or sells debts securities that are generically called bonds.

A bond is normally an interest-only loan, meaning simple interest charged on the loan is paidevery period and the principal is paid in full only at the end of the loan duration. Suppose a

corporation wishes to borrow $1000 for 30 years. The interest rate on similar debts issued by

similar corporations is 12%. So our corporation decides to pay 0.12 x $1000 = $120 in interest

each of the 30 years. At the end of the 30 years, the corporation will also have to repay the

initial $1000.

The $120 that the corporation promises to make in regular interest payments is termed as the

bonds coupons. Constant annual coupon payments characterize a level coupon bond. The

amount that will be repaid at the end of the loan tenure is called the bondsface value or par

value. In the example cited above, the face value is $1000. When a bond sells for its face value

or par value, it is termed as apar value bond. Dividing the annual coupon by the face value of a

bond gives you the coupon rate on the bond. The coupe on rate in the example cited above is

$120/$1000 = 12%. The number of years remaining till the face value of the bond is paid is

termed as the time to maturityof the bond.

Interest rates change in the market place but the cash flows from a bond however, stay the

same. The value of a bond tends to fluctuate as a result. A rise in market interest rates results in

a decline of the present value of a bonds remaining cash flows, decreasing the value of the

bond. When interest rates fall, there is an increase in the present value of the bonds remainingcash flows and the bond is worth more.

To determine the value of a bond at a particular point in time, we need to know the number of

periods remaining until maturity, the face value, the coupon and the market interest rate for

bonds with similar features. The interest rate required on a bond in the market is termed as the

bonds yield to maturity.

Suppose a corporation were to issue a bond with a maturity of 10 years and an annual coupon

of $80. Similar bonds are known to have a yield to maturity of 8%. How much would this bond

sell for?

We know the corporation will pay $80 per year for the next 10 years in coupon interest. And in

10 years, the corporation will pay the owner of the bond $1000.

The coupons constitute an annuity component whilst the face value to be paid at maturity is a

lump sum.


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This suggests that the value of the bond can be determined as follows, ifCis the coupon paid

per year and Fis the face value of the bond.

Bond Value = C[ 1 (1/(1 + r)t)]/r + F/ (1 + r)t

This is equivalent to the sum of the present value of the coupons and the present value of face value ofthe bond. Applying this to the above example, we obtain

Bond Value = $ 80 [1 (1 / (1 + 0.08)10)]/0.08 + $ 1000 / (1 + 0.08)10

= $ 80 (1 1/2.1589)/0.08 + $ 1000 / 2.1589

= $ 80 x 6.7101 + $ 1000 / 2.1589

= $ 536.81 + $ 463.19

= $ 1000

The bond sells for exactly its face valued. This is similar to an interest-only loan issue at 8% for 10 years.

To illustrate what happens when interest rates increase, let us assume that after a year, the market

interest rate has risen to 10%. The time to maturity is now 9 years and the new bond value is given by:

Bond Value = $ 80 [1 (1 / (1 + 0.10)9)]/0.10 + $ 1000 / (1 + 0.10)9

= $ 80 x 5.7590 + $ 1000 / 2.3579

= $ 460.72 + $ 424.10

= $ 884.82

It can be seen that the present value of the bond now declines. The market rate is now 10%, meaning

that the bond now pays less than the going rate. Investors are therefore now willing to lend something

less than the $1000 promised repayment. A bond that sells for less than its face value is called a discount

bond.

Let us now consider a scenario in which the yield to maturity of the bond declines. To illustrate what

happens when interest rates decreases, let us assume that in the previous scenario instead of the

market interest rate rising after a year, it drops to 6%. The time to maturity is still 9 years and the new

bond value is given by:

Bond Value = $ 80 [1 (1 / (1 + 0.06)9)]/0.06 + $ 1000 / (1 + 0.06)9

= $ 80 x 6.8017 + $ 1000 / 1.6895

= $ 544.14 + $ 591.89

= $ 1136.03


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The value of the bond now is about $136 in excess of the par value. The bond has a coupon rate of 8%

when the yield to maturity is 6%. Investors are therefore willing to pay premium to get the amount in

excess of the coupon that should be offered at the prevailing yield to maturity. A bond that sells at a

premium sells for more than the face value is called a premium bond.

Interest rate risk and bonds

Generally, all other things being equal:

1. The longer the time to maturity of a bond, the greater the interest rate risk.2. The lower the coupon rate, the greater the interest rate risk.

The other thing to note about interest rate risk is that, like most elements in finance and economics, it

increases at a decreasing rate. If we compare a 10-year bond to a 1-year bond, it becomes evident that

the 10-year bond has a much higher interest rate risk. If we were to compare a 20-year bond to a 30-

year bond, we would find that the 30-year bond has a somewhat greater interest rate risk than the 20-

year bond though the difference between the interest rate risks of the 20-year bond and that of the 30-year bond will not be as pronounced as the difference between the interest rate risks of the 1-year bond

and the 10-year bond.

If two bonds have different coupon rates with the same maturity, the value of the one with the lower

coupon is proportionately more dependent on the face value to be received at maturity. As a result, the

value of the bond with the lower coupon will fluctuate with changes in interest rate, all other things

being equal. Another way of explaining this is to say that the bond with the higher coupon has a larger

cash flow earlier on in its life, rendering its value less sensitive to changes in the discount rate.

Some useful terms in the analysis of bonds

Indenture: The indenture is the written agreement between a corporation (borrower) and its creditors.

It is sometimes referred to as the deed of trust. Usually a trustee (such as a bank) is appointed by the

corporation to represent bondholders. The trust company must:

1. Make sure the terms of the indenture are obeyed.2. Manage the sinking fund3. Represent the bondholders in default, ie, if the company defaults on its payments to the

bondholders.

A bond indenture is a legal document and usually includes the following provisions:

1. The basic terms of the bond2. The total amount of bonds issued3. A description of the property used as security.4. The repayment arrangements5. The call provisions6. Details of the protective covenant.


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A bond can either be in a registered form or a bearer form.

Registered Form: In registered form, the registrar of the company records the ownership of each bond;

payment is made directly to the owner of the record.

Bearer Form: the form of bond issue in which the bond is issued without record of the owners name;

payment is made to whoever holds the bond.

Collateral:This is a general term that frequently means securities such as bonds and stocks that are

pledged as security for payment of debts. For example, collateral trust bonds often involve a pledge of

common stock held by the corporation. However, the term collateral is more commonly used to refer to

any asset pledged on a debt.

Mortgages security: are securities secured by a mortgage on the real property of the borrower. The

property involved is usually real estate such as land or buildings. The legal document that describes the

mortgage is called a mortgage trust indenture or trust deed. Sometimes mortgages are on specific

properties, for example, a railroad car. More often blanket mortgages are used. A blanket mortgagepledges all the real property owned by the company.

Debentures: are unsecured debts usually with a maturity of 10 years or more.

A Note is an unsecured debt usually with a maturity of less than 10 years.

Sinking Fund: An account managed by the bond trustee for early bond redemption.

Call provision: An agreement giving the corporation the option to repurchase the bond at a specified

price prior to maturity.

Call premium: The amount by which the call price exceeds the par value of the bond.

Deferred Call Provision: A call provision prohibiting the company from redeeming the bond prior to a

certain date.

Call protected bond: A bond which, during a certain period, cannot be redeemed by the issuer.

Protective Covenant: A part of the indenture limiting certain actions that might be taken during the

term of the loan, usually to protect the lenders interest.


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Types of Bonds

Government Bonds: these are bonds that are issued by government. In America, there are treasury

issues and there are municipal bonds. Municipal bonds have varying degrees of default risk and are

rated much in the same manner as corporate bonds. Coupons on municipal bonds are also exempt from

federal income taxes not state income taxes making them attractive to high-income, high-tax bracketinvestors. Because of the enormous tax break they receive, the yields on municipal bonds are much

lower than those on taxable bonds.

Example, suppose taxable bonds are currently yielding 8%, while at the same time, municipal bonds of

comparable risk and maturity are yielding 6%. Which is more attractive to an investor in a 40% tax

bracket? What is the break-even tax rate? How do you interpret this rate?

For an investor in a 40% tax bracket, a taxable bond yields 8 x (1 0.4) = 4.8% after taxes, making the

municipal bond more attractive. The break-even tax rate is the tax rate at which an investor is

indifferent between a taxable and a non-taxable issue. If we let tstand for the break-even tax rate, then

we can solve for it as follows.

0.08 x (1t) = 0.06 => t= 0.25

Implying, an investor in a 25% tax bracket would make 6% after taxes in either scenario.

Zero coupon bonds: is a bond that pays no coupons at all. Such bonds are offered at price that is far less

than their stated value.

Floating Rate bonds: have adjustable coupon payments. The adjustments are tied to an interest rate

index such as the Treasury bill interest rate or the 30-year Treasury bond rate. The value of a floating-

rate bond depends on exactly how the coupon payment adjustments are defined.

Inflation and Interest Rates

Real rates of interest are rates that have been adjusted for inflation. Nominal rates of interest are rates

that have not been adjusted for inflation.

Let us assume that prices are rising by 5% annually. We say the rate of inflation is 5%. If there is an

investment that costs $100 today and will be worth $115.50 in one year, the investment will have a

15.5% rate of return. This percentage does not account for the effect of inflation. Assuming bread costs

$5 apiece at the beginning of the year, we can buy 20 loaves with our initial investment of $100. A 5%

rate of inflation brings the price of bread to $5.25 at the end of the year. The $115.50 that our $100

grows to after being invested for a year, can buy 115.50/5.25 = 22 loaves of bread at the end of the year.

This is a 10% increase on the initial amount of 20 loaves that could be bought at the beginning of the


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year. In this scenario, our 15.5% is the nominal rate of return, which has not been adjusted for inflation.

The 10% by which our buying power actually goes up is termed the real rate of return.

Another way of looking at this is to say that at 5% inflation, each nominal dollar that we now have is

worth 5% less in real terms. So the real dollar value in an investment of 115.50 is: $115.50/1.05 = $110.

The nominal rate on an investment is the percentage change in the amount of money the number of

say, dollars in hand at the end of the investment period. The real rate on an investment is the

percentage change in how much you can buy with the dollars in hand the percentage change in the

buying power.

The Fisher Effect goes further to shed light on the real relationship between the nominal and real rates

of interest or return. Let R stand for the nominal rate and rstand for the real rate. According to the

Fisher effect, the relationship between the two rates can be written as:

1 + R = (1 + r) (1 + h), where h is the inflation rate.

Referring back to the example involving the bread, we have: 1 + 0.1550 = (1 + r) (1 + 0.05) => r= 10%

The following terms are of significance when discussing the determinants of bond yields:

Term Structure of Interest Rates: refers to the relationship between short- and long-term interest rates.

It refers to the relationship between nominal interest rates on default-free, pure discount securities and

time to maturity ie, the pure time value of money.

Inflation premium: The portion of a nominal interest rate that represents compensation for expected

future inflation.

Interest rate risk premium: The compensation investors demand for bearing interest rate risk.

Default risk premium: The portion of a nominal interest rate or bond yield that represents

compensation for the possibility of default.

Taxability premium: The portion of a nominal interest rate or bond yield that represents compensation

for unfavorable tax status.

Liquidity premium: the portion of a nominal interest rate or bond yield that represents compensation

for lack of liquidity.


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STOCK VALUATION

A share of common stock is more difficult to value in practice than a bond for at least three reasons:

1. Not even the promised cash flows are known in advance.2. The life of the investment is essentially forever, since common stock has no maturity.3. There is no way to easily observe the rate of return that the market requires.

Three simplifying assumptions about the pattern of future dividends can be made to enable us come up

with a value for the stock. The assumptions include the following:

1. The dividend has a zero growth rate.2. The dividend grows at a constant rate.3. The dividend grows at a constant rate after some length of time.

Zero Growth Rate: For a zero growth share of common stock, such as a share of preferred stock, the

dividend Dthas zero growth and is constant through time. Therefore D1 = D2= D3= Dt= D =constant

The value of the stock at time 0, P0 = [D1/(1+R)] + [D2/(1+R)2] + [D3/(1+R)

3] + ..

Therefore, P0 = D/R, where R is the required rate of return

Constant Growth Rate: Suppose the dividend for some company always grows at a steady rate g. If D0 is

the dividend just paid, then the next dividend D1 is given by D1 = D0 (1 + g). The dividend in two periods

is D2 = D1 (1 + g) = D0 (1 + g) (1 + g) = D0 (1 + g)2. Accordingly, the dividend at time t is given by:

Dt= D0 (1 + g)t

An asset with cash flows that grow at a constant rate forever is called a growing perpetuity. Assuming

the Ashanti Corporation has just paid a dividend of $3 per share and the dividend of this company grows

at a steady rate of 8% per annum, what will the dividend be in five years. Here, tis 5 and g is 0.08 and D0

is $3.

Therefore, D5 =$ 3 x (1 + 0.08)5 = $4.41

In the case of constant growth, assuming a rate of return R, P0, the price/value of the stock at the

present, becomes:

P0 = [D1/(1+R)] + [D2/(1+R)2] + [D3/(1+R)

3] +

= [D0 (1 + g)/(1+R)] + [D0 (1 + g)2/(1+R)2] + [D0 (1 + g)

3/(1+R)3] +

= D0 (1 + g) / R g

= D1/( R g)


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The final result is what is termed as the dividend growth model. And it can be expressed thus:

Pt = Dt (1 + g) / R g = Dt+1/( R g)

Suppose = D0 is $2.30, R is 13% and g is 5%. The price per share in this case becomes:

P0 = D0 (1 + g) / R g

= $2.30 x (1 + 0.05) / (0.13 0.05)

= $30.19

D5 = D0 (1 + g)5 = $2.30 x 1.055 = $2.935

P5 = D5 (1 + 0.05) / 0.13 0.05 = $2.935 x 1.05 / 0.08 = $38.53

Suppose the next dividend a company will pay is $4 and investors require 16% return on such companies

as the one we are considering. The companys dividend increases by 6% every year. Based on the

dividend growth model, what is the value of the companys stock today? What is the value in four years?

Nonconstant growth: For a simple example of nonconstant growth, assume a company is currently not

paying dividends. It is predicted that in five years the company will pay a dividend for the first time to

the tune of $0.50 per share. It is expected that this dividend will grow at a rate of 10% per year

indefinitely. The required rate of return on companies such as this one is 20%. What is the price of the

stock today?

The first dividend will be paid in 5 years and will grow steadily from then on so using the dividend

growth model we can say that the price in four years will be:

P4 = D4 (1 + g) / R g = D5 / R g = $0.05/ (0.2 0.1) = $5

If this is how much the stock will be worth in four years, then we can calculate the present value by

discounting this price back four years at 20%.

P0 = $5 /(1 + 0.2)4 = $2.41

What happens if the dividends are not zero for the first few years. Let us suppose that the expected

dividend after the first year is $1, $2 after the second year and $2.50 after the third year. After the third

year, dividends will grow at a constant 5% per annum. The required rate of return is 10%. What is the

value of the stock today?


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A most important thing to notice in this question, is when constant growth starts. It starts at Time 3,

implying the stock price at Time 3, P3, can be determined using the constant growth model.

P3 = D3 (1 + g) / R g = $2.50 x (1 + 0.05)/(0.10 0.05) = $52.50

The total value of the stock can now be determined as the present value of the first three dividends plus

the present value of the price at Time 3, P 3.

P0 = [D1/(1+R)] + [D2/(1+R)2] + [D3/(1+R)

3] + [P3/(1+R)3]

= [$1/(1+0.1)] + [$2/(1+0.1)2] + [$2.50/(1+0.1)3] + [$52.50/(1+0.1)3]

= $0.91 + $1.65 + $1.88 + $39.44

= 43.88

Components of a required return

From the formula P0 = D1 / R g , we derive R = D1 / P0 + g. R is therefore said to have two components.

The expression D1 / P0 is termed as the dividend yield. Because the dividend yield is calculated as the

expected cash dividend divided by the current price, it is conceptually similar to the current yield on a

bond. Is the dividend growth rate as well as the rate at which stock price grows. Thus this growth rate

can be interpreted as the Capital gains yield ie the rate at which the value of the investment grows.

Therefore R = Dividend yield + Capital gains yield.

If a stock sells for $20 per share and the next dividend will be $1 per share and the dividend is expected

to grow by 10% per year more or less indefinitely. What return does this stock offer if the expected

growth rate is correct?

The total return, R = Dividend yield + Capital gains yield = D1 / P0 + g

Therefore R = $1/20 + 10% = 5% + 10% = 15%. The stock therefore has an expected return of 15%.

Some terms used in the Stock Market

Primary market: the market in which new securities are originally sold to investors.

Secondary market: The market in which previously issued securities are traded among investors.

Dealer: An agent who buys and sells securities from inventory. He maintains an inventory and stands

ready to buy or sell at any time.

Broker: An agent who arranges security transactions among investors.


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RETURN, RISK AND THE SECURITY MARKET LINE

Suppose there are two different stocks, namely Stock A and Stock B, with expected returns of 20% and

25% respectively, questions arise as why any investor will choose Stock A over B. The reason depends on

the level of risk of each of these investments. The return on B, though expected to be 15% could actually

end up being much lower.

Suppose there are only two possible situations in the economy; a boom and a recession, both of which

are equally likely to happen. Assuming the rate of return on Stock A is 30% during a recession and 10%

during a boom, the expected rate of return on Stock A is: E(RA) = 0.50 x 30% + 0.50 x 10% = 20%. If the

rate of return on Stock B is 20% in a recession and 70% in a boon, the expected rate of return on Stock

B is: E(RB) = 0.50 x20% + 0.50 x 70% = 25%

In the market, there is a reward, on average, for bearing risk. The reward is commonly known as the risk

premium. The risk premium is the difference between the return on a risky investment and that on a

risk-free investment. When projected returns are used, we determine the expected risk premium as the

difference between the expected return on a risky investment and the certain return on a risk-free

investment. Risk premiums are generally larger for riskier investments.

Suppose risk-free investments are currently offering 8% ie the risk-free rate, Rfis 8%, and the expected

return on Stock A is 20%,

Risk premium = Expected return Risk-free rate

= 20% 8%

= 12%

Similarly, the risk premium on Stock B is: 25% 8% = 17%.

To calculate the variance of the returns on Stock A and Stock B, we first determine the squared

deviations from the expected return and then sum the products of each squared deviation and its

corresponding probability. The standard deviation is always the square root of the variance.

In the case of Stock A, the variance will be: 0.5 x (30% 20% )2 + 0.5 x (10% 20%)2 = 0.01.

The standard deviation will be the square root of the variance: 0.01 = 0.1 = 10%

The variance for Stock B will be: 0.5 x ( 20% 25%)2 + 0.5 x (70% 25%)2 = 0.2025

And the standard deviation for Stock B will be: 0.2025 = 0.45 = 45%

From this, it is evident that Stock B may have the higher expected return, but it also has the higher risk


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The Expected Return on a Portfolio

Let us now consider a portfolio of assets. A portfolio weight is the percentage of the portfolios total

value that is invested in a particular asset. Assuming we have a portfolio with half its portfolio weight

invested in Stock A and the other half invested in Stock B. Suppose the economy enters a recession. In

this case, the return on the half of the money invested in Stock B is 20% and the returns on the otherhalf of the money invested in Stock A is 30%. The portfolio return in a recession, is therefore:

RP = 0.50 x 20% + 0.50 x 30% = 5%

In case there is a boom, the portfolio return will be: RP = 0.50 x 70% + 0.50 x 10% = 40%

The expected return on the portfolio can therefore be found as the sum of the products of each

portfolio return and the probability of the corresponding state of the economy that yields it.

And therefore, E(RP) = 0.50 x 5% + 0.50 x 40% = 22.5%

Alternatively, the expected return on the portfolio can be found as the sum of the product of the

expected return of each asset in the portfolio and its corresponding portfolio weight. In this instance,

the portfolio expected return is: E(RP) = 0.50 x E(RA)+ 0.50 x E(RB) = 0.5 x 20% + 0.5 x 25% = 22.5%

Letxibe the percentage of the total value of our portfolio invested in Asset iin a portfolio consisting ofn

assets. The expected return on our portfolio is then given by:

E(RP) =x1 E(R1) +x2 E(R2) +x3 E(R3) +x4 E(R4) +. +xn E(Rn)

Suppose we have a portfolio consisting of Stocks A, B and C. The economy has a 60% chance of going

into recession and a 40% chance of experiencing a boom. In a boom, the returns on each of Stocks A, B

and C are 10%, 15% and 20% respectively. In a recession, the returns on Stocks A, B and C are 8%, 4%

and 0% respectively. What would be the expected return on the portfolio if the portfolio is equally

weighted, ie the portfolio has equal investments in each asset or the portfolio weights are the same for

all assets?

We would first have to determine the expected return on each asset.

For Stock A: E(RA) = 0.40 x 10% + 0.60 x 8% = 8.8%

For Stock B: E(RB) = 0.40 x 15% + 0.60 x 4% = 8.4%

For Stock C: E(RC) = 0.40 x 20% + 0.60 x 0% = 8.0%


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Because the portfolio weights are the same and there are three assets in this portfolio, the portfolio

weight is . The expected return on the portfolio can therefore be calculated as:

E(RP) =xA E(RA) +xB E(RB) +xCE(RC)

E(RP) = x 8.8% + x 8.4% + x 8.0%

= 8.4%

Assuming instead of the portfolio being equally weighted, half of the portfolio were in Stock A and the

remaining weight were equally divided between Stock B and Stock C?

In that case, xAwill be 2 ,xBwill be 4 andxCwill be 4 and

E(RP) = 2x 8.8% + 4x 8.4% + 4 x 8.0% = 8.5%

The Variance of a Portfolio

We refer back to Stock A with returns of 30% during a recession and 10% during a boom and an

expected rate of return of 20%; and Stock B, with the return of 20% in a recession and 70% in a boom

and an expected rate of return of 25%. Let us assume that this time round, 9/11 of the portfolio is

invested in Stock A and 2/11 of the portfolio is invested in Stock B. During a recession, the portfolio will

now have a return of:

RP = 2/11 x 20% + 9/11 x 30% = 20.91%

During a boom, we expect the portfolio to have a return of:

RP = 2/11 x 70% + 9/11 x 10% = 20.91%

The return does not change in spite of what happens. No further calculations are needed. This portfolio

has zero variance. Combining assets into portfolios can substantially alter the risks faces by the investor;

a point whose implications we will explore as we go along.

The variance on a portfolio is not a simple combination of the variances of the assets in the portfolio.

Referring back to the portfolio consisting of Stocks A, B and C. The economy has a 60% chance of goinginto recession and a 40% chance of experiencing a boom. In a boom, the returns on each of Stocks A, B

and C are 10%, 15% and 20% respectively. In a recession, the returns on Stocks A, B and C are 8%, 4%

and 0% respectively. What would be the variance on the portfolio if half of the portfolio is invested in

Stock A and the remaining half is equally split between Stock B and Stock C?


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In a boom, the portfolio return will be: RP= 2x 10% + 4x 15% + 4 x 20% = 13.75%

In a recession, the portfolio return will be: RP= 2x 8% + 4x 4% + 4 x 0% = 5%

The expected return of the portfolio was initially calculated to be 8.5%

The variance of the portfolio is thus given as: 0.4 x (0.1375 0.085) + 0.6 x (0.05 0.085) = 0.0018375

Systematic and Unsystematic RiskThe unanticipated part of the return; that portion resulting from surprises, is the true risk of any

investment. Investments will be rendered risk-free if our expectations of the market were always met.

The risk of owning an asset comes from surprises; the occurrences that we do not anticipate. There are

however, important differences among the various sources of risk. There are certain events whose

implications may be specific to a corporation. On the other hand, announcements about interest rates,

of GDP are clearly important for nearly all companies, whereas developments involving the president of

a corporation, the corporations research or its sales, is of specific interest to the corporation.

The type of unanticipated development that affects a large number of assets, each to a greater or lesser

extent, is termed as systematic risk. Owing to the fact that systematic risks usually result in effect that

impact the market in its entirety, they are often termed as market risks.

An unsystematic riskis one that affects a single asset or a small group of assets. Because these risks are

unique or specific to individual assets, they are sometimes termed as unique or asset-specific risks.

Actual return in the business environment is said to consist of the expected component and the surprise

component. Therefore, R = E(R) + U. The total surprise component, however, has a systematic portion

and an unsystematic portion so:

R = E(R) + Systematic portion + Unsystematic portion

The significant thing about this equation is that it isolates the portion of the total surprise that is specific

to a company or an asset or a small group of assets.

Diversification and Portfolio Risk

It was seen earlier that in principle, the risk of a portfolio can be quite different from the risk of the

assets that make up the portfolio. Some of the risk associated with individual assets can be eliminated

by forming portfolios. The process of spreading investments across assets and thereby forming a

portfolio is known as diversification. The principle of diversification tells us that spreading an

investment across a number of assets will eliminate some of the risk. The degree of risk that can be


19/37

eliminated by diversification is termed as diversifiable risk. There is however a minimum level of risk that

cannot be eliminated simply by diversifying. This minimum level is oftentimes referred to as

nondiversifiable risk. Together, these suggest that diversification reduces risk, but only up to a point.

Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost

no unsystematic risk. For this reason, the terms unsystematic risk and diversifiable risk are often usedinterchangeably.

Systematic risk on the other hand cannot be eliminated by diversification because by definition,

systematic risk affects all assets in the market to some degree. Systematic risk is sometimes referred to

as nondiversifiable risk.

Systematic Risk and Beta

The Systematic Risk Principle states that the reward for bearing risk depends only on the systematic of

an investment. The reason for this is that because unsystematic risk can be eliminated at virtually no

cost by diversification, there is no reward for bearing it. The market does not reward risk that is borne

unnecessarily. The implication of the systematic risk principle is that the expected return on an asset

depends only on the systematic risk of the asset. It follows from this that no matter how much total risk

an asset has, only the systematic portion is relevant in determining the expected return and thereby the

risk premium on that asset.

A specific measure used to determine the degree of systematic risk in an asset is the beta coefficient,

which is represented with the Greek letter . The beta coefficient gives an indication of how much

systematic risk a particular asset has relative to an average asset. By definition, an average asset has a

beta coefficient or simply beta of 1.0, relative to itself. An asset of 0.50, therefore has half as much

systematic risk as an average asset, and an asset with a beta of 2.0 has twice as much systematic risk as

an average asset. A risk-free asset has a beta of 0.

The expected return on an asset, as well as its risk premium, both depend only on its systematic risk.

Because assets with larger betas have greater systematic risks, they will have greater expected returns.

Consider two securities, Security A and Security B, with standard deviations of 40% and 20% respectively

and betas of 0.50 and 1.50 respectively. Security A has the greater total risk since it has the larger

standard deviation but it has substantially less systematic risk. Total risk consists of systematic andunsystematic risk, implying that Security A has the greater unsystematic risk. By virtue of the systematic

principle, we expect Security B to have a higher risk premium and a greater expected return, despite the

fact that it has less total risk.


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Portfolio Beta

The beta for a portfolio is calculated in much the same way as the expected return for a portfolio.

Assume a portfolio consists of equal weights of Lions Corporation stock and Dealers Company Limited

stock. Lions has a beta of 0.80 and Dealers has a beta of 1.65. What will be the beta of this portfolio?

Let Lbe the beta for Lions and Dbe the beta for Dealers. The beta for the portfolio, P, is given as:

P= 0.5 x L+ 0.5 x D = 0.5 x 0.8 + 0.5 x 1.65 = 1.225

To obtain the beta of a portfolio, we sum the products of each assets beta and its corresponding

portfolio weight.

Suppose a portfolio consists of $1000 invested in Stock A, with an expected return of 8% and a beta of

0.8, $2000 invested in Stock B, which has an expected return of 12% and a beta of 0.95, $3000 invested

in Stock C with an expected return of 15% and a beta of 1.1 and $4000 invested in Stock D with an

expected return of 18% and a beta of 1.4, what is the expected return on this portfolio? Does this

portfolio have more or less systematic risk than the average asset?

The portfolio weights for the Stocks A, B, C and D are, 10%, 20%, 30% and 40% respectively. The

expected return on the portfolio E(RP), is therefore:

E(RP) = 0.1 x E(RA) + 0.2 x E(RB) + 0.3 x E(RC) + 0.4 x E(RD)

E(RP) = 0.1 x 8% + 0.2 x 12% + 0.3 x 15% + 0.4 x 18%

= 14.9%

The portfolio beta Pis determined as: P= 0.1 x A+ 0.2 x B+ 0.3 C+ 0.4 x D

P = 0.1 x 0.8 + 0.2 x 0.95 + 0.3 x 1.1 + 0.4 x 1.4

= 1.16 > 1.0

Because the beta is larger than 1, this portfolio has greater systematic risk than an average asset.


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The Security Market Line

Beta and the Risk Premium

Consider a portfolio made up of an Asset A, with an expected return of 20% and a beta of 1.6 and a risk-

free asset. Suppose the risk-free rate is 8%. Different values for the expected return and beta of this

portfolio can be derived, by altering the portfolio weights of one of the two assets in the portfolio. If we

assume the portfolio weight for Asset A is 25%, then the portfolios expected return becomes:

E(RP) = 0.25 x E(RA) + (1 0.25) x Rf

= 0.25 x 20% + 0.75 x 8%

= 11%

The beta on the portfolio would be: P= 0.25 x A + (10.25) x f

P = 0.25 x 1.6 + 0.75 x 0

= 0.4

Assuming an investor has $50, all of which he invests in Asset A, and then proceeds to borrow $25 at the

risk-free rate, all of which he also invests in Asset A, the total investment in Asset A now becomes $75,

or 150% of the investors wealth. The expected return in this case is:

E(RP) = 1.5 x E(RA) + (1 1.5) x Rf

= 1.5 x 20% 0.5 x 8%

= 26%

The beta on the portfolio would be: P= 1.5 x A + (1 1.5) x f

P = 1.5 x 1.6 0.5 x 0

= 2.4

It can be seen from the above calculations that as the portfolio weight for Asset A increases, the

expected return on the portfolio increases and so does the beta. Plotting the portfolio expected return

against the portfolio betas, we obtain a graph of the form below.


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The slope of the line obtained from plotting the portfolio expected returns against the portfolio betas

can be determined as the rise over the run. It should be noted that this ratio is equivalent to dividing

the risk premium on Asset A, by Asset As beta.

The slope of the line is: (E(RA)Rf) / P

= (20% 8%) / 1.6

= 7.5%

The slope is accordingly termed as the reward-to-risk ratio for Asset A.

Assuming second asset, Asset B is available in the market, offering a beta of 1.2 and an expected return

of 16%, whiles Asset A still offers an expected return of 20% at a beta of 1.6. Investors are always going

to choose Asset A over B every time because Asset B offers less compensation for its level of systematic

risk, relative to Asset A.

Let us consider a portfolio consisting of 25% investment in Asset B and a risk-free asset, given a risk-free

interest rate of 8% as in the case of Asset A.

E(RP) = 0.25 x E(RB) + (1 0.25) x Rf

= 0.25 x 16% + 0.75 x 8%

= 10%


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The beta on the portfolio would be: P= 0.25 x B + (10.25) x f

P = 0.25 x 1.2 + 0.75 x 0

= 0.3

Should we plot the portfolio expected returns against the portfolio betas like we did in the case of the

portfolio consisting of Asset A and a risk-free asset, we will end up with a straight line just like we did in

the case of Asset A. But the slope for the line obtained for the portfolio expected returns and betas for

Asset B will be

(E(RB)Rf) / P

= (16% 8%) / 1.2

= 6.67%

Therefore, Asset B has a reward-to-risk ratio of 6.67%, which is less than the 7.5% that was obtained inthe case of Asset A. The situation for Assets A and B cannot persist in a well-organised, active market

because investors would be attracted to Asset A every time and away from Asset B, till Asset As price

rises as a result and Asset Bs price falls. The expected return on Asset A would then decline and that for

Asset B would increase. The phenomenon will continue until the two assets plot exactly on the same

line. In an active, competitive market, we must have the situation that:

(E(RA)Rf) / P = (E(RB)Rf) / P

This is the fundamental relationship between risk and return. No matter the number of assets

considered, this is the conclusion that would always be reached. The reward-to-risk ratio must be the

same for all the assets in the market.

The line that results when expected returns are plotted against beta coefficients; the line used to

describe the relationship between systematic risk and expected return in financial markets is referred to

as the Security Market Line.

A market portfolio is a portfolio consisting of all the assets in the market. The expected return on a

market portfolio is denoted E(RM). All assets in the market must plot on the Security Market Line (SML),

so a market portfolio must also plot on the SML, since it consists of all assets in the market. Being

representative of all the assets in the market, the market portfolio must have average systematic risk

and therefore a beta of 1.0. The slope of the SML is therefore given as:

(E(RM)Rf) / P = (E(RM)Rf) / 1 = E(RM)Rf

The derived term E(RM)Rfis referred to as the market risk premium, because it is the risk premium on

a market portfolio.


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Capital Asset Pricing Model

Let E(Ri) and ibe the expected return and beta, respectively, on any asset in the market. The asset must

plot on the SML and as a result, the reward-to-risk ratio is the same as the overall markets. Therefore

(E(Ri)Rf) / i = E(RM)Rf

Rearranging the above equation, making E(Ri) the subject, we obtain:

E(Ri) = Rf + [E(RM)Rf] i

The resulting equation is the Capital Asset Pricing Model (CAPM). The CAPM shows that the expected

return for a particular asset depends on three things:

1. The pure time value of money: As measured by the risk-free rate, Rf, is the reward for merelywaiting for your money, without taking any risk.

2. The reward for bearing systematic risk: As measured by the market risk premium, E(RM)Rf, thereward the market offers for bearing average systematic risk in addition to waiting.

3. The amount of systematic risk: As measured by i , this is the amount of systematic risk presentin a particular asset or portfolio, relative to that in an average asset.

CAPM is equally applicable to individual assets as it is to portfolios. Suppose the risk-free rate is 4%, the

market premium is 8.6% and a particular stock has a beta of 1.3. Based on CAPM, what is the expected

return on this stock? What would the expected return be if the beta were to double?

With a beta of 1.3, the risk premium for the stock is 1.3 x 8.6% = 11.18%. The risk-free rate is 4%, so the

expected return would be 15.8%. If the beta were to double to 2.6, the risk premium would double to

22.36 and the expected return would be 26.36%.


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FOREIGN EXCHANGE MARKETS AND EXCHANGE RATES

The foreign exchange market is the worlds largest financial market. It is the market where one countrys

currency is traded for another. Most of the trading takes place in a few currencies such as the U. S dollar

and the Euro. It is an over-the-counter market, which implies that there is no single location where

traders get together. Instead market participants are located I major commercial and investment banks.They communicate using computer terminals, telephones and other telecommunication devices.

An exchange rate is simply the price of one countrys currency expressed in terms of another countrys

currency. In practice, almost all trading of currencies takes place in terms of the U. S. dollar. For

example, both the Swiss franc and the Japanese yen are traded with their prices quoted in US dollars. So

for the purposes of this course, an exchange rate is simply the price of the US dollar expressed in terms

of another currency.

Suppose the exchange rate in terms of yen per dollar is 119.6 and you have a $1000, how much will your

$1000 get you?

$1000 x 119.6 per dollar = 119,600

If the exchange rate in terms of dollars per euro is given as 0.8883 and we need 100,000 to pay off a

debt, we will need 100,000 x $0.8883 per euro = $88,830.

Cross Rates and Triangle Arbitrage

A cross rate is the exchange rate for a non-US currency expressed in terms of another currency. For

example, suppose we observe that the exchange rate for euro is 1 per dollar and that of the Swiss franc

is SF2 per dollar. Suppose the cross-rate is quoted as per SF = 0.4, this would mean the cross rate is

inconsistent with the exchange rates, as at the given exchange rates, we should get per SF = 0.5.

Let us assume we have $100. If we convert this to Swiss francs at the given rate of SF2 per dollar, we

receive: $100 x SF2 per dollar = SF200. And if this is converted to euros at the cross rate, we get:

SF2 x per Swiss franc = 80.

However, if we just convert the $100 to euros without first changing it into Swiss francs, we get:

$100 x per dollar = 100.

It is evident from this scenario that the euro has two prices. 1 per $1 and 0.8 per $1, with the price we

pay, depending on the steps taken in getting the euros. We could make some money by buying low and

selling high. The important step here is to note that euros are cheaper when purchased with dollars

because you get 1 per $1 instead of 0.8 per $1. In order to make some money with our initial $100,

we could proceed as follows:


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1. Buy 100 for $100.2. Use the 100 to buy Swiss francs at the cross-rate. Because it takes 0.4 to buy a Swiss franc, we

will receive 100/0.4 = SF 250

3. Use the Swiss francs to buy dollars. Because the exchange rate is SF2 per dollar, we will receiveSF250/2 = $125, for a concluding profit of $25.

4. We can repeat step s 1 through 3 to make more money.

This particular activity is called triangle arbitrage because the arbitrage involves moving through three

different exchange rates. To prevent such opportunities from existing, because a dollar will buy you

either 1 or SF2, the cross-rate must be: (1/$1) / (SF2/$1) = 1/SF2 = 0.5/SF. The cross rate must be

1 per SF2 as anything else would result in an arbitrage opportunity.

Types of Transactions

There are two basic types of trade in the foreign exchange market: spot trades and forward trades. A

spot trade is an agreement to exchange currency within two business days. The exchange rate on a spot

trade is called the spot exchange rate. Implicitly, the exchange rates and transactions that we have

discussed so far have all referred to the spot market.

Aforward trade is an agreement to exchange currency at some time in the future. The exchange rate

that will be used is agreed upon in the present and is called theforward exchange rate. A forward trade

will normally be settled sometime within the next 12 months. Assuming the spot exchange rate for aSwiss franc today is SF1 = $0.5871 and the 180-day (6 months) forward exchange rate is SF1 = 0.5887, it

implies that you can buy a Swiss franc today or $0.5871 or you can agree to take delivery a Swiss franc in

six months and pay $0.5887. It should be noted that the Swiss franc is more expensive in the forward

market. This is because the forward market allows businesses to lock in a future exchange rate today,

thereby eliminating any risk from unfavorable shifts in the exchange rate.

When a currency is more expensive in the future than it is today, ie if it requires more dollars to buy a

unit of that particular currency today than it does in the future, it is said to be selling at a premium. In

this scenario, because the dollar sells for more now than it will in the future, the dollar is said to be

selling at a discount relative to the Swiss franc.


27/37

Purchasing Power Parity

A question that may have occurred to us repetitively up to this point, may be that of how the level of

spot exchange rates are determined. In addition, we may want to know what is responsible for the rate

of change in exchange rates. Part of the answer in both cases is what may be termed aspurchasing

power parity (PPP), the idea that the exchange rate adjusts to keep purchasing power constant amongcurrencies. There are two forms of purchasing power parity; absolute purchasing power parityand

relative purchasing power parity.

Absolute Purchasing Power Parity: The basic idea behind this concept is that a commodity costs the

same regardless of the currency used to purchase it or where it is selling. In other words, if a beer costs

2 in London and the exchange rate is 0.6, then a beer costs 2/0.6 = $3.33 in New York. Absolute

purchasing power parity implies that any amount in dollars will buy you the same quantity of a specific

product anywhere in the world.

Let S0 be the spot exchange rate between the British pound and the US dollar today. It should be noted

that we are quoting exchange rates as the amount of foreign currency per dollar. Let PUS and PUKbe the

current US and British prices, respectively on a particular commodity, say bread. Absolute PPP says that:

PUK= S0 x PUS

Implying that the British price for something is equal to the product of the exchange rate and the US

price for that same commodity. It is worth noting that if PPP does not hold, arbitrage opportunities

would be possible if one commodity was transported from one country to another.

Suppose apples are selling in New York for $4 per bushel, whereas in London they are selling for 2.40per bushel. Absolute PPP implies that:

PUK= S0 x PUS

2.40 = S0 x $4

S0 = 2.40 / $4 = 0.6

That is, the implied spot exchange rate is 0.60 per $1. Equivalently, a pound is worth $1/0.60 = $1.67

Suppose the actual exchange rate is 0.50. Starting with $4, a trader could buy a bushel of apples in New

York, ship it to London and sell it there for 2.40. Our trader would then convert the 2.40 into dollars at

the prevailing exchange rate, S0 = 0.50, yielding a total of 2.40/0.50 = $4.80. The round-trip gain would

be 80 cents. If such profit potential existed, forces would be set in motion to change the exchange rate

or the price of the apples. In our example, apples would begin moving from New York to London at such

a rate as to reduce the supply of apples in New York. The reduction in the supply of apples in New York

would raise the price of apples in New York and the increased supply of apples in Britain would lower

the price of apples in London. In addition, apple traders would be busily converting pounds back into


28/37

dollars to buy some more apples. This would increase the supply of pounds and simultaneously increase

the demand for dollars. We would expect the value of the pound to fall as the dollar increases in value.

Because the exchange rate is quoted as pounds per dollar, the exchange rate would increase from

0.50.

For absolute PPP to hold absolutely, several things must hold:

1. The transactions costs of trading the particular commodity concerned shipping, insurance,spoilage and others must be zero.

2. There must be no barriers to the trading of the commodity concerned no tariffs, taxes or otherpolitical barriers such as voluntary restraint agreements.

3. Finally, the concerned commodity must be identical in the geographic areas relevant to thetrade. There would be no point in exporting a prohduct to a location where it is of no use.

Relative Purchasing Power Parity: This concept does not tell us what determines the absolute level ofthe exchange rate. Rather, it tells us what determines the change in the exchange rate over time.

Suppose the exchange rate for the British pound is currently S0 = 0.50 and that the inflation rate in

Britain is predicted to be 10% over the coming year and that in the United States is predicted to be zero.

What would the exchange rate be in a year?

With 10% inflation, we would expect prices in Britain generally to rise by 10%. So we expect the price of

a dollar to increase by 10% and the exchange rate should be 0.50 x 1.1 = 0.55. If the inflation rate in

the United States is not zero, then we need to be concerned with the relative inflation rates in the two

countries. For example, suppose the inflation rate is 4%, relative to prices in the United States, prices in

Britain are rising at a rate of 10% 4% = 6% per year. So we expect the price of the dollar to rise by 6%

and the predicted exchange rate is 0.50 x 1.06 = 0.53.

Relative PPP says that the change in the exchange rate is determined by the difference in the inflation

rates of the two countries. To be more specific, we will use the following notation:

S0 = Current (Time 0) spot exchange rate (foreign currency per dollar)

E(St) = Expected exchange rate in tperiods

hUS = inflation rate in the United States

hFC= inflation rate in the foreign country

Relative PPP says that the expected percentage change in the exchange rate over the next year,

[E(S1)S0], is: [E(S1)S0] / S0 = hFChUS. Relative PPP says that the expected percentage change in the

exchange rate is equal to the difference in inflation rates. If we rearrange this slightly, we get:

E(S1) = S0 [1 + (hFChUS)]


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Back to the example involving Britain and the United States, relative PPP says that the exchange rate will

rise by hFChUS = 10%4% = 6% per year. Assuming the difference in inflation rates doesnt change, the

expected exchange rate in two years, E(S3), will therefore be:

E(S2) =E(S1) (1 + 0.06) = 0.53 x 1.06 = 0.562

It should be noted that: E(S2) =E(S1) (1 + 0.06) = 0.53 x 1.06 = (0.5 x 1.06) x 1.06 = 0.5 x 1.062

Generally, relative PPP says that the expected exchange rate at some time tin the future : E(St), is:

E(St) = S0 [1 + (hFChUS)]t

In reality, we expect only relative PPP to hold so it will be the focus of the course from this point

onwards.

Suppose the Japanese exchange rate is currently 105 per dollar. The inflation rate in Japan over thenext three years will run, say 2% per year, whereas the US inflation rate will be 6%. Based on relative

PPP, what will the exchange rate be in three years?

Because the US inflation rate is higher, we expect that a dollar will become less valuable. The exchange

rate change will be 2% 6% = 4 per year. Over three years, the exchange rate will fall to:

E(S3) = S0 [1 + (hFChUS)]3

= 105 x [1 + (0.04 )]3

= 92.90


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Interest Rate Parity, Unbiased Forward rates and the International Fischer Effect

Covered Interest Arbitrage

A relationship exists between the spot exchange rate, forward exchange rates and interest rates. T get

started, we need some additional notation:

Ft= Forward exchange rate for settlement at time t

RUS = the nominal risk-free interest rate in the United States

RFC= the nominal risk-free interest rate in a foreign country

S0 = Current (Time 0) spot exchange rate (foreign currency per dollar)

Suppose RUS is the T-bill rate and that we observe the following about US and Swiss currency in the

market:

S0 = SF2 Ft= SF1.90 RUS = 10% RS =5%

where RS is the nominal risk-free interest rate in Switzerland. Assuming one decided to invest a dollar in

risk-free venture in the US, come a years time, at the prevailing interest rate, the $1 will be worth $1.1.

Alternatively, one could decide to invest in the Swiss risk-free investment by converting the dollar into

Swiss francs and simultaneously executing a forward contract to trade francs back to dollars in a year. It

can be done as follows.

1. Convert $1 to Swiss francs; $1 x S0 = SF22. At the same time, enter into a forward agreement to convert Swiss francs back to dollars in a

year. Because the forward rate will be SF1.90, you will get $1 for every SF1.90 instead of every

SF2.

3. Invest the SF2 in Switzerland at RS; in one year SF2 (1 + RS) = 2 x 1.05 = 2.104. Convert the SF2.10 obtained back to dollars at the agreed-upon rate of SF1.90 = $1; so that

SF2.1/1.90 = 1.1053 is obtained.

This is higher than the 10% that is obtained by investing in a risk-free investment in the United

States. This activity is referred to as covered interest arbitrage. Coveredbecause, we are covered in

the event of a change in the exchange rate because we lock in the forward exchange rate today.


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Interest Rate Parity

Assuming that significant covered arbitrage opportunities do not exist, there must be some

relationship between spot exchange rates, forward rates and relative interest rates. From our initial

example of the risk-free dollar investment, we obtained 1 + RUS for every dollar invested. By

investing in the Swiss risk-free investment, we obtained S0 x (1 + RS) / F1 for every dollar invested.With no arbitrage present, these two investments must yield the same returns so:

1 + RUS = S0 x (1 + RS) / F1

Rearranging this equation gives the condition known as the interest rate parity(IRP):

F1/ S0 = (1 + RFC) (1 + RUS)

If we define the percentage forward premium or discount as (F1 S0)/ S0, interest rate parity says

the percentage premium or discount is approximatelyequal to the differences in interest rates:

(F1 S0)/ S0 = RFCRUS

In general, if we have tperiods instead of just one, the IRP approximation is written as:

Ft = S0 [1 + (RFCRUS)]t

Suppose the exchange rate for Japanese yen, S0 , is currently 120 = $1. If the interest rate in the

United States is RUS = 10% and the interest rate in Japan is RJ = 5%, then what must the forward rate

be to prevent covered interest arbitrage?

Ft = S0 [1 + (RFCRUS)]t

F1 = S0 [1 + (RJRUS)]1

F1 = S0 [1 + (RJRUS)]

= 120 [1 + (0.05 0.10)]

= 120 x 0.95

= 114

The Unbiased Forward Rates condition says that the forward rate, Ft , is equal to the expected

future spot rate, thereby, outlining the relationship between the forward rate and the expected

future spot rate. The condition states that the current forward rates an unbiased predictor of the

future spot exchange rate. In other words

Ft= E(St)


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Now, since Ft= E(St), we could proceed to figure out the relationship between the expected future

spot rate and the risk-free nominal interest rate prevailing in the United States and the risk-free

nominal interest rate prevailing in other countries:

E(St) = S0 [1 + (RFCRUS)]t

This important relationship is known as the uncovered interest parity(UIP). It is the condition stating

that the expected percentage change in the exchange rate is equal to the difference in the interest

rates.

The International Fisher Effect

Relative PPP states that E(St) = S0 [1 + (hFChUS)]t .

And UIP states that E(St) = S0 [1 + (RFCRUS)]t

So S0 [1 + (hFChUS)]t = S0 [1 + (RFCRUS)]

t and

hFChUS = RFCRUS

This tells us that the difference between the returns between the United States and foreign country

is just equal to the difference in inflation rates. Rearranging this equation gives:

RUShUS = RFChFC

This equation is known as the international fisher effect (IFE). The theory states that real interest

rates are equal across countries. This conclusion is rather basic economics because if real returns

were higher in say, Mexico than in the United States, money would flow out the United States

financial markets into Mexican markets. Asset prices in Mexico would rise and their returns would

fall, while asset prices in the United States would fall and their returns would rise. The process

would act to equal returns.

International Capital Budgeting

Assuming a US-based international company is evaluating an overseas investment. The companys

exports of meat have increased to such a degree that it is considering building a distribution center

in Britain. The project is expected to cost 2 million to launch. The cash flows are expected to 0.9

million a year for the next three years. The current spot exchange rate is 0.5. the risk-free rate in


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the United States is 5% and the risk-free rate in Britain is 7 %. The interest rates are observed in

financial markets, not estimated. The company requires a 10% rate of return. Should the company

take this investment? Whether or not this investment is worthwhile depends on the net present

value. So we need to calculate the net present value of this investment in US dollars. There are 2

basic ways to go about doing this.

1. The home currency approach requires you to convert all pound cash flows into dollars, and thendiscount at 10% to find the net present value in dollars. For this approach, we have to come up

with the future exchange rates to convert the future projected pound-cash flows into dollars.

2. The foreign currency approach requires you to determine the required return on the poundinvestments and then discount the pound investments to find the net present value in pounds.

Then convert the pound net present value to a dollar net present value. This approach requires

you to somehow convert the 10% dollar required return to the equivalent pound required

return.

The difference between these two approaches is primarily a matter of when the pounds areconverted to dollars. In the first case, the pounds are converted before estimating the net

present value. In the second case, the conversion is done after estimating the net present value.

It might appear that the second method is superior because for it, we only have to come up with

one number, the pound discount rate. Furthermore, because the first approach required us to

forecast future exchange rates, it probably seems that there is greater room for error with this

approach.

The Home Currency Approach

The expected exchange rate at time t, E(St) = S0 x [1 + (R RUS)]t

Where Rstands for the nominal risk-free rate in Germany = 7%

RUS = 5% and S0 = 0.5

And therefore E(St) = 0.5 x [1 + (0.07 + 0.05)]t= 0.5 x 1.02

t

Year Expected Exchange Rate

1 0.5 x 1.021 = 0.5100

2 0.5 x 1.022 = 0.5202

3 0.5 x 1.023 = 0.5306

Using these exchange rates along with the appropriate current exchange rate, all of the Euro

cash flows are converted to dollars


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Year (1)Cashflow in mil (2) Expected

Exchange Rate

Cashflow in $ mil

(1)(2)

0 2.0 0.5000 $ 4.00

1 0.9 0.5100 $1.76

2 0.9 0.5202 $1.73

3 0.9 0.5306 $1.70

Platinum Dealers require a rate of 10% on this investment so:

NPV$ = $4 + $1.76/(1 +0.1) + $1.73/(1 +0.1)2 + 1.70/(1 +0.1)3

= $ 0.3 million

And therefore, the project is profitable

The Foreign Currency Approach

The company requires a 10% nominal rate of return on the dollar denominated cash flows. This

would have to be converted to a rate suitable for pound-denominated cash flows. Based on the

international fisher effect, the nominal rates can be determined as:

RRUS= hhUS

= 7% 5% = 2%

The appropriate discount rate for estimating the pound cash flows from the project is approximately

equal to 10% plus an extra 2% to compensate for the greater pound inflation rate.

If we calculate the net present value of the pound cash flows at this rate, we get:

NPV = 2 + 0.9/(1 +0.12) + 0.9/(1 +0.12)2 + 0.9/(1 +0.12)3 = 0.16 million

The net present value of this project is 0.16 million. Taking the project makes the company 0.16

million richer today. What would this be in dollars? Because the exchange rate today is 0.5, the dollar

net present value of the project will be:

NPV$ = NPV/S0 = $0.3 milliion.


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OPTIONS, HEDGING AND PRICE VOLATILITY

An option is a contract that gives its owner the right to buy or sell some asset at a fixed price on or

before a given date. An option on a building might give the holder of the option the right to buy the

building for say $1million anytime on or before the Saturday prior to the third Wednesday of January

2010.

The act of selling or buying the underlying asset via the option contract is referred to as exercising the

option. The fixed price specified in the option contract at which the holder can buy or sell the underlying

asset is called the strike price or exercise price. An option usually has a limited life. The option is said to

expire at the end of its life. The last day on which the option may be exercised is called the expiration

date. An American option may be exercised anytime up to and including the expiration date. A European

option on the other hand, may be exercised only on the expiration date.

Options come in two basic types. A call option gives the owner the right to buy an asset at a fixed price

curing a particular time period. A put option gives the holder the right to sell the asset for a fixed

exercise price.

If an investor sells a call option, the investor receives money up front and has the obligation to sell that

asset at the exercise price if the option holder so desires. Similarly, an investor who sells a put option

receives cash upfront and is then obligated to buy the asset at the exercise price if the option holder

demands it.

Fundamentals of Option Valuation

Some notation that comes in handy when determining the value of a call option at expiration, are S1,the stock price at expiration in one period, S0, the stock price today, C1, the value of the call option o the

expiration date in one period, C0 , the value of the call option today, and E, the exercise price of the

option. In determining the value of a call option, two quantities are of interest; the upper bound of the

call options value and the lower bound of the call options value. The upper bound of the value of the

call option is C0 < S0, implying that the call option can never be worth more than the stock. However, in

the case of the lower bound, in order to prevent arbitrage, the value of the call today must be greater

than the stock price less the exercise price. Therefore, the lower bound becomes C0 > S0E.

Putting these two conditions together, we have C0 > 0 ifS0E< 0 and C0 > S0 EifS0 E > 0. These

conditions simply state that the lower bound on the calls value is either 0 or S0E, whichever is bigger.

The lower bound is called the intrinsic value of an option and it is simply what the option would be

worth if it were about to expire.

The value of a call option is given as the difference between the stock value and the present value of the

exercise price. Hence,

C0= S0E/(1 + Rf)t, where tis the time to expiration of the call


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Let us consider a call option on a stock, with an exercise price of $20. The stock currently sells for $35.

Its future price in one period will either be $25 or $50. If the risk-free rate is 10%, what is the value of

this call option.

C0= S0E/(1 + Rf)t, in this case, t is 1.

= $35 20/(1 + 0.1)

= $16.82

In the above example, it can be seen that either way, there is no way that the stock sells for a price

lower than or equal to its exercise price. In this case, we say the option finishes in the money. When an

option. Assuming the exercise price was $30, the option would be worth $0, when the price of the stock

one period from now is $25. In that case, we say the option finishes out of the money.

If we take a look at the expression, we realize that the value of the call depends on four things:

1. The stock price: The higher the stock price, (S0) is, the more the call is worth. This should comeas no surprise because the option gives us the right to buy the stock at a fixed price.

2. The exercise price: The higher the exercise price (E) is, theless the call is worth. This holdsbecause the exercise price is what we have to pay to get the stock.

3. The time to expiration: The longer the time to expiration (t) is, the more the option is worth.Because the option gives us the right to buy for a fixed length of time, its value goes up as that

length of time increases.

4. The risk-free rate: The higher the risk-free rate (Rf) is, the more the call is worth. Normally, wethink of asset values as going down as rates rise. In this case, the exercise price is a cash outflow;

a liability. The current value of that liability goes down as the discount rate goes up.

5. There is however, a fifth factor that determines the value of a call, which is the variance of theunderlying asset. The greater the variance is, the more the call is worth. This is because

increasing the variance of the possible future prices on the underlying asset does not affect the

options value when the option finishes out of the money.

HEDGING AND PRICE VOLATILITY

Hedging refers to the act of reducing a firms exposure to price or rate fluctuations. It is also

termed as immunization. There is usually no direct way of hedging any particular risk. It is

usually the financial managers job to create a way by using available financial instruments to

create innovative ways. The process of coming up with ways to hedge risks is known as financial

engineering. Corporate risk management involves the buying and selling ofderivative securities.

A derivative security is a financial asset that represents a claim to another financial asset. A

stock option for example, fives the owner the right to buy or sell stock, which is a financial asset.

Stock options are therefore derivative securities.


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There are three specific areas of particular importance to businesses in which price volatility can

increase dramatically. These three are interest rates, exchange rates and commodity prices.

Exchange rate risk will usually comprise ofshort-run exposure, long-run exposure and

translation exposure.

The basic tool for identifying and measuring a firms exposure to financial risk is known as the

risk profile. It is a plot showing how the value of the firm is affected by changes in prices of

goods, some particular service or rates. To illustrate, let us consider a firm that is into the

production of wheat and another firm that buys wheat for the purposes of its commercial

activities. The diagram below shows how each party is affected by an increase in wheat prices.

If the two firms manage to come to an agreement that at a set date in the future, the grower

will deliver a certain quantity of wheat and the processor will pay a set price for the wheat, then

much risk would have been eliminated. Once the agreement is signed, both firms would have

locked in a price of wheat for as long as the contract is in effect and both their risk profiles with

regard to the wheat prices will be completely flat during that time. However, in reality, a firm

that hedges financial risk usually will not be able to create a completely flat risk profile. The

wheat grower cannot know the size of the harvest of wheat ahead of time. If the harvest isbigger than the quantity that has been hedged, some portion of the produce will not be hedged.

If the crop is smaller than the quantity that has been hedged, then the grower will have to buy

more to fulfill the contract and thereby will be exposed to the risk of price fluctuations. There is

still some exposure to wheat price fluctuations, however by hedging that exposure is sharply

reduced.

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Notes on basic finance