CORPORATE FINANCE MODULE - Chinhoyi University of Web viewword publishers2016corporate finance modulecuac 207time value of money conceptsj.mashonganyikachinhoyi university of technologyc

W o r d p u b l i s h e r s

2016

CORPORATE FINANCE MODULECUAC 207

TIME VALUE OF MONEY CONCEPTS

J.MASHONGANYIKA

CHINHOYI UNIVERSITY OF TECHNOLOGYC

Fundamentals of Corporate Finance Page 1

TIME VALUE OF MONEY CONCEPTS

Introduction

The concept of the time value of money is an integral concept in the study of financial management. This is the focus of this unit which you have to study now before you proceed to other topics. The discussion may appear to be very "technical", however, you are advised to make an effort to grasp the concepts that are covered in this unit as they will come up from time to time in our study of the other topics.

For example, you need the concepts covered in this unit in order to study Capital Budgeting, Valuation of Shares, the Cost of Capital and many other issues covered in corporate financial management. Additionally, most of the concepts covered in this unit will come in handy in your advanced studies of the subject.

Time Value of Money

The notion that money has a time value is one of the most basic concepts in finance and investment analysis. Making decisions today regarding future cash flows requires understanding that the value of money does not remain the same throughout time.

A dollar today is worth less than a dollar sometime in the future for two reasons.

Reason No. 1: Cash flows occurring at different points in time have different values relative to any one point in time. One dollar one year from now is not as valuable as one dollar today. After all, you can invest a dollar today and earn interest so that the value it grows to next year is greater than the one dollar today. This means we have to take into account the time value of money to quantify the relation between cash flows at different points in time.

Reason No. 2: Cash flows are uncertain. Expected cash flows may not materialize. Uncertainty stems from the nature of forecasts of the timing and/or the amount of cash flows. We do not know for certain when, whether, or how much cash flows will be in the future. This uncertainty regarding future


cash flows must somehow be taken into account in assessing the value of an investment.

Translating a current value into its equivalent future value is referred to as compounding. Translating a future cash flow or value into its equivalent value in a prior period is referred to as discounting. We are going to deal with e basic mathematical techniques used in compounding and discounting.

An investment of money has different values on different dates. The adjustment in time value is a function of the following factors: time, inflation rate, risk.

A lender will need compensation from a borrower for delaying payment and this compensation will be determined by above three factors. This compensation is the interest rate, which represents the opportunity cost of funds. Let’s now discuss the following:

Future Value Present Value Simple Interest Simple Discount Compound Interest

Simple Interest

Remark: Interest is the price paid for the use of borrowed money.

Interest is paid by the party who uses or borrows the money to the party who lends the money. Interest is calculated as a fraction of the amount borrowed or saved (principal amount) over a certain period of time. The fraction, also known as the interest rate, is usually expressed as a percentage per year, but must be reduced to a decimal fraction for calculation purposes. For example, if we’ve borrowed an amount from the bank at an interest rate of 12% per year, we can express the interest as:

12% of the amount borrowed

or 12/100 of the amount borrowed

or 0,12× the amount borrowed.

When and how interest is calculated result in different types of interest.


For example, simple interest is interest that is calculated on the principal amount that was borrowed

or saved at the end of the completed term.

Remark: Simple interest is interest that is computed on the principal for the entire term of the loan, and is therefore due at the end of the term. It is given by

I = PrtWhere;I-is the simple interest (in $) paid at the end of the term for the use of the moneyP - is the principal or total amount borrowed (in $) which is subject to interest (P is also known as the present value (PV ) of the loan)r- is the rate of interest, that is, the fraction of the principal that must be paid each period (say, a year) for the use of the principal (also called the period interest rate)t - is the time in years, for which the principal is borrowed

NB: Interest is earned only on the original investment; no interest is earned on interest

Example

Suppose you have $10 000 to invest in a bank savings account at a simple interest of 20% per annum. How much will you have at the end of the year?

Remark: The amount or sum accumulated of Future Value (S) (also known as the maturity value, accrued principal) at the end of the term t, is given by

S = Principal value + Interest

S = P + I

S = P + Prt

S = P(1 + rt).

Remark:The date at the end of the term on which the debt is to be paid is known as the due date or maturity date.

Example


Suppose you deposit $10000 today in an account that pays simple interest of 20% per annum. How much will you have at the end of3 years?

Example

You borrow $18 000 for a simple rate of 22% per annum for 125 days. How much will you have to pay to the lender?

Practice Questions

1. Calculate the simple interest and sum accumulated for $5 000 borrowed for 90days at 15% per annum. ($5 185)

2. Calculate the sum accumulated at the end of 3 years, 4 months and 17 days on a deposit of$20 000 and an interest rate of 18.27% per year.

Present Values [discounting]

Sometimes we not only consider the basic formula I = Prt but also turn it inside out and upside down, as it were, in order to obtain formula for each variable in terms of the others. Of particular importance is the concept of present value P or PV, which is obtained from the basic formula for the sum or future value S, namely

S = P(1 + rt)

Dividing by the factor (1 + rt) givesP= S

1+rt.How do we interpret this result? We do this as follows: P is the amount that must be borrowed now to accrue to the sum S, after a term t, at interest rate r per year. As such it is known as the present value of the sum S.

Remark:Discounting is a process of moving the future value of an obligation/investment back to the present/today.

For Simple Interest = P= S(1+rt)

For Compound Interest=P=S

(1+i)n


t- term

r =Interestrate

P or PV

FV or S= P(1 +rt)

ExampleA promissory note with a future value of $12000, simple interest rate is 12% per annum is sold 3 months prior to its due date. What is the Present Value on the day it is sold?Remark:A promissory note is a written promise by a debtor (called the maker of the note) to pay a creditor (called the payee) a stated sum of money (the so-called “maturity value”) on a specific date (the due date), and stating a specific rate of interest. Such notes can be bought and sold, that is, they are negotiable. Obviously with such transactions it is the present value of the note that counts.

Time linesA time line is a useful way of representing interest rate calculations graphically. Time flow is represented by a horizontal line. Inflows of money are indicated by an arrow from above pointing to the line, while outflows are indicated by a downward pointing arrow below the time line.

For a simple interest rate calculation, the time line is as follows:

At the beginning of the term, the principal P (or present value) is deposited (or borrowed) – that is, it is entered onto the line. At the end of the term, the amount or sum accumulated, S (or future value) is received (or paid back). Note that the sum accumulated includes the interest received.

Remember that

FV or Sum accumulated (S) = Principal + Interest received that is S = P + Prt = P(1 + rt)

or equivalently


Future value = Present value + Interest received.

Negotiable Certificates of Deposit (NCDs).

The concept of simple interest is often applied to financial instruments found on the money market (the short-term market ). An important instrument on this market is the negotiable certificate of deposit (NCD). NCDs arise when banks solicit large deposits from investors for a fixed period of time during which the money cannot be withdrawn. The investor is then given a certificate which is negotiable. This means that the investor can sale or negotiate the certificate in the money market at any stage before the maturity date of the deposit.

The amount invested is the nominal, or face value of the instrument upon which interest is calculated at the period of the deposit using simple interest. To find out how this interest is calculated, let us look at the following example.

S = P + IS = P ( 1+ rt )

Example

Suppose on 1 May 2012, you purchase an NCD with a maturity date of 31 July, 2012, a face value of $1 000 000 and an interest rate of 34.65% that is payable on maturity. How much will you receive on the maturity date?

Note that, when a security is issued for the first time, it is issued on the primary market. Subsequently, it starts trading on the secondary market, on which it acquires a value which may not necessarily be equal to the face value. NCDs are traded in the secondary market on a yield basis, that is the price is determined on the basis of a yield. When calculating the market value, or the consideration to be paid when the NCD is being negotiated, we need to know the number of days remaining to maturity.

Counting days


The convention is that to determine the exact number of days between the two relevant term dates, we include the day the money is deposited or lent and exclude the day the money is repaid (or withdrawn). . The reasoning behind this is the simple fact that if you deposit money on the 12th of June and withdraw it on the 13th of June, there is only one day between the two dates, not two. However, when a security is issued and held until maturity, we include the day on which it was issued.

Let us look at the following example.

Example Calculate the number of days between 25 May and 17 August.You must remember that some months have 31 days while others have 30 days. You should be able to get the number of days by a simple count of your fingers:

Month DaysXxxXxxxxxTotal days

Now, let’s look at the following examples.

Example

On 1 May 2012 you purchase an NCD with a maturity date of 31 July 2012, nominal value of $1 000 000 and an interest rate of 34.65%. Subsequently, on that same day, the yield on similar securities falls to 33%. You then decide to sell the NCD. How much should you expect?

Practice Exercise

1. On 1 May 2012, you purchase an NCD with a maturity date of 31 July 2012, nominal value of $1 000 000 and an interest rate of 34.65%. On 18 June 2012 you then sell the NCD at a yield of 32% pa. How much do you receive?

2. Determine the number of days between 19 March and 11 September.

3. Suppose an investor wishes to purchase a treasury bill (with a par value, that is face value, of $100 000) maturing on 2 July 2012 at a discount rate of 16,55% per annum and with a settlement date of 13


t- term

d= discount rate

P or PV

S=

May 2012. What would the required price be (this is present or discount value – also referred to as the consideration)? What is the equivalent simple interest rate of the investment?

You can see from these examples and the exercise that you have done, that the values of an NCD increases when the market yield decreases relative to the interest rate.

Simple Discount

Remark: is interest calculated on the face (future) value of a term and paid at the beginning of the investment term. You will receive interest in advance

Previously, we emphasised the interest that has to be paid at the end of the term for which the loan (or investment) is made. On the due date, the principal borrowed plus the interest earned is paid back.

In practice, there is no reason why the interest cannot be paid at the beginning rather than at the end of the term. Indeed, this implies that the lender deducts the interest from the principal in advance. At the end of the term, only the principal is then due. Loans handled in this way are said to be discounted and the interest paid in advance is called the discount. The amount then advanced by the lender is termed the discounted value. The discounted value is simply the present value of the sum to be paid back and we could approach the calculations using the present value technique as before.

Expressed in terms of the time line of the previous section, this meansthat we are given S and asked to calculate P.


The discount on the sum S is then simply the difference between the future and present values. Thus the discount (D) is given by

D = S − P.

The discount D is also given by

D = Sdt

(compare to the formula for simple interest I = Prt) where d= simple discount rateand the discounted (or present) value of S is

P = S − D= S − SdtP= S(1 − dt)

or

Present Value = Future Value − Future Value × discount rate × time.

PV = FV − FV × d × t= FV (1 − dt)

(compare to the formula for the accumulated sum or future value for simple interest)

S = P(1 + rt).

Example

Suppose the government floats Treasury billsof facevalue $10 at a discount of 10%. Lisa wants to subscribe and has t $10. The tenureof the TB is 1 year. How much does Lisa Pay now and how much will she get at the end of 1 year.

Solution

Example

A treasury bill with a tenure of 90 days and a face value of $100 000 is issued at a discount of 18%. At what consideration is it being issued?


Example

A customer signs a promissory note agreeing to pay $100000 in 3 months’ time. He then decides to discount the note with a bank at a discount rate of 22%. How much will he receive from the bank now?

NB. Money-market instruments that are traded on a discount basis are bankers’ acceptances (referred to as BAs) and treasury bills. The value appearing on the acceptance or bill, the so-called “face value”, is what the owner thereof will receive on the maturity date. On the other hand, the price paid is the present value, which is calculated as described above using the current rate as set by the market.

Practice QuestionSuppose that a discount security has a nominal value of $1000 but is issued at $945 with a tenor of 90 days. Calculate the discount rate, d.

Equivalent Simple Interest Rate

It establishes a relationship between Simple Discount and Simple Interest. The calculation of the discount rate is based on the assumption that the security is held to its full tenor. If an investor buys a security, he may not necessarily hold it to its full tenor. The investor may opt to sell the security before it matures. The yield will be the difference between what the investor gets when he sells the security and what he paid for it. This is also called the equivalent simple interest rate. When a note is discounted, the interest rate which is equivalent to the discount rate will be greater than the actual discount rate. This difference arises from the fact that the Discount Rate is calculated on the Face Value whereas Interest is calculated on the Present Value.

Example

Determine the discount, discount value and the equivalent simple interest rate on a loan of $35000 due in 9months with a discount rate of 26%?


NBNote the considerable difference between the interest rate of 32% and the discount rate of 26%. This emphasises the important fact that the interest rate and the discount rate are not the same thing. The point is that they act on different amounts, and at different times – the former acts on the present value, whereas the latter acts on the future value.

Practise Question.

1. Determine the Discounted Value on a promissory note of $3000 due in 8months at a discount rate of 15%. What is the equivalent Simple Interest rate?

2. A bank’s simple discount rate is 18%. If you sign a promissory note to pay$4 000 in six months’ time, how much would you receive from the bank now?What is the equivalent simple interest rate?

3. Determine the simple interest rate that is equivalent to a discount rate of

(a) 12% for three months(b) 12% for nine monthsHint: Let S = 100 and use the appropriate formula to set up an equation for r.

Cardinal Rules of Time Value

Remark: A particular investment has different values on different dates. This is linked through the Future Value and Present Value by applying relevant interest rates whether simple or compound.

For example, $1 000 today will not be the same as $1 000 in six-months’ time. In fact, if the prevailing simple interest rate is 16% per annum, then, in six months, the $1 000 will have accumulated to $1 080.

⇒ 1 000 × (1 + 0,16× 1/2) = 1 080

On the other hand, three months ago it was worth less – to be precise, it was worth $961,54.

PV = 1 000 Fundamentals of Corporate Finance Page 12

now

1 080

$1 000t=3/12 t=6/12

916.54

16%

1 000

1 + 0,16×3/4 = 961,54

Represented on a time line, these statements yield the following picture:

This is summarised as below:

1. To move money forward(determining the Face Value)a. Where simple interest is applicable you inflate the relevant sum by

multiplying (1+rt)b. Where compound interest is applicable you inflate by (1+i)n

2. When you want to move money backwards(determining the present value)

a. Where simple interest is applicable, we deflate by (1+rt)b. Where compound interest is applicable you deflate the relevant sum

by (1+i)n

The point is that the mathematics of finance deals with dated values of money. This fact is fundamental to any financial transaction involving money due on different dates. In principle, every sum of money specified should have an attached date.

Practice Question

Jack borrows a sum of money from a bank and, in terms of the agreement, must pay back $1 000 nine months from today. How much does he receive now if the agreed rate of simple interest is 12% per annum? How much does he owe after four months? Suppose he wants to repay his debt at the end of one year. How much will he have to pay then?


Interest and date values

Payments and obligations of different dates

The value of a sum of money is determined by the date at which it is paid or received

Example

If you owe $2000 to be paid in 10months time at an interest of 27%. How much would you pay?

Given that S=P (1+rt)

=20000 (1+0.27×10/12)

=$24500

Example

If you want $20000 today, how much should you have invested done 5months ago at the same interest rate of 27%.

The above examples are represented in a time line as following:

PV?t= 5/12 t=10/12

-5 0 10 months

20000 FV?

Given that PV= S(1+rt )

PV= 20000(1+0.27∗5/12)

PV=$17977, 53


Example

Suppose you owe $100000 to be paid 4months from now, $120000 to be paid 7months from now. You then negotiate to pay all the amounts owed 10months from now. How much will you eventually pay? (Use a simple interest rate of 22% for the evaluation purpose)

Time line presentation is as follows;

T=6 months

T=3 months

r=22% PV=? 100 000 120 000

0 4 months 7 months 10 months

Future Value??

So we need to calculate the values of the new obligation (t=6months for $100000 and t=3months for $120000) at a time period 10months at a simple interest rate of 22%.

NB-For comparison purpose, all date values must be brought to the same date. Only cashflows evaluated at the same date are comparable.

New Obligation

1) S ($100000for 6 months ) =P(1+rt)=100000(1+0.22×6/12)=$111000

2) S ($120 000for 3 months) =P(1+rt)

=120000(1+0.22×3/12)

=$126600

⇒Therefore Total obligation owing will be (Obligation 1 + Obligation 2)

= (111000+126600)

=$237600

Suppose you offered to pay $20000 now in part settlement of the debt, this amount cannot simply be deducted from the amount ($237600). The


$20000 must be extended (evaluated for time value at an appropriate rate) for 10months for comparison purposes (inflating- finding the future value).Then find the final amount needed to liquidate the resultant obligation.

Time value concept illustrated below in a timeline;

t=10 months

P=20 000 S ?? @10 months

Given that S= P(1+rt) ; S@ 10 months= 20000(1+0.22×10/12)

S=$23667

To find final owing, at the final due date, the Total obligations should equal the Total payments.

Total Owing = Total Payments

Total Owing =Part Payment + Final Payment (say X to be determined)

⇒What he owes $237600 less what hepaid $23667 (time value adjusted) gives what he has to pay to level off the debt(X).

Their fore Final payment (X) =$237600-$23667

X=$213933

From time to time a debtor may wish to replace a set of financial obligations with a single payment on a given date. In fact, this is one of the most important problems in financial mathematics. It must be emphasised here that the sum of a set of dated values due on different dates has no meaning. All dated values must first be transformed to values due on the same date(normally the date on which the payment that we want to calculate is due). The process is simply one of repeated application of the key rules of time value as the following example illustrates:

Example


Lisa owes Tracy $5000 due in 3months and $2000 due in 6months. Lisa offers to pay $3000 immediately, if she can pay the balance in one year. Tracy agrees that they use simple interest rate of 16% per annum. They also agreed that the $3000 paid now will also be subject to the same rate of 16% for evaluation purposes. How much will Lisa pay at the end of the year?

Time Line

5000 t=9/12 r=0.16

2000 t=6/12 r=0.16

0 3 6 12 months

3000 t=12/12 r=0.16 ????

Finding values of Obligation at Final Due Date

a) S(5000@ 12 months) =P (1+rt)r= 0.16 t=9/12

=5000(1+0.16×9/12)

=$5600

b) S(2000@ 12 months) =P (1+rt) r=0.16 t=6/12

=2000(1+0.16×6/12)

=$2160

Total obligations = (a) + (b)

=$5600+$2160

=$7760

Finding the values of Payments

S(3000@12 months) =P (1+rt) given r=016 t= 1 year

= 3000(1+0.16×1)

=$3480

At the Final duedate value ,which is 12 months;


Total obligation=Total payments ( i.e. part payments + final payment[X])

This way we find what Lisa owes Tracy at the end of 12 months

⇒ 7760 = 3480+X

⇒ X=7760-3480

⇒X=$4280

Lisa owes $4 280.

Practice Questions

1. Noma owes 8 500 due in 10 months. For each of the following cases, what single payment will repay her debt if money is worth 15% simple interest per annum?a) nowb) six months from nowc) in one year

2. LK owes AT $20 000 due in six months and $6 000 due in 11 months. LK offers to pay $10 000 immediately if he can pay the balance in two year. AT agrees, on condition that they use a simple interest rate of 18% per annum. They also agree that for settlement purposes the $10 000 paid now will also be subject to the same rate. How much will LK have to pay at the end of the two years? (Take the comparison date as one year from now!)

1. Mufaro must pay the bank $2 000 which is due in one year. She is anxious to lessen her burden in advance and therefore pays $600 after three months, and another $800 four months later. If the bank agrees that both payments are subject to the same simple interest rate as the loan, namely 14% per annum, how much will she have to pay at the end of the year to settle her outstanding debt?

Compound Interest

Compound interest arises when, in a transaction over an extended period of time, interest due at the end of a payment period is not paid, but added to the principal. Thereafter, the interest also earns interest, that is, it is


compounded. The amount due at the end of the transaction period is the compounded amount or accrued principal or future value, and the difference between the compounded amount and the original principal is the compound interest. Essentially, the basic idea is that interest is earned on interest previously earned.

Examples

You deposit $1000 at 10% per annum into a savings account, how much will you have at the end of 4 years if interest is compounded once per year.

Year/Period Beginning Amount

Interest factor

Ending amount Interest

1 1000 0.1 1100 1002 1100 0.1 1210 1103 1210 0.1 1331 1214 1331 0.1 1464,10 133,10

Compound interest=

464,10

Compound Interest = Ending amount (1464.10)- Beginning amount (1000) = 464.10

As shown in the example, compound interest in fact is just the repeated application of simple interest to an amount that is at each stage increased by the simple interest earned in the previous period. It is, however, obvious that where the investment term involved stretches over many periods, compound interest calculations along the above lines can become tedious.

To remedy this we use a formula for calculating the amount generated for any number of periods.

S (FV) =P (1+i)n

wheren, is number of periods and i, compound interest rate per period and P is Present value

It also follows when that when given the future value amount S, you can find the Present value ,P by the process of discounting as follows;

PV=S/ (1+i)n


Example

Find the present value of $170000 which should be received at the end of 8years when the interest rate is 22.67% compounded once a year.

Solution

Timeline

PV?? t=8

0 8 years

170K

Given that PV= S(1+i)n

PV=170000/ (1+0.2267)8

PV=$33154

Practice Questions

1. What is the Present Value of the following yearly successive cash flows given that the interest rate is 26.61% p.a compounded once per year?

CFs : $12000,$15000,$16900,$26950

Given∑i=1

n

CF /(1+i)n=¿($40757, 37)

2. Find the compounded amount on $5 000 invested for ten years at 7.5% perannum compounded annually.

3. How much interest is earned on $9 000 invested for five years at 8% per annum and compounded annually?


Compounding More than Once a Year

Perhaps you have noticed that we have been careful to use the phrase “compounded annually” in the above examples and exercises. This is because the compound interest earned depends a lot on the intervals or periods over which it is compounded.Financial institutions frequently advertise investment possibilities in which interest is calculated at intervals of less than a year, such as semi-annually, quarterly, monthly or even on “daily balance”. What difference does this make? A few examples should help us answer this question.

To find the Future Value when interest is paid more than once per year we use the following relationship :

S=P (1+ jmm

)tm

Where S ≡ the accrued amount, also known as the future valueP ≡ the initial principal, also known as the present valuei≡jm/m, the annual interest rate compounded m times per yearn≡ t × m, = number of compounding periodst≡ the number of years’ of investmentm≡ the number of compounding periods per yearjm≡ the nominal interest rate per year

The above equation is same as: S=P (1+i)n

Where i= jm/m

n=tm

Example

Find the future value of $40 000 deposited into an account that earns 12.62% per annum for 6 years, compounded:

i. Once per yearii. Semi-annuallyiii. Quarterlyiv. Monthlyv. Daily


Solution

Other Formulas useful in Time Value calculations

You don’t need to cram these but you can deduce them by yourself by rearranging the above formulas for time value .

Check for yourself if you come up with these;

t=ln ( S

P)

mln (1+i )wherei can be replaced by jm/m

jm=m [( SP )1 /tm

−1]

n=ln ( s

p)

ln (1+i)

Practice Questions

1. How much time does it take for and investment will double e.g. from 5 000 to 10 000 @ 10% compounded twice a year.?

2. At what interest rate per annum must money be invested if the accrued principal must treble in ten years?

Nominal and Effective Annual Rates

Remark:In cases where interest is calculated more than once a year, the annual rate quoted is the Nominal rate.

Effective Annual Rate [EAR]

Is the actual interest earned per year calculated and expressed as a percentage of the relevant principal. This is the equivalent annual rate of interest – that is, the rate of interest actually earned in one year if compounding is done on a yearly basis.


Example

Calculate the [EAR] Effective Rate of Interest when the nominal rate of interest is 15% per annum compounded on the following basis:

I. YearlyII. Semi-yearlyIII. QuarterlyIV. MonthlyV. Daily

Solution

Given the following formula defined before;

S=P (1+i)n or thatS=P (1+ jmm

)tm

From the above example you should note that, in order to calculate the effective rate, we do not require the actual principal involved. In fact, it is convenient to use P = 100, since the interest calculated then immediately yields the effective rate as a percentage.

The EARs formulation is as follows:

EAR∨ jeff=[(1+ jmm )m

−1]

The Effect of increasing the number of Compounding times per annum

As we increase the number of times the interest is paid in a year implies that, effectively we are increasing the interest rate or return earned on an investment. Notice that the Future Value is increasing as we increase m, the number of compounding times p.a. The Future Value increases at an increasing rate then tails off to a certain upper limiting value as m approaches positive infinity ( as well the interest rate increases at an


increasing rate then tails off to a limiting value as m approaches positive infinity). As m tends to positive infinity, the EAR turns to an upper limit.

What is the significance of this behaviour of EAR?

It protects the creditor against the lender e.g. banks, in that the return on an investment cannot be infinitesimally increased by increasing the rate at which interest is earned per annum If such a limiting value did not exist, it would have meant that the future value of an investment (or debt) could be made arbitrarily large by increasing the compounding frequency. If it does exist, we know that there is an upper limit to the accrued value of an investment or debt over a limited time period.

Plot of EAR and m

EAR EAR curve

Number of compounding times,m

Continuous Compounding

There is a limit regarding the effects of increasing m , on the accumulated Future Value or EAR. The limit exists when m is so large that it approaches infinity is equal to letter e which is the base of a natural logarithm which is equal to 2.1782


e=2.1782

limm→∞ (1+ jmm )

n

=e jm

We can write the effective interest rate (as a percentage) as:

Jeff= 100( ejm− 1).

This is the effective interest rate when the number of compounding periods tends to infinity. Therefore we will use the symbol J∞ to identify it, and now define

J∞ = 100(ejm− 1).

EAR for continuous compounding [ j∞]

The case where interest is compounded an almost infinite number of times as continuous compounding at a rate c, and to J∞ as the effective interest rate expressed as a percentage for continuous compounding.Thus

J∞ = 100(eC– 1)

NB. Thus, finally, with continuous compounding at rate c and for principal P, we can deduce that:

The FV in one year isS = Pec.

The FV in t years will then simply be

S = Pect

Derivation of usable Formulae involving continuous compounding

Suppose that we have a sum P that we invest for one year, on the one hand, at a nominal annual rate of jm compounded m times per year and, on the other, at a continuous compounding rate of c. In these two cases the sum accumulated in one year is then respectively

S=Pec.

S=P (1+ jmm

)m


The question is: What must the continuous rate c be for these two amounts to be equal? It must be

ec=(1+ jmm

)m

since the principal is the same in both cases.

We can solve for c by taking the natural logarithm of both sides.

[stages left here]

This gives the following results

c= m ln(1+ jmm )And

jm=m (ecm−1)

Remark: We use the above formulae to convert a continuous compounded interest rate to an equivalent nominal interest rate that is compounded periodically, or vice versa. The two rates obtained in this way are equivalent in the sense that they will yield the same amount of interest, or give rise to the same effective interest rate.

Practice Questions

1. An investor buys a security that pays an interest of 20% per annum compounded continuously. What is the EAR?

2. Suppose $12 000 was invested on 15 November 20X0 at a continuous rate of 16%. What would the accumulated sum be on 18 May 20X1? (Count the days exactly.)

3. You have two investment options:(a) 1975% per annum compounded semi-annually(b) 19% per annum compounded monthlyUse continuous rates to decide which is the better option.

Equations of Value


From time to time, a debtor (the guy who owes money) may wish to replace his set of financial obligations with another set. On such occasions, he must negotiate with his creditor (the guy who is owed money) and agree upon a new due date, as well as on a new interest rate. This is generally achieved by evaluating each obligation in terms of the new due date, and equating the sum of the old and the new obligations on the new date. The resultant equation of value is then solved to obtain the new future value that must be paid on the new due date.

It is evident from these remarks that the time value of money concepts must play an important role in any such considerations, even more so than they did in the simple interest case, since the investment terms are generally longer in cases where compound interest is relevant.

Example

You decide now that when you graduate in four years’ time you are going to treat yourself to a car to the value of 20 000. If you can earn 17% interest compounded monthly on an investment, calculate the amount that you need to invest now.

Solution

Example

An obligation of $50 000 falls due in three years’ time. What amount will be needed to cover the debt if it is paid(a) in six months from now(b) in four years from nowif the interest is credited quarterly at a nominal rate of 12% per annum?

Solution

[Draw the relevant time line.]


As we noted above that we would concern ourselves here with replacing one set of financial obligation with another equivalent set. This sounds complicated, but is really just a case of applying the above rules for moving money back and forward, keeping a clear head and remembering that, at all times, the only money that may be added together (or subtracted) is that with a common date.

Example

Tenesmus Sithole foresees cash flow problems ahead. He borrowed $10 000 one year ago at 15% per annum, compounded semi-annually and due six months from now. He also owes $5 000, borrowed six months ago at 18% per annum, compounded quarterly and due nine months from now. He wishes to pay $4 000 now and reschedule his remaining debt so as to settle his obligations 18 months from today. His creditor agrees to this, provided that the old obligations are subject to 19% per annum compounded monthly for the extended period. It is also agreed that the $4 000 paid now will be subject to this same rate of 19% for evaluation purposes. What will his payment be in 18 months’ time?

Solution

Self Test Problems

1) At what rate of simple interest will $600 amount to $654 in nine months?

2) A promissory note dated 1 April 2006 for $1 500, borrowed at 16% per annum, which is due on 1 October 2006 is sold on 1July 2006. What is the maturity value of the note? What is the present value on the date of sale?


3) The simple discount rate of a bank is 16% per annum. If a client signs a note to pay $6 000 in nine months time, how much will the client receive? What is the equivalent simple interest rate?

4) Calculate the sum accumulated if a fixed deposit of $10 000 is invested on 15 March 2003 until 1 July 2005 and interest is credited annually on 1 July at 15.5% per annum.

5) You are quoted a rate of 20% per annum compounded semi-annually. What is the equivalent continuous interest rate?

6) You have two investment options:

a. 19.75% per annum compounded semi-annually.b. 19% per annum compounded monthly.

Use continuous rates to decide which the better investment option is.

7) Determine the effective rates of interest if the nominal rate is 18% and interest is calculated:

a. Half-yearly.b. Monthly.

8) A small businessman borrowed some money from the bank under the following conditions:

$500 000 to be paid after 3 months from the date of the loan. $800 000 to be paid one 1 year from the date of the loan. $900 000 to be paid 1 year 6 months from the date of the loan.

The businessman has found things to be tough this year and fails to make any payments.

6 months from now he makes a payment of $300 000. 9 months later he pays $250 000.The bank accepts this arrangement provided that the balance is to be paid on the last date as agreed. If simple interest is charged at 22% per year, how much is to be paid by the businessman? Illustrate in a time line.

9) A lender quotes an interest rate on loans at 22% per annum with continuous compounding, but interest is actually paid quarterly. Find the amount of interest on a loan of $250 000 after 1 year.

10) Compare the amounts accumulated on a principal of $10 000 if invested from 10 March 2003 to 1 July 2005 at 16.5% per annum compounded semi-annually, and credited on 1 July and 1 July, if:


a. Simple interest is used for the odd period and compound interest for the rest of the term.

11) Paul owes Winston $1 000 due in 3 years and $8 000 due in 5 years. He wishes to reschedule his debt so as to pay two sums on different dates, one say X, in one year and the other, which is twice as much (i.e. 2X), five years later. Winston agrees provided that the interest rate is 18% per annum compounded quarterly. What are Paul’s payments? Illustrate in a time line.

12) Determine the future value of an annuity after five payments of $600 each, paid annually at an interest rate of 10% per annum.

13) Mrs. Dudley decides to save for her daughter’s higher education and, every year from the child’s first birthday onwards, puts away $1 200. If she receives 11% interest annually, what will the amount be after her daughter’s 18th birthday?

14) Determine the amount and the present value of an ordinary annuity with payments of $200 per month for five years at 18% per annum compounded monthly. What is the total interest paid?

15) Suppose the annuity just described above is not an ordinary annuity but an annuity due. What would the amount and present value be then, and what would be the interest paid?

16)Peter Penniless owes Wendy Worth $5 000 due in two years from now, and $3 000 due in five years from now. He agrees to pay $4 000 immediately and settle his outstanding debt completely three years from now. How much must he pay then if they agree that the money is worth 12% per annum compounded half-yearly?


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CORPORATE FINANCE MODULE - Chinhoyi University of Web viewword publishers2016corporate finance modulecuac 207time value of money conceptsj.mashonganyikachinhoyi university of technologyc