TIME VALUE OF MONEY CONCEPTS

Introduction

The concept of the time value of moneyis 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, inflationrate, 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 borrowedor 12/100 of the amount borrowedor 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 borrowedor 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 interestExample

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?

Given that I=Prt

I = 10000×0.20×1 I =$2000 is the interest due (I) Simple

InterestTherefore Total amount dueS=Interest+Principal Payment=2000+10000

= 12000

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

S = Principal value + Interest

S = P + IS = 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?

S =P (1+rt)

S= 10000(1+20%×3)S =10000(1+0.20×3)=$16000

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?

t=125365………note that t- is always in years so set it as

a fraction of number of days in a year.P=18000r =0.22

so applying , S=P (1+rt)

S=18000(1+0.22×125/365)S=$19356

This is the Future Value of the amount to be paid 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 = Prtbut 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

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?

S=$12000 r=0.12 t=3/12

Given that ,P= S(1+rt)

t- term

r =Interestrate

P or PV

FV or S= P(1 +rt)

Then PV= 12000(1+0.12×3 /12)

= $11 650Remark: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 moneymarket ( the short-term market ). An important instrument on this market is the negotiablecertificate 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?

We have already seen that S = P (1 + rt ), therefore the maturity value, S, is going to be :

S = 1 000 000 (1 + 0.3465 x 91 / 365 ) = $1 086 387.67

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 includethe day the money is deposited or lent and excludethe day the moneyis 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 itwas 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 DaysMay (including 25 May) 7June 30July 31August (excluding 17 August) 16Total 84 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?

Since we have seen that S= P (1+rt ), we can deduce that the consideration, or market value, P, is given by the following relationship:

P = S/( 1 + rd )

Where:S is the maturity value,dis the days remaining to maturity,ris the yield per year.

Therefore, the consideration, P, will be equal to:P =1 087 336.99(1 + 0.33 x 91 /365)

P = 1 087 336.991.082273973

P = $1 004 678.13

Practice Exercise

1. On 1 May 2012, you purchase an NCD with a maturity date of 31 July 2012, nominalvalue of $1 000 000 and an interest rate of 34.65%. On 18 June 2012 you then sell theNCD 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 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 anNCD 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 termfor which the loan (or investment) is made. On the due date, the principal borrowedplus the interest earned is paid back.

t- term

d= discount rate

P or PV

S=

In practice, there is no reason why the interest cannot be paid at the beginning ratherthan at the end of the term. Indeed, this implies that the lender deducts the interestfrom the principal in advance. At the end of the term, only the principal is thendue. Loans handled in this way are said to be discounted and the interest paid inadvance is called the discount. The amount then advanced by the lender is termed thediscounted value. The discounted value is simply the present value of the sum to bepaid 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

When Lisa subscribes to the issue she pays $9 and at the end of the tenure she will get $10 from Treasury.

Discounted Value = S (1-dt) or (S-D)

Discount = Sdt

= 10× 0.10×1 =$1

Therefore PV given above is P=S-D = $10-$1 = $9

Or better still, Discounted Value = S (1-dt)= 10 (1-0.10×1) = $9

Which is the amount paid by Lisa to be paid back $10 in one years’ time.

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?

PV=S (1-dt)PV=100000[1-(0.18×90/365)]PV=$95561

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?

PV=S (1-dt)

PV=100000[1-(0.22×3/12)]PV=$94500

The person receives $94500 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 acceptanceor bill, the so-called “face value”, is what the owner thereof will receive on the maturitydate. On the other hand, the price paid is the

present value, which is calculatedas 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%?S= $35 0000 d= 26% t= 9/12⟹Discount (D) =Sdt

=35000×0.26× 9/12= $6 825

⇒ Discounted Value (PV) =S-D or S (1-dt)=35000-6825=$28175

The discounted value is $28175. In order to determine the equivalent interest rate r,we note that 28 175 is the price now and that 35 000 is paid back nine months later.

I = S − P= 35000 – 28 175= 6825

The interest is thus 6825. The question can thus be rephrased as follows: What simple interest rate, when applied to a principal of $ 28 175 , will yield $6825 interest in nine months?But remember

I = Prtand with substitute we get

6825 = 28 175 ×r × 9/12If we make r subject , we get

r = 0.32298 =32%

Thus the equivalent simple interest rate is 32 % per annum.

NB

Note the considerable difference between the interest rate of 32% and the discountrate of 26%. This emphasises the important fact that the interest rate and thediscount rate are not the same thing. The point is that they act on differentamounts,and at different times – the former acts on the present value, whereas the latter actson 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 anequation forr.

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.

now

1 080

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

916.54

16%

1 000

⇒ 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 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, wedeflate 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 receivedExample 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 months20000 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 monthsT=3 months

r=22% PV=? 100 000 120 0000 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 Obligation1) 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 monthsGiven that S= P(1+rt) ; S@ 10 months= 20000(1+0.22×10/12)

S=$23667To 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, ifshe 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=0.162000t=6/12 r=0.16

0 3 6 12 months3000 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=$4280Lisa 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

AmountInterest 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 valueIt 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

TimelinePV?? t=8

0 8 years170K

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 value

P ≡ the initial principal, also known as the present value

i≡jm/m, the annual interest rate compounded m times per year

n≡ 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

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

)tm

We will have the following results with t=6 ,P= $40 000 , jm

=0.1262 and changing value of mi. S =40 000(1+0.1262/1)6 where m=1

=ii. S =40 000(1+0.1262/2)6×2 where m=2

=115 487.42

iii. S =40 000(1+0.1262/4)6×3 where m=3

=192 779.87

iv. S =40 000(1+0.1262/12)6×12 where m=12

=511 095.58

v. S =40 000(1+0.1262/365)6×365 where m=365

=14630 261.50

Other Formulasuseful 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 interestactually 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. DailySolution

Given the following formula defined before;

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

)tm

I. Yearly case: assuming that P=100 , t=1 ,m=1S=P (1+ jm

m)tm

S=100(1+ 0151

)1∗1

=115.00Interest=Future Value(S)-Present Value (P)

=115-100=15

EAR= InterestPrincipal=

15100×100

=15% EAR =15% yearlyII. Half yearly case : m=2 ,t=1, P=100

S=100(1+ 0152

)1∗2

=115.56 semi-yearlyI=S-P =115.56-100=15.56

EAR=15.56100 ×100

=15.56%

III. Quarterly case : m=4 , t=1, P=100

¿100(1+ 0154

)1∗4

=115.865 (1/4 yearly)I=S-P=115.865-100=15.87

EAR=15.87100 ×100

=15.87% (quarterly)IV. Monthlycase : m=12, t=1 ,P=100

S=100(1+ 01512 )1∗12

=116.075

I =S-P=116.075-100 =16.08

EAR=16.08100 ×100

=16.08 %V. Daily Case :m=365 ,t=1 ,P=100

S=100(1+ 015365

)1∗365

=116.179I =S-P =116.179-100 =16.179

EAR=16.18100 ×100

=16.18 %

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 tendsto positiveinfinity, the EAR turns to an upper limit.

What is the significance of this behaviour of EAR?

It protects the lender 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 increasingm , on the accumulated Future Value or EAR. The limit exists whenm is so large that it approaches infinity is equal toletter ewhich is thebase 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 jmcompounded 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+ jm

m)m

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

Use 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[include time line presentation]

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.]

(a) To determine the debt if it is paid in six months (ie two quarters), we must discount the debt back two-and-a-half-years from the due date to obtain the amount due.

S = P (1 + i)n

P = S (1 + i)-n

with m = 4, t = 2,5 and jm = 0,12.

P = S (1 + i)-n

P=?????

= 37 204,70The present value of the debt six months from now is $37 204,70.

(b) To determine the debt, if it is allowed to accumulate for one year past the due date, we must move the money forward one year to obtain.

S = P (1 + i)n

S=????

= 56 275,44The future value of the debt four years from now is $56 275,44

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

Example

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?

CHAPTER 3

ANNUITIES

Annuities

An annuity is a series of equal payments made at fixed intervals for a specified number of periods. For example, a promise to pay $ 1 000 a year for 3 years is a 3-year annuity. There are two types of annuities : annuity due and ordinary (deferred) annuity.

Diagram for Ordinary Annuity

R R R R R R R

FVIf the payments are made at the end of each period, that is, they are made at the same time that interest is credited, it is an ordinary annuity. If the payments are made at the beginning of each period, the annuity is known as an annuity due.

Diagram for Ordinary due

R R R R R R R

FV

If the payments begin and end on a fixed date, the annuity is known as an annuitycertain. On the other hand, if the payments continue for ever, the annuity is knownas perpetuity.

The FV of an annuity is the sum of all payments made and the accumulated interest at the end of the term.

The PV is the sum of payments, each discounted to the beginning of the term, that is, the sum of the present value of all payments.

Future Value of an Ordinary Annuity

ExampleSuppose you deposit $ 100 at the end of each year for 3 years in a savings account thatpays 5% interest per year, how much will you have at the end of 3 years?

In this example, each payment is compounded out to the end of period n, and the sum ofthe compounded payments is the future value of the annuity

Timeline presentation of the problem:

100 100 100

0 1 2 3100 105 110.25315.25

Thus, the future value of the annuity = [100 (1.05)2 + 100 (1.05)1 + 100] = $315.25.You should notice that this type of calculation becomes tedious if the investment term spans over a very long period of time, so we engage a formula, the derivation of which shall be beyond the scope of this course/module.

Future Value Interest Factor Annuity [FVIFA]

The future value of an annuity of $1 for a period of n years at an interest rate of i is given by the following formula:

FVIFAn , i%=(1+ i)n−1

i

For the above example;FVIFA 3,5%=

(1+0 .05 )3−10 .05

= 3.1525To find the FV, we multiply the FVIFA by the size of periodic payment

FV= 3.1525×100 = 315.25Generally, The FV of an ordinary annuity is given by the following formulation;

FV OD=[(1+i )n−1

i]×R

Where iis interest rate i per payment interval, R is annuity payment amount per periodn is the number of payment intervals of the annuity

NB. The annual interest rate jmcompounded m times per year jm/mis denoted by i [i= jm/m]While the number of interest compounding periods tm is denoted by n.

As such;

FV OD=[ (1+ jmm )tm

−1

jmm

]∗R

Practice Questions1. Mrs Thodes 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?2. What is the accumulated amount (future value) of an annuity with a payment of $600 four times per year and an interest rate of 13% per annum compoundedquarterly at the end of a term of five years?

Future Value of an Annuity Due.

If the $100 payments had been made at the beginning of each year, this would be an annuity due and the time line would look as shown below :

100 100 100

0 1 2 3105 110.25115.76

331.01

Thus, the future value of the annuity due is [100 (1.05) + 100 (1.05)2 + 100 (1.05)3 ] = $331.01.

Since the payments occur earlier, more interest is also earned, thus the future value of an annuity due is larger ($331,01) than that of an ordinary annuity ($315.25).

To get the future value of an annuity due we compound the future value of an ordinary annuity by an extra period.

Future value of annuity due [FVAD] = R× FVIFAOD (1+i)

In our example, the future value of the annuity due is 100 (3.1525) (1.05) = $331.01.

Present Value of an Ordinary Annuity.

Remark: The present value of an annuity is the amount of money that must be invested now, at i percent, so that n equal periodic payments may be withdrawn without any money being left over at the end of the term of n periods.

Example

Suppose an investor has the following alternatives : a three year annuity of $1 000.00 or a lump sum payment today. What must the lump sum payment be to make it equivalent to the annuity if the interest rate is 10% ?

If the payments come at the end of each year, the annuity is an ordinary annuity and the time line will look as shown below :

1000 1000 1000

0 1 2 3

909.10 826.40 751.302 486.80

Thus, the present value of the annuity is given by :[ (1 000 ) / (1.10) + (1 000 ) / (1.10)2 + (1 000 )(1.10)3] = $2 486.80

This is the present value of three payments of $1 000 each deposited at the end of each year.

Present Value Interest Factor Annuity , (PVIFAOD)

The present value of an annuity of $1 for a period of n years at an interest rate of i is given

by the following formula:

PVIFAOD=(1+ i )n−1i (1+ i)n

Where iis interest rate i per payment interval, R is annuity payment amount per periodn is the number of payment intervals of the annuity

NB. The annual interest rate jmcompounded m times per year jm/mis denoted by i [i= jm/m], while the number of interest compounding periods tm is denoted by n.[tm=n]

To find the PV of the annuity , we can make use of the formula: Multiply the PVIFA by the periodic payment,R. So we have;

PVOD=PVIFAOD× R

PV OD=[ ( I+i )n−1i (1+i )n ]×R

or where interest is earned more than once

PV OD=[ (1+ jmm )tm

−1

jmm (1+ jmm )

tm ]∗RFor the above example, this would be;

PV OD=[ (1+0 .10 )3−10 .10 (1+0 .10 )3 ]×1000

= $2 486.80

Present Value of an Annuity Due

We can establish a relationship between the present value of an annuity due and an ordinary annuity, namely

PV (of annuity due with n periods)= firstpayment +PV ( of ordinary annuity with (n − 1) periods)

Thus the present value of an annuity due is given by

Present Value of Annuity[PVAD] = R× PVIFAOD

(1+i)

Example

If the monthly rental on a building is $1 200 payable in advance, what is the equivalent yearly rental? Interest is charged at 12% per annum compounded monthly. How much interest is paid?

SolutionWhat the question in fact asks is: What rental must be paid as a lump sum at the beginning of the year instead of monthly payments? In other words, this is a present value-type calculation. There is one payment at the beginning but not at the end. This means that the first payment does not have to be discounted and that the remaining 11 payments form an ordinary annuity. Thus the equivalent yearly rental (ER) is given by

ER = First payment + Present value of remaining 11 payments

Solution:[use timeline]

Or use the following relationship

Present Value of Annuity[PVAD] = R× PVIFAOD

(1+i)

Working:

Summary

1. Present Value of Ordinary Annuity [PVOD] = R× PVIFAOD

2. Present Value of Annuity Due [PVAD] = R× PVIFAOD (1+i)3. Future Value of Ordinary Annuity[FVOD] = R× FVIFAOD

4. Future Value of Annuity Due [FVAD] = R× FVIFAOD (1+i)

Annuity Payment Date differs from Interest Payment Date

In practice the payment of interest does not to coincide exactly with the annuity payment date. When the two does not coincide, it presents practical difficulties on deciding which interest rate to use for which period since they differ. If this happens we have to make use of the concept of continuous compounding as noted earlier in our discussions, to convert the mismatching interest rate through continuous compounding to make the interest payment date (IPD) and the annuity payment date (APD) coincide. [ i.e. marry/match/coincide IPD and APD}.

What we will be doing is to replace the specified interest rate and period with an equivalent interest rate that corresponds to the period of the payments.

Example

An investor makes a payment of $10 000 at the end of every 4 months for the next 6 years. Calculate the Future Value of the payments if interest is compounded semi-annually at 19.5%.

Timeline presentation

XXXX

Lets establish notation here:

mi(The frequency of Interest Payment) = 2 mp(frequency of annuity payment) = 3

We see mi ≠ mp thus giving the need to convert the semi- annually compounded rate to be applicable to the periods of annuity payment ie every 4 months

Step 1Convert the semi-annual compounding of interest to three time per year to coincide with the frequency of payments per year of annuity. We go through the equivalent continuous effective rate.

Jm= 0.195 mi =2We know that ;

c=m ln(1+ jmm )

c=2 ln (1+0 .1952 )

= 18.60697%

Step 2Is to calculate the equivalent nominal rate, jm , to the continuous rate calculated in step 1 , coinciding with mp =3.

We also know that;

jm=m(ecm−1)

Given c = 18.60697% and mp=3We will have;

jm=3(e0.18603 −1)

=19.1961%This rate is an annual equivalent nominal rate compounded 3 times p.a , which makes now applicable to the period of payement which is also 3 time p.a.

Step 3We can now aplly this rate to find the Future value of the investment as follows.

FVOD = R× FVIFAOD

FV OD=[ (1+ jmm )tm

−1

jmm

]∗R

Thus;

FV OD=[ (1+ 0.19193 )3×6

−1

0.19193

]×10000 = $320 923.78

, which is the FV of the investment.

Alternative method to find the Equivalent Periodic Nominal Rate

We can use the following formula

jn=n[(1+ jmm )mn−1]

, jn compounded n times, equivalent to jm compounded m times

n- is where you want to go ie the frequency of paymentm- is the where you are coming from ie the frequency of interest compounding

Try use the formulae and see if you get the same coincided periodic nominal rate.

Practice Questions

1. Calculate the present value of an annuity that provides $1 000 per year forfiveyears if the interest rate is 12,5% per annum.

2. Max puts $3 000 down on a second-hand car and contracts to pay the balance in 24 monthly instalments of $400 each. If interest is charged at a rate of 24% per annum, payable monthly, how much did the car originally cost when Max purchased it? How much interest does he pay?

3. Determine the present value of an annuity with semi-annual payments of $800 at 16% per year compounded half-yearly and with a term of ten years.