Chapter 4

Money – Time Relationships & Equivalence

HarrisWidya Adi NugrohoFauzanVery BudimanErny Apriany Sylwana

Why consider return the capital?

Interest and profit pay the providers of capital for forgoing its use during the time the capital is being usedInterest and profit are payment for the risk the investor takes in permitting another person, or organization to use his/her capital

Introduction of Capital

Interest

Compound interest arises when interest is added to the principal, so that, from that moment on, the interest that has been added also earns interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid

In the earliest instances, interest was paid in money for the use of grain or other commodities that were borrowed; consequently interest taking again became viewed as an essential and legal part of doing bussiness

Equivalence & Cash Flow

• Relative attractiveness of different alternatives can be judged by using the technique of equivalence

• We use comparable equivalent values of alternatives to judge the relative attractiveness of the given alternatives

• Equivalent present values can be used in conjunction with the cash flow chart report to determine the cash flow characteristics of the payment plan.

Upward arrows - positive cash flow (receiving the loan) Downward arrows - negative cash flow (pay off)

Single Cash Flow

Present Equivalent Values• Formula:

F=P(1+i)N

• Functional notation:

F=P(F/P,i,%N)

Future Equivalent Values• Formula:

P=F(1/(1+i))N = F(1+i)-N

• Functional notation:

P=F(P/F,i,N)

Where: i = effecive interest rate per interest periodN = number of compounding periodsP = present sum of moneyF = future sum of money

Interest Value Formulas Relating A Uniform Series (Annuity) to Its Present and Future

Equivalent Values

A = Uniform Amounts (Given)N = Number of Interest Period

A A A A A A

1 2 3 4 N -1 N

P = Present EquivalentF = Future Equivalent

P Fi = Interest Rate per Period

Relationship of F & A Relationship of P & A

Interest Factor Relationship

EQUATIONS

GRAPHS

Interest Formulas for Discrete Compounding and Discrete Cash Flows

Discrete compounding means that the interest is compounded at the end of each finite-length period, such as

a month or a year

TO Find Given Factor by wich to Multiply Given

Factor Name Factor Functional

Symbol

Single Cash Flows

F P Single payment compound amount

(F/P, r%,N)

P F Single payment present worth

(P/F, r%,N)

Uniform Cash Flows

F A Uniform series compound amound

(F/A, r%,N)

P A Unform series present worth

(P/A, r%,N)

A F Sinking fund (A/F, r%,N)

A P Capital recovery (A/P, r%,N)

1 2 J - 1 J J + 1 J + 2 J + 3 NN - 1

Deffered Annuities (Uniform Series)

Period i %

Deferred annuities means that the cash flow does not begin until J periods

P0 = A (P/A, i%, N – J)(P/F, i%, J)

Equivalent Calculation Involving Multiple Interest Formulas (Calculation of P, F, A

Values)

P0 = $1203.82

$100(F1)

$200(F2)

$500(F3)

$400(F4)

0 1 2 3 4 5 6 7 8

P0 = F1(P/F, 20%,1) + F2(P/F, 20%,2) + F3(P/F, 20%,3) + A(P/A,20%,5) x (P/F, 20%,3)

P0 = $100(0.8333) + $200(0.6944) + $500(0.5787) + $400(2.9900) x (0.5787)

P0 = $1203.82

$400(F5)

$400(F6)

$400(F7)

$400(F8)

i = 20%

Calculation of Present Equiv. Expend. (P0)

F8 = $5176.19

$100(F1)

$200(F2)

$500(F3)

$400(F4)

0 1 2 3 4 5 6 7 8

F0 = P0(F/P, 20%,8)

F0 = $1203.82 (4.2998)

F0 = $5176.19

$400(F5)

$400(F6)

$400(F7)

$400(F8)

i = 20%

Calculation of Future Equiv. Expend. (F8)

P = F(1 + i)-N Symbol: F = P(F/P, i%,N) F = P(1 + i)NSymbol: P = F(P/F, i%,N)

P0 = $1203.82

A

0 1 2 3 4 5 6 7 8

i = 20%

Calculation of Annual Equiv. Expend. (A)

F8 = $5176.19

A A A A A A A

A = $313.73

A = P0(A/P, 20%,8)A = $1203.82(0.2606) = $313.73

i(1 + i)N

(1 + i)N - 1

Symbol: (A/P, i%,N) P Given

A = P

i

(1 + i)N - 1

Symbol: (A/F, i%,N) F Given

A = F

Or

Equivalent Calculation Involving Multiple Interest Formulas (Calculation of P, F, A

Values)

A = F8(A/F,20%,8)A = $5176.19(0.0606) = $313.73

OR

0 1 2 3 4 N-2 N-1 N

F = G(F/A,i%,N-1) + G(F/A,i%,N-2) + ...... + G(F/A,i%,2) + G(F/A,i%,1) G Given:

i = Effective interest rate per periode

Uniform Gradient Increasing by G per period

Interest Formulas for Uniform (Arithmetic) Gradient (G) of Cash Flow (Calculation of P, F,

A Values , G given)

G

2G

3G

(N-3)G

(N-1)G

(N-2)G

End of Period

N

(1 + i)N - 1

A = G1

i

Symbol: G(A/G, i%,N) G Given

0 1 2 3 4 N-2 N-1 N

P = G(P/G,i%,N) G Given

i = Effective interest rate per periode

Uniform Gradient Increasing by G per period

Interest Formulas for Uniform (Arithmetic) Gradient (G) of Cash Flow (Calculation of P, F,

A Values , G given)

G

2G

3G

(N-3)G

(N-1)G

(N-2)G

End of Period

N

(1 + i)N

P = G (1 + i)N - 1

i(1 + i)N

1

i

Geometric Series

Sometimes economic equivalence problems involve projected cash flow pattern that are changing at an average rate (constants rate) ,each period this is a geometric gradient series.

Cash Flow Diagram For a Geometric Sequence Of Payments Increasing at a Constant Rate of per Period

We can find the present value of a geometric series by using the appropriate formula below

Where is the initial cash flow in the series

Interest Rates That Vary with Time

Interest rates on a loan can vary with time. It is necessary to take this into account when determining the future equivalent value of the loan

It is becoming common to see interest rate “escalation riders” on some types of loan

The present equivalent of future cash flow subject to varying interest rates, a procedure similar to the preceding one would be utilized. In general the present value of cash flow occurring at the end of period N can be computed with the equation below, where ik is the interest rate for the kth period.

If F4 = $2,500 and i1=8%, i2=10%, and i3=11%, then

Nominal and Effective Interest Rates.

Vary often the interest period, or time between successive compounding, or the interest period, is less than one year.

The annual rate is known as a nominal rate.

A nominal rate of 12%, compounded monthly, means an interest of 1% (12%/12) would accrue each month, and the annual rate would be effectively somewhat greater than 12%.

The actual or exact rate of interest earned on the principal during one year is known as the effective rate.

Let r be the nominal, annual interest rate and M the number of compounding periods per year. We can find, i, the effective interest by using the formula below.

Interest Problems With Compounding More Often Than Once Per Year

Single Amount. If a nominal interest rate is quoted and the number of compounding period per year and number of years are known, any problem involving future, annual or present equivalent value can be calculated by straightforward use the equation respectively.

Uniform series and Gradient Series. When there is more than one compounded interest period per year, the formulas and tables for uniform series and gradient series can be used as long as there is cash flow at the end of each interest.

Interest Formulas For Continuous Compounding And Discrete Cash Flow

Interest is typically compounded at the end of discrete periods.

In most companies cash is always flowing, and should be immediately put to use.

We can allow compounding to occur continuously throughout the period.

The effect of this compared to discrete compounding is small in most cases.

We can use the effective interest formula to derive the interest factors.

As the number of compounding periods gets larger (M gets larger), we find that

Continuous compounding interest and discrete cash flow

The other factors can be found from these.

Interest Formulas for Continous Compounding and Discrete Cash Flows

To Find Given Factor by wich to Multiply Given

Factor Name Factor Functional

Symbol

Single Cash Flows

F P erN Cont. Compounfinf compound amount

(F/P, r%,N)

P F e-rN Cont. Compounding present equivalent

(P/F, r%,N)

Uniform Cash Flows

F A erN-1er-1

Cont. compounding compound amount

(F/A, r%,N)

P A erN-1erN(er-1)

Cont. compounding present equivalent

(P/A, r%,N)

A F er-1erN-1

Cont. compounding sinking fund

(A/F, r%,N)

A P erN(er-1)erN-1

Cont. compounding capital recovery

(A/P, r%,N)

How to Use It

Given: Present loan (P), interest rate(r), and period time what is Annual value (A) after (N) years of period?

So, A would be:A=P(A/P, r%,N)

To do:Find A (from P) to table and then subtitute it, if

there is no table find P from A and do inverse.

Solved Problem

Given:

Loan Principal (P) = $10,000

Interest rate (r) = 8%,

Duration of loan (N) = 3 years

Annual Payment (A) = 3,880

End Of Year Interest Paid Principal repayment

1 $800 ?

2 $553.60 $3326.4

3 ? ?

$3,880-$800=3,080

Equivalent Series

Interest rate = 10%$1000

$800$600

$400

$200

1 2 3 4 5

$100

$X

1 2 3 4 5

$X $X $X

$1000+$800(P/A,10%,4) - $200(P/G,10%,4)= 100 + X(P/A, 10%,4)

X = …

Economic equivalence

-Convert all commponent to annual-Count cash flows

Example:

Salary: $3,200

Debt repayment :

AStudentLoan = $20000(A/P, i%, N)

AcreditCard = $5000(A/P, i%, N)

Aetc = ..Savings :

ACondo = $Salary(A/F, i%, N)