Chapter 3 Interest. Simple interest Compound interest Present value Future value Annuity ...

Chapter 3 Interest

Simple interest Compound interest Present value Future value Annuity Discounted Cash Flow

Simple Interest

flat rate of interest

Simple interest

Simple interest is when the interest is calculated only on the principal, so the same amount of interest is earned each year.

YearPrincipal

at firstInterest Earned Total Value

Principal at end

1 $100 $100×10% = $10 $100 + $10 = $110 $100

2 $100 $100×10% = $10 $110 + $10 = $120 $100

3 $100 $100×10% = $10 $120 + $10 = $130 $100

$100 , 10% p.a. 3 years simple interest

Itotal = P × R × T

A = P + Itotal

= P + P × R ×T

= P ×(1 + RT)

Principal Present Value

Total Value Future Value/ Accumulated Value

/Maturity Value

Formula transformation

A = P ×(1 + RT)

P =

R =

T =

RT)(1

A

RP

P-A

T

1PA

TP

I

or

RP

I

or

TR

I

or

Bills of ExchangePromissory note

Used by businesses and government as a form of loan contract over a short period of time. At the end of the period (date of maturity) the principal (face value) of the loan is repayable with interest accrued to that date.

Maturity Value(M)=Face Value (F) + Interest(I)

Bills of ExchangePromissory note

Maturity Value(M)=Face Value (F) + Interest(I)

I = F × R × T

M = F + FRT

M = F (1+RT)

Borrowing Money at Simple Interest $10,000, 10% p.a., simple interest, repay

quarterly over two years

1)How much will he pay in total?

2)How much interest is paid together?

3)How much is his quarterly installment?

4)How much interest is paid in each quarter?

Borrowing Money at Simple Interest $10,000, 10% p.a., simple interest, repay

quarterly over two years

FV =

8

)21.01(000,10$

)21.01(000,10$

Payment =

Itotal = $10,000 × 0.1 × 2

Ipayment =

8

21.0000,10$

= $1,500

= $2,000

= $250

= $12,000

FV = P (1+RT)

Itotal = P×R×T

①

②

③

④

•$10,000, 10% p.a., simple interest, repay quarterly over

two years

Payment Number

Balance at Beginning

PaymentInterest Component

Principal Component

Balance at End

1 $10,000 $1,500 $250 $1,250 $8,750

2 $8,750 $1,500 $250 $1,250 $7,500

3 $7,500 $1,500 $250 $1,250 $6,250

4 $6,250 $1,500 $250 $1,250 $5,000

5 $5,000 $1,500 $250 $1,250 $3,750

6 $3,750 $1,500 $250 $1,250 $2,500

7 $2,500 $1,500 $250 $1,250 $1,250

8 $1,250 $1,500 $250 $1,250 $0

total $12,000 $2,000 $10,000

Compound Interest

Interest on Interest

Compound Interest

Paid on the original investment plus any interest previously accrued, and will increase each period as the investment grows.

$100 , 10% p.a. 3 years compound interest compounded annually

FV1 = PV (1+ i)

FV2 = PV (1+ i) (1+ i)

FV3 = PV (1+ i) (1+ i) (1+ i)

FV = PV (1 + i)n

YearPrincipal

at firstInterest Earned Total Value

Principal at end

1 $100 $100×10% = $10 $100 + $10 = $110 $110

2 $110 $110×10% = $11 $110 + $11 = $121 $121

3 $121 $121×10% = $12.1 $121 + $12.1 = $133.1 $133.1

FV1 = $100(1+10%) = $110

FV2 = $100(1+10%)(1+10%) = $121

FV3 = $100(1+10%)(1+10%)(1+10%)=$133.1

FV1 = PV(1+i)1

FV2 = PV(1+i)2

FV3 = PV(1+i)3

Interest compounding more than once per annum

$5,000 6% p.a. compounding monthly, 2 years

FV = PV (1+i)n

FV = $5,000 (1+6%/12)12×2 =$5,635.80

FV = $5,000 (1+6%)2 =$5,618

Interest compounding more than once per annum

$5,000 6% p.a. compounding monthly, 1 years

FV = PV (1+i)n

FV = $5,000 (1+6%/12)12 =$5,308.39(1+6%/12)12

Nominal interest rateAnnual Percentage Rate(APR)

6%

Real interest

rate6%/12

FV = PV (1+i/m)m×n

Effective Interest Rate (EIR)

FV = PV (1+i/m)m(1+i/m)m

ie = (1+i/m)m-1

ieFV = PV (1+i )1(1+ie)

=

Effective Annual Rate of Interest(EAR)

Formula Manipulation

FV = PV (1+i)n

i = 1PV

FV n

1

FV = PV (1+i)n

(1+i)n =

1 + i =

i =

PV

FV

n

PV

FV

1PV

FV n

1

Formula Manipulation

FV = PV (1+i)n

n =

FV = PV ×(1+i)n

lnFV = lnPV + ln(1+i)n

lnFV - lnPV = ln(1+i)n

lnFV - lnPV = nln(1+i)

n = i)ln(1lnPV-lnFV

i)ln(1PVFV

ln

i)ln(1PVFV

ln

FVIF=

FV = PV (1+i)n

PV =

Formula Manipulation

FV (1+i)-n

Further application

FV = PV (1+i)n

PV = FV (1+i)-n $5,000 now

$7,000 in 4 years,

10% p.a., payable

quarterly

Package 1:

Package 2: P1: $5,000 P2:

$7,000 ×

(1+0.1/4)-(4×4)

= $4,715.38

FV = PV (1+i)n

FVIF (Future Value Interest Factor)t

FVIFi, n

$1,000 12% 5

PVIF (Present Value Interest Factor)

PVIFi, n

$1,000, 12%, 5

PV = FV (1+i)-nPV

FVFVIF

FV

PVPVIF

Check Tables

Exercises

Interpolation

FVIF = 1.9738 Interest rate = 12% n= 6

Interpolation

FVIF = 3 Interest rate = 10% n?

Interpolation

0 11 n1 12

3.1384

3

2.8531

n

FVIF

Interpolation

0 100 x1 300

600

400

200

x

y

200600

200400

100300

1001

x

x1= 200

Interpolation

FVIF = 3 Interest rate = 10% n?

8531.21384.3

8531.23

1112

11

n

n = 11.515

Annuity

A series of payments or receipts of a fixed amount for a specific number of periods. Payments are made at fixed intervals.

Annuity

Ordinary annuity Annuity due Deferred annuity Perpetuity

Ordinary Annuity

An ordinary annuity is one in which the payments or cash flows occur at the end of each interest period.

Deposit $100, end of each month, one year, annual nominal interest of 12% paid per month

FVA(Future Value of an Annuity) =

…$100(1+1%)11+$100(1+1%)10+ + $100(1+1%)1+$100(1+1%)0

0 1 2 n-2 n-1 n

A A A A A

A(1+i)0

A(1+i)1

A(1+i)2

A(1+i)n-2

A(1+i)n-1

+

+

+

+

+

FVA:

$100(1+1%)11+$100(1+1%) 10+…+ $100(1+1%)1 +$100

S ×(1+i) - S = a(1+i)n - a

S (1+i-1) = a(1+i)n - a

S =i

aia n )1(

i

ia

n 11

S = a + a(1+i)1 + + a(1+i)n-1…

S ×(1+i) = a(1+i)1 + + a(1+i)n-1 + a(1+i)n…

FVA

i

ia

n 11

FVIFA (Future Value Interest Factor of an Annuity)

$1,000 1% 12

FVIFA i, n

PMT

FVA 1 1

ni

PMTi

Annuity Amount (Sinking Fund)

PMT = 11 ni

iFVA

FVA 1 1

ni

ai

Period(n)

ln 1

ln 1

FVA iai

n =

(1 ) 1nFVA i

a i

(1 ) 1nFVA i a i

1 (1 )nFVA i

ia

ln 1 ln(1 )nFVA i

ia

(1 ) 1nFVA ii

a

ln 1 ln(1 )FVA i

n ia

What is the present value of $100 to be received at the end of each month for the next 12 months, nominal interest rate 12%

PVA (Present Value of an Annuity)=

…$100(1+1%) -1 +$100(1+1%) -2 + +$100(1+1%) -12

0 1 2 n-1 n

A A A A

A(1+i)-1

A(1+i)-2

A(1+i)-(n-1)

A(1+i)-n

+

+

+

+

S ×(1+i) - S = a - a(1+i)-n

S (1+i-1) = a - a(1+i)-n

S =i

iaa n )1(

i

ia

n11

S = a(1+i)-1 + a(1+i)-2 + a(1+i)-(n-1) + a(1+i)-n…

S ×(1+i) = a + a(1+i)-1+ + a(1+i)-(n-2) + a(1+i)-(n-1)…

PVA

i

ia

n11

PVIFA (Present Value Interest Factor of an Annuity)

$1,000 1% 12

PVIFA i, n

PMT

Annuity Amount (Periodic repayment)

a = 1 1n

PVA i

i

PVA

i

ia

n11

PVA

Period(n)

ln 1

ln 1

PVA iai

n =

1 (1 ) nPVA i

a i

1 (1 ) nPVA i a i

1 (1 ) nPVA ii

a

1 (1 ) nPVA ii

a

ln 1 ln(1 )PVA i

n ia

i

ia

n11

1 (1 ) nPVA ii

a

-

Borrowing Money at Compound Interest

You borrow $5,000 to be repaid over the next 5 years with equal annual installments. Interest on the loan is 12% p.a.

1) What are the annual repayments?2) How much will be owing on the loan after the

third installment is paid? (principal, interest)3) If you want to liquidate the loan in the 4th

period, how much interest will you save?4) Calculate the breakdown of interest and

principal from the 3rd to the 4th period.

Borrowing Money at Compound Interest•$5,000, 12% p.a., compound interest, repay annually over the next 5 years

1)What are the annual repayments?

1 (1 ) niPVA a

i

51 (1 12%)

$5,00012%

a

a= $1,387.05

•$5,000, 12% p.a., compound interest, repay annually

over the next 5 years

Payment Number

Opening Balance of Principal

Repayment Amount

Interest Component

Principal Component

Closing Balance of Principal

1 $5,000.00 $1,387.05 $600.00 $787.05 $4,212.95

2 $4,212.95 $1,387.05 $505.55 $881.49 $3,331.46

3 $3,331.46 $1,387.05 $399.78 $987.27 $2,344.19

4 $2,344.19 $1,387.05 $281.30 $1,105.75 $1,238.44

5 $1,238.44 $1,387.05 $148.61 $1,238.44 $0.00

Borrowing Money at Compound Interest•$5,000, 12% p.a., compound interest, repay annually over the next 5 years

2) How much will be owing on the loan after the third installment is paid? (principal, interest)

1 (1 ) niPVA a

i

= $2,344.19

21 (1 12%)$1,387.05

12%

Interest:$2,344.19 ×12% = $281.30

Principal:$1,387.05 - $281.30= $1,105.75

•$5,000, 12% p.a., compound interest, repay annually

over the next 5 years

Payment Number

Opening Balance of Principal

Repayment Amount

Interest Component

Principal Component

Closing Balance of Principal

1 $5,000.00 $1,387.05 $600.00 $787.05 $4,212.95

2 $4,212.95 $1,387.05 $505.55 $881.49 $3,331.46

3 $3,331.46 $1,387.05 $399.78 $987.27 $2,344.19

4 $2,344.19 $1,387.05 $281.30 $1,105.75 $1,238.44

5 $1,238.44 $1,387.05 $148.61 $1,238.44 $0.00

Borrowing Money at Compound Interest•$5,000, 12% p.a., compound interest, repay annually over the next 5 years

3) If you want to liquidate the loan in the 4th period, how much interest will you save?

1 (1 ) niPVA a

i

= $2,344.19

21 (1 12%)$1,387.05

12%

Save: $1,387.05×2 - $2,344.19 = $429.91

•$5,000, 12% p.a., compound interest, repay annually

over the next 5 years

Payment Number

Opening Balance of Principal

Repayment Amount

Interest Component

Principal Component

Closing Balance of Principal

1 $5,000.00 $1,387.05 $600.00 $787.05 $4,212.95

2 $4,212.95 $1,387.05 $505.55 $881.49 $3,331.46

3 $3,331.46 $1,387.05 $399.78 $987.27 $2,344.19

4 $2,344.19 $1,387.05 $281.30 $1,105.75 $1,238.44

5 $1,238.44 $1,387.05 $148.61 $1,238.44 $0.00

Borrowing Money at Compound Interest•$5,000, 12% p.a., compound interest, repay annually over the next 5 years

More…

4) Calculate the breakdown of interest and principal from the 3rd to the 4th period.

•$5,000, 12% p.a., compound interest, repay annually

over the next 5 years

Payment Number

Opening Balance of Principal

Repayment Amount

Interest Component

Principal Component

Closing Balance of Principal

1 $5,000.00 $1,387.05 $600.00 $787.05 $4,212.95

2 $4,212.95 $1,387.05 $505.55 $881.49 $3,331.46

3 $3,331.46 $1,387.05 $399.78 $987.27 $2,344.19

4 $2,344.19 $1,387.05 $281.30 $1,105.75 $1,238.44

5 $1,238.44 $1,387.05 $148.61 $1,238.44 $0.00

Annuity Due An annuity due is one in which the payments

or cash flows occur at the beginning of each interest period.

0 1 2 n-2 n-1 n

A A A A A

A(1+i)1

A(1+i)2

A(1+i)n-2

A(1+i)n-1

A(1+i)n

FVA (Due)

+

+

+

+

+

)1(

11i

i

ia

n

S

n

t

tnia1

1)1(

= a × FVIFA(i, n) ×(1+i)

S = a(1+i)n + a(1+i)n-1+ + a(1+i)2+ a(1+i)1…

0 1 2 n-1 n

A A A A

A(1+i)0

A(1+i)-1

A(1+i)-2

A(1+i)n-1

PVA (Due)

+

+

+

+

n

t

tia1

1)1(S

)1()1(1

ii

ia

n

= a × PVIFA(n, i) × (1+i)

S = a(1+i)0 + a(1+i)-1+ + a(1+i)n-2+ a(1+i)n-1…

Deferred Annuity The first payment is deferred for a number

of periods. Special case of ordinary annuity

0 1 2 0 1 2 n-1 n

A A A A

A(1+i)0

A(1+i)1

A(1+i)n-2

A(1+i)n-1

m m+1 m+2 m+n-1 m+n

FVA (Deferred)

+

+

+

+

0 1 2 0 1 2 n-1 n

A A A A

A/(1+i)m+1

m m+1 m+2 m+n-1 m+n

A/(1+i)m+2

A/(1+i)m+n-1

A/(1+i)m+n

PVA (Deferred)

+

+

+

+

P = P(m+n) –Pm

=A × PVIFA(i, m+n) – a × PVIFA(i, m)

Pm = A × PVIFA(i, n)

= Pm × (1+i)-m

Approach 1:

Approach 2:

Perpetuity

PVA

i

a

Where n

PVA

i

ia

n11

Discounted Cash Flow

Discounted cash flow is the result of the effect of time on the outflows and inflows of a financial arrangement (time value of money).

NPV (Net Present Value) IRR (Internal Rate of Return

Internal Reward Rate)

Net Present Value

It reflects the net income a project can bring.

End of year Cash ($)0 -$6,0001 $4,0002 $3,0003 -$2,0004 $5,000

Project A is expected to have the following cash flows for it over the next four years.

The initial cost is $6,000, followed by an inflow of $4,000 at the end of year 1, then a $3,000 inflow at the end of year 2 and an outflow of $2,000 at the end of year 3 with a final inflow of $5,000 at the end of year 4.

End of year Cash ($)0 -$6,0001 $4,0002 $3,0003 -$2,0004 $5,000

Given that the cost of capital is 10%, is the project viable?

1 2 3 4$$4 000 $3 000 $2 000 $5 000 6 000

(1 0.1) (1 0.1) (1 0.1) (1 0.1)NPV

， ， ， ， ，

$2,028.14

1 21 2

...(1 )(1 ) (1 )

nCF CF CFNPV I

rr r

n

1 21 2(1 ) (1 ) ... (1 ) nNPV CF r CF r CF r I n

CFt = cash flow generated by project in period t

(t = 1,2,3, …..,n)I = initial cost of the projectn = expected life of the projectr = required rate of return (cost of capital)

= discount rate

1 1

ntt

t

CFNPV I

r

End of year Cash ($)0 -$20,0001 $11,8002 $13,240

End of year Cash ($)0 -$20,0001 $8,0002 $8,0003 $8,000

End of year Cash ($)0 -$20,0001 $9,0002 $8,0003 $7,000

Project A:

Project B:

Project C:

Given that the cost of capital is 10% , which project is the most viable?

Project A:

Project B:

1 2$11,800 (1 10%) $13,240 (1 10%) $20,000 $1,669NPV

1

2

3

$9,000 (1 10%) $8,000 (1 10%)$7,000 (1 10%) $20,000

$52.59

NPV

Project C:

1

2

3

$8,000 (1 10%) $8,000 (1 10%)$8,000 (1 10%) $20,000

$105.18

NPV

Internal Rate of Return

The highest rate of return a project can reach.

1

01

ntt

t

CFI

r

0NPV

Company A intends to invest $200,000 to buy cars for rent. The project is expected to have a steady inflow of $122,000 in the coming two years. What is the IRR of the project? Suppose the cost of capital is 10%, is it viable?

End of year Cash ($)0 -$200,0001 $122,0002 $122,000

End of year Cash ($)0 -$200,0001 $122,0002 $122,000 1

01

ntt

t

CFI

r

8

1

$122$200 0

1t

t r

,2$122 $200 0rPVIFA

,2

$2001.6393

$122rPVIFA

14% ~ 15%r

Interpolation

0 15% r1 14%

1.6467

1.6393

1.6257

r

PVIFA

14% ~ 15%r

15% 1.6393 1.6257

14% 15% 1.6467 1.6257

r

14.35%r

To be specific:

14.35%>10%, the project is viable.

Exercise

15% 1.6393 1.6257

14% 15% 1.6467 1.6257

r