Upload
shanna-ray
View
242
Download
0
Embed Size (px)
Citation preview
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