interest rate conversion effective interest rate (i%) 03b.pdf22 engineering economy – © 2016 dr....

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Engineering Economy – © 2016 Dr. Tareq Albahri – Kuwait University

Interest Rate Conversion

Unless stated otherwise, given interest rate is (% per year compounded annually)

Effective Interest Rate (i%)

Interest rate is effective when it is compounded once in the interest period

(i.e. Interest period and compounding frequency must be the same)

% per year compounded annually

% per semester compounded semiannually

% per quarter compounded quarterly

% per month compounded monthly

% per weak compounded weekly

% per day compounded daily

Nominal Interest Rate (r%)

When the interest period & compounding frequency are not the same. Then we need to

convert the interest rate from nominal to effective.

A. Converting from nominal effective (1 step)

The Actual (Effective) interest rate = i %

11

.

BA

A

ri

Use this table to change compounding frequency

From Compounding A To Compounding B

daily 365 daily 1/365

weekly 52 weekly 1/52

monthly 12 monthly 1/12

quarterly 4 quarterly 1/4

semi-annually 2 semi-annually 1/2

annually 1 annually 1

r = Annual (nominal) APR

i = Actual (effective)

B = Length of time interval (yrs)

A = Number of compounding periods in 1 yr

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A. 1. To change compounding frequency only

EX:

If the interest rate (r) is 12 % per year compounded monthly A = 12

We want to calc the interest % per year compounded annually. B = 1

anuually compundedyear per % 12.68or 0.1268112

12.0111

112.

xBA

A

ri

EX:

If the interest rate (r) is 18 % per year compounded weekly A = 52

We want to calc the interest % per year compounded annually. B = 1

anuually compundedyear per % 19.68or 0.1968152

18.0111

152.

xBA

A

ri

EX:

If the interest rate (r) is 14 % per semester compounded monthly A = 12

We want to calc the interest % per semester compounded semiannually. B = 1/2

lysemiannual compoundedsemester per % 7.21or 0.0721112

14.0111

2/12.

BA

A

ri

الفائدة تحسب شهريا ولكن تدفع سنويا

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EX:

If the interest rate (r) is 10 % per semester compounded weekly A = 52

We want to calc the interest % per semester compounded semiannually. B = 1/2

lysemiannual compoundedsemester per % 5.12or 0.0512152

10.0111

2/52.

BA

A

ri

A. 2. To Change interest periods only [shortcut]

EX:

Given interest 6% per year compounded monthly

Want to calculate the effective interest rate % per month compounded monthly

There are 12 months in one year so we divide by 12

12

6

12

ri = ½ (% per month compounded monthly)

And vice versa

EX:

Given interest i = ½ % per month compounded monthly

r = (½ %) 12 = 6% per year compounded monthly

Same compounding

frequency

Same

compounding

frequency

25


B. Converting from effective effective (2 steps)

To change both interest period and frequency

Step 1 – change the interest period

Step 2 – change the compounding

EX: from monthly to annual لتحويل الفائدة الشهرية الي سنوية

If the interest rate (im) is 1.5 % per month compounded monthly (Effective monthly)

We want to calc the interest (ia) % per year compounded annually. (Effective annual)

First: convert % per month to % per year using shortcut, (Compounding remains the same)

r = (im) (12 months in one year) = (1.5 %) (12) = 18 % per year compounded monthly

Second: change compounding from monthly to annually using the equation

So, from 18 % per year compounded monthly A = 12

Calculate ia % per year compounded annually. B = 1

0.19561015.01112

18.0111

12

12.

BA

aA

ri

ia = 19.56% per year compounded annually

to generalize

1112 ma ii annually compoundedyear per % 19.56or 0.19561015.1

12

والعكس صحيح – لتحويل الفائدة السنوية الي شهرية

Rearranging the above equation

1112/1 am ii monthly compoundedmonth per % 1.5or 0.01511956.1

12/1

Note: to convert from annual to monthly using detailed procedure, you

must change the compounding first then the period. If you start with the

period first then you will get r = % per month compounded annually which

does not make any sense!

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EX: from quarterly to annual لتحويل الفائدة الربع سنوية الي سنوية

If the interest rate (iq) is 4.5 % per quarter compounded quarterly (Effective quarterly)


First: convert % per quarter to % per year using shortcut, (Compounding remains the same)

r = (iq) (4 quarters in one year) = (4.5 %) (4) = 18 % per year compounded quarterly

Second: change compounding from quarterly to annually using the equation

So, from 18 % per year compounded quarterly A = 4


0.1925191045.0114

18.0111

4

4.

BA

aA

ri


to generalize

114 qa ii annually compoundedyear per % 19.2519or 0.1925191045.1

4

والعكس صحيح – لتحويل الفائدة السنوية الي ربع سنوية


114/1 aq ii quarterly compoundedquarter per % 4.5or 0.0451192519.01

4/1

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EX: from daily to annual لتحويل الفائدة اليومية الي سنوية

If the interest rate (id) is 0.1 % per day compounded daily (Effective daily)


First: convert % per day to % per year using shortcut, (Compounding remains the same)

r = (id) (365 days in one year) = (0.1 %) (365) = 36.5 % per year compounded daily

Second: change compounding from daily to annually using the equation

So, from 36.5 % per year compounded daily A = 365


0.440251001.011365

365.0111

365

365.

BA

aA

ri


to generalize

11365

da ii

annually compoundedyear per % 44.025or 0.440251001.01365

والعكس صحيح – لتحويل الفائدة السنوية الي يومية


11365/1

ad ii daily compoundedday per % 0.1or 0.001144025.01365/1

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EX: from semiannual to annual لتحويل الفائدة النصف سنوية الي سنوية

If the interest rate (is) is 1% per semester compounded semiannually (Effective semiannual)


First: convert % per semester to % per year using shortcut, (Compounding remains the same)

r = (is) (2 semesters in one year) = (1%) (2) = 2% per year compounded semiannually

Second: change compounding from semiannual to annually using the equation

So, from 2 % per year compounded semiannually A = 2


0.0201101.0112

02.0111

2

2.

BA

aA

ri


to generalize

112 sa ii annually compoundedyear per % 2.01or 0.0201101.01

2

والعكس صحيح – لتحويل الفائدة السنوية الي نصف سنوية


112/1 as ii lysemiannual compoundedsemester per % 1or 0.0110201.01

2/1

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EX: from weekly to annual لتحويل الفائدة األسوويية الي سنوية

If the interest rate (iw) is 0.25 % per week compounded weekly (Effective weekly)


First: convert % per week to % per year using shortcut, (Compounding remains the same)

r = (iw) (52 weeks in one year) = (0.25 %) (52) = 13 % per year compounded weekly

Second: change compounding from weekly to annually using the equation

So, from 13 % per year compounded weekly A = 52


0.1386410025.01152

13.0111

52

52.

BA

aA

ri

ia = 13.864 % per year compounded annually

to generalize

1152 wa ii annually compoundedyear per % 13.864or 0.1386410025.1

52

والعكس صحيح – لتحويل الفائدة السنوية الي أسوويية


1152/1 aw ii weeklycompoundedper week % 0.25or 0.0025113864.01

52/1

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EX: from daily to monthly لتحويل الفائدة اليومية الي شهرية

If the interest rate (id) is 0.1 % per day compounded daily (Effective daily)

We want to calc the interest (im) % per month compounded monthly. (Effective monthly)

First: convert % per day to % per month using shortcut, (Compounding remains the same)

r = (id) (30 days in one week) = (0.1%) (30) = 3 % per month compounded daily

Second: change compounding from weekly to annually using the equation

So, from 3 % per month compounded daily A = 365

Calculate ia % per month compounded monthly B = 1/12

112

001.01111111

12/365

.

..

BA

d

BA

d

BA

m BiA

ABi

A

ri = 0.00253783

0.002510000822.011365

03.0111

42.30

12/365.

BA

mA

ri

ia = 0.25% per month compounded monthly

to generalize

11 AB

dm Bii monthly compoundedmonth per % 0.25or 0.00251130

12

001.0

1130

12 di

mi

والعكس صحيح – لتحويل الفائدة الشهرية الي يومية


11/11

AB

mBd ii daily compoundedday per % 0.1or 0.00110025.0112365/12

1112365/12

md ii

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To generalize

11 12

ABBii

Table: to convert from effective interest rate to another effective interest rate

From 1 A To 2 B

daily 365 daily 1/365

weekly 52 weekly 1/52

monthly 12 monthly 1/12

quarterly 4 quarterly 1/4

semi-annually 2 semi-annually 1/2

annually 1 annually 1

From daily to monthly 112

1

12/365

d

m

ii

From weekly to monthly 112

1

12/52

w

m

ii

From daily to annually 11365

da ii Annual to daily 11365/1

ad ii

From weekly to annually 1152 wa ii Annual to weekly 11

52/1 aw ii

From monthly to annual 1112 ma ii Annual to monthly 11

12/1 am ii

From quarterly to annual 114 qa ii Annual to quarterly 11

4/1 aq ii

From semiannually to

annual 11

2 sa ii Annual to

semiannual 11

2/1 as ii

The year actually contains 365.25 days,12 months, 30.44 days

per month, 7 days per week, 52.18 weeks per year

Example:

Convert the interest rate from 0.1% per day compounded daily

to % per week compounded weekly

Solution

First convert to annually

11365

da ii = 1001.01365

= 0.44025 (44.025% per year compounded annually)

Then convert to weekly

1152/1 aw ii = 144025.01

52/1 = 0.00704 (0.704% per week compounded weekly)

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For continuous compounding

1 rei

EX: If the interest rate is 18% per year compounded continuously

i = exp (0.18) - 1 = 0.1972 or (19.72% per year compounded annually)

EX: If the interest rate is 1% per month compounded continuously

i = exp (0.01) - 1 = 0.01005 or (1.005% per month compounded monthly)

For periods shorter than one year, use

A

yearper rateinterest nominalr and n = A (number of years)

Where, A = number of periods per year

EX:

If the interest rate (r) is 15 % per year compounded continuously

We want to calc the interest % per month compounded monthly.

A = 12 months per year

lycontinuous compoundedmonth per % 1.25or 0.012512

%15mr (still nominal)

monthly compoundedmonth per 1.26%or 0126.010125.0 eim (effective)

To convert (from annual) to % per month compounded continuously just divide by 12

To convert (from annual) to % per weak compounded continuously just divide by 52

and so on

r = nominal (annual) APR - % per year compounded continuously (given).

ia = effective (actual) - % per year compounded annually.

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Effect of compounding on interest rates:

If the nominal interest rate is

Number of periods per

year (m)

Per year compounded

annually

18% per year compounded

annually

1 18%


semiannually

2 %81.181

2

18.01

2


quarterly

4 %2517.191

4

18.01

4


monthly

12 %5618.191

12

18.01

12


weekly

52 %6843.191

52

18.01

52


daily

365 %7164.191

365

18.01

365


continuously

∞ exp (0.18) – 1 = 19.7217%

Example:

Which is more desirable?

a) 16% compounded annually

b) 15% compounded monthly

Solution:

Convert 15 % per year compounded monthly → A = 12

to % per year compounded annually → B = 1

112

15.01

12

ai = 0.1608 (or 16.08 % per year compounded annually)

Therefore, option (b) is better

Note: although I chose to convert option (b) then compare with option (a) which is easier, the

opposite is also possible; convert option (a) to % per year compounded monthly then

compare with (b).

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Home Work Problems: (due in one week) draw the cash flow diagram for all problems.

1. What amount will be owed in 5 years if $5,000 is borrowed now at 10% per year

simple interest? Answer: $7,500

2. What is the principal amount if the principal plus interest at the end of 4 ¼ years is

$14,000 for a simple interest rate of 12% per annum?

3. A person lends $10,000 at 8% simple interest for 5 years. At the end of this time, the

entire amount (principal plus interest) is invested at 12% compounded annually for 10

years. How much will accumulate at the end of the 15-year period? Draw a cash flow

diagram for the loan and investment situation described from the viewpoint of the

person making the loan and the subsequent investment. Answer: $43,484

4. What will be the amount accumulated by the following present investments? $5,000

in 8 years at 13% compounded annually. Answer: $13,290

5. What is the present value of the following future receipt? $18,000 5 years from now at

8% compounded annually. Answer: $12,251

6. What is the accumulated value of the following series of payments? $600 at the end of

each year for 5 years at 10% compounded annually. Answer: $3,663

7. What equal series of payments must be put into a sinking fund to accumulate the

following amount? $65,000 in 15 years at 15% compounded annually when payments

are annual. Answer: $1,365

8. What is the present value of the following series of prospective receipts? $1,500 a

year for 15 years at 15% compounded annually. Answer: $8,771

9. What series of equal payments is necessary to repay the following present amount?

$5,000 in 5 years at 15% compounded annually with annual payments. Answer:

$1,492

10. What annual equal payment series is necessary to repay the following increasing

series of payments? A series of 7 end-of-year payments that begins at $2,000 and

increases at the rate of $100 a year with 10% interest compounded annually. Answer:

$2,262

11. What equal-annual-payment series is necessary to repay the following decreasing

series of payments? A series of 10 end-of-year payments that begins at $6,000 and

decreases at the rate of $200 a year with 12% interest compounded annually.

12. What is the present value of the following geometrically decreasing series of

payments? A first-year base of $9,000 decreasing by 10% per year, to year 10 with an

interest rate of 17%. Answer: $30,915

13. What is the value of n if F = $5,000, P = $1,000, and i = 8% compounded annually?

14. How many years will be required for an investment of $3,000 to increase to $6,939 if

interest is 15% compounded annually? Answer: 6 years

15. What effective annual interest rate corresponds to the following?

a. Nominal interest rate of 12% compounded semiannually.

b. Nominal interest rate of 12% compounded monthly.

c. Nominal interest rate of 12% compounded quarterly.

d. Nominal interest rate of 12% compounded weekly.

e. Nominal interest rate of 12% compounded daily.

16. The “Square Deal Loan” Company offers money at 0.3% interest per week

compounded weekly. What is the effective annual interest rate?

17. Find the nominal interest rate (% per year compounded continuously) for an annual

effective rate 12%; Answer: 11.33%.

interest rate conversion effective interest rate (i%) 03b.pdf22 engineering economy – © 2016 dr....

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