Amortization (3)

8/2/2019 Amortization (3)

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AMORTIZATION

yIs a means of repaying a debt by a

series of equal payments at equaltime interval. The periodicpayments from an annuity in which

the present value is the principal ofan interest-bearing debt.


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AMORTIZATIONy Formulas:

A = R [1 (1 + i)-n /i]

and

R = Ai/ 1 (1 + i)-n

where

A = principal,R = periodicpayment,

i = interestperperiodand

n = totalnumberofpaymentperiods


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AMORTIZATIONyExample:

yAn obligation of PhP21,000 with interest of8% compounded semi-annually must be paid atthe end of every 6months for 4 years. a)Find the size of periodicpayment. b) Find the

remaining liability justafter making the 5th

payment. c) Prepare theamortization table.

yGiven Data:

A = PhP 21,000m = 2

i = 0.04j = 8%

t = 4years

n = 8


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Solution:

R = Ai/ 1 (1 + i)-n

= 21,000 (0.04)/ 1 (1.04)-8= 3,119.08

A = R [1 (1 +i)-n /i]

= 3,119.08 [1 (1.04)-3 /0.04

= 8,655.73

- Theremainingliabilityafterthe5th paymentisthepresentvalueoftheremainingperiodicpayments.


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Period Balance Payment Interest Paid Payment for

Principal

1 21,000.00 3,119.08 840.00 2,279.00

2 18,720.92 3,119.08 748.84 2,370.24

3 16,350.68 3,119.08 654.03 2,465.054 13,885.63 3,119.08 555.43 2,563.65

5 11,321.98 3,119.08 542.88 2,666.20

6 8,655.78 3,119.08 346.23 2,772.85

7 5,882.93 3,119.08 235.72 2,883.768 2,999.16 3,119.08 119.97 2,999.11

Total 24,952.64 3,953.10 21,000.00*

c) Amortization Table

* The actual value is PhP 20,999.94. This is due to rounding error.


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AMORTIZATION

yStudent Activity # 1:Atsumi borrows acertain amount to buy a bicycle at 12%compounded monthly. The debt will bedischarged by paying PhP 400 monthlyfor 1 year. a) What is the cash value ofthe bicycle? b) How much of her 7th

payment is interest and how much goesto repayment of principal? c) Constructthe amortization table.


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SINKING FUND

yRefers to a fund created by makingperiodic deposits to anticipate the need

of paying a large amount of money atsome future dates.

yThe amount of fund at any time is the

sum of an ordinary annuity accumulated by equal periodicpayments at equal intervals of time,and the amount of interest earned.


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SINKING FUND

yAsinking fund schedule illustrates how thefund accumulates every payment period,and to determine the amount in the fund atany given time, the following geometricprogression formulas for ordinary annuityare used.

ySO

= R [(1 + i)n 1]/i

yR = SO

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


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SINKING FUND

y Exampley A fund is created by making

equal monthly deposits ofPhP 3,000.00 at 9% convertedmonthly.

y Determine the sum after halfyear.y What is the amount in the

fund after the 4th deposit?y Construct the sinking fund

schedule for a 6-month

period.

y Given:R = 3,000j = 9%or0.09m = 12

y Solution:SO = R [(1 + i)

n 1]/i= 3,000 [(1.0075)6 1]/

0.0075= 18,340.89

* n = 4, SO= 12,135.66


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Number of

Payment

Periodic

Deposit

Interest of

Fund

Increase in

Fund

Amount in

Fund

1 3,000 0 3,000.00 3,000.00

2 3,000 22.50 3,022.50 6,022.50

3 3,000 45.17 3,045.17 9,067.67

4 3,000 68.01 3,068.01 12,135.68

5 3,000 91.02 3,091.02 15,226.70

6 3,000 114.20 3,114.20 18,340.90

Sinking fund schedule


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yStudent Activity # 2: Three years from

now, Mr. Tan needs PhP 30,000.00 toliquidate a certain debt, at 6%converted semi-annually.

yHow much must he deposit at the endof every 6 months to provide for thepayment of the debt?

yPrepare a sinking fund table showingthe growth of the fund for 3 years.


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It is a written contract by a debtor to pay a final redemptionvalue on an indicated redemption date, or maturity date, and to paya certain sum

Face value/ par value is the borrowed principal and

describes the payments as periodic payments of interest at aspecified nominal rate called the bond rate.

Asmall dated coupon is attached to the bond correspondingto each payment. Thus, a coupon is a contract to pay on acorresponding date.

The bond owner will detach each coupon when it becomes adue. This is then presented for payment through the bank. Thepayments is called coupon annuity. A bond is named after itsvalue and bond rate.


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BONDS1.Bond Price on a Coupon

Date- is when a coupon of a bond

becomes due- A bond can be sold at any

time and if it is sold on acoupon date, the seller getsthe coupon which is alreadydue.

2. Premium Equation- The premium or discount in

the purchase of a bond can becomputed even without firstcomputing the price.

3. Amortizing a Premium- If the price of a bond is greater

than the redemption value, we say that the bond isbought at a premium wherethe price redemption valueas the premium beingamortized by the couponpayments.

4. Accumulation of aDiscount- The bookkeeping methods of

adding the unpaid interest tothe value of the bond.


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BONDS

5. Price of Bonds for Sale- In buying or selling of

bonds, what is mostconsidered is the couponpayment as being earned

or growing continuouslyduring the correspondingperiod although paymentthereof is due at the end ofthe period.

- Prices of bond are

controlled by the law ofsupply and demand.Considering that bondsmay be sold at any date.

6. Flat Price between InterestDates

- TofindtheflatpriceP andtheand interest price q ofabondonadaybetweensuccessiveinterestdatesAandi toyield

theinvestmentrate i,wehaveto1)ComputetheflatpriceP1 ofthebondtoyieldtheratei atA.

2) AddtoP1 thesimpleinterestonP1 attheratei forthetimefrom

AtoB toobtaintheflatpricePatB toyieldtheratei.

3)SubtracttheaccruedinterestationdayB fromP tofindtheand-interest-priceqofthebondatBtoyieldtheratei.


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BONDS

yStudent Activity # 3: Constructa formula and the sequence ofthe solution of the flat pricebetween interest dates through

the given deadlines. Usesymbols only.


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BONDS

7. Approximate Bond Yield- computetheaverageinvested

capital (V + Q /2)- computethetotalinterest

receivedbyinvestorinthenyearsfrom 1 or2anddividebyn

tofindtheaverageinterestperyear-Anestimatedyieldrepresented

byJ is:J = Average AnnualInterest/

AverageInvestment

- Example: A 1,000,6%bondwillberedeemedattheendof10years.Estimatetheinvestorsyieldforbuyingthebondwhenitisquotedat85andaccruedinterest.

Solution:a.Thebookvalueofthebondchanges

from850to 1,000duetotheaccumulationofthediscount (85means85%of1,000 = 850)

b.Theaverageinvestedcapitalis:850 + 1,000/2 = 925

c.Thediscountonthesellingdateis:1,000 850 = 150d.Thetotalofcouponpaymentsin 10

yearsis:1,000 (0.06)x 10years = 600

e.Totalinterestin 10yearsis:600 + 150 = 750

f. Averageannualinterestis:750/ 10 = 75

g. Approximateyieldis:75/ 925 = 0.0811 or8.11%


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BONDS

8. Valuation of Various Contractsa. Serial Issue bonds which are redeemed on installment basis.

b. Serial Bond is one whose face value is redeemable in installments with interest payable periodically as it becomes due on outstanding

principal. It is common in the sale of real state. It is essentiallycomposed of several bonds combined in one contract. On any date, theflat price of a serial bond issue is the sum of the corresponding pricesof all bonds of the issue still unredeemed.

c. Annuity Bond a contract promising the payment of an annuitywhose present value is H at the bond rate. When H and bond rate are

given, the periodic payment Rof the bond can be computed. At anygiven date, the price of the annuity bond can be obtained bycomputing the present value of the future payments of the bond at theinvestors interest rate.

Amortization (3)

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