3 equation of value

8/9/2019 3 Equation of Value

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v) Equation of Value

Sometimes there is a need to replace a set of debts by another set of diere

amounts due at dierent times. For example, a single payment on the thyesettle these t!o debts"

a) #$%% due in & yearsb) # '%% due in ( years

n order to *nd the single payment, an equation of value should be set up.

n an equation of value, the sum of the values of one set of debts on a compdate equals the sum of the values of another set of debts on the same date.sum on the comparison date is obtained by either accumulating or discountidepending on !hen the obligation and the comparison dates fall.

Example " Francis o!es # $%%, due in & years, and # '%%, due in ( years. - allo!ed to settle these obligations by a single payment on the thyear. Findmuch he has to pay on the thyear if money is !orth / 0m 1 2).


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Example " Francis o!es # $%%, due in & years, and # '%%, due in ( years. - allo!ed to settle these obligations by a single payment on the thyear. Find much he has to pay on the thyear if money is !orth / 0m 1 2).

Solution"Step " 3a4e a time diagram"

5ebts 6ate" 0/, m12) # $%% 75 # '%%

% 2 & $ ( ' 0years) x 8ayment

Step 2" 7hoose a comparison date 0usually the payment date, to simplify

computation). -ere, the th

year is used as comparison date 075).

Step &" 9ring all values to the comparison date by either accumulating ordiscounting. :rite do!n the equation of value, !hich can be brie;y stated

all payments 1 all debts


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5ebts" '%% x 1 1 # ,=.2


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Example 2" 3arian o!es # 2,$%% due in 2 years and # ,%%% due in $ years.settle these obligations, she agrees to pay # ,%%% on the *rst year and anpayment on the thyear. -o! much is the payment on the thyear if mone!orth / compounded quarterlyA

Example &" :hat t!o equal payments at the end of 2 years and $ years !illequitably replace the follo!ing interestBbearing debtsA a) # 2%%% due in & years !ith accumulated interest from today at %/ compounded quarterly. b) # %%% due in ( years !ith accumulated interest from today at =/ compounded semiBannually.3oney is !orth '/ eective rate.


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Example 2" 3arian o!es # 2,$%% due in 2 years and # ,%%% due in $ years.settle these obligations, she agrees to pay # ,%%% on the *rst year and anpayment on the thyear. -o! much is the payment on the thyear if mone!orth / compounded quarterlyA

Solution" Cet x be the payment on the thyear. 7hoose years as 75.

5ebts 6ate" 0/, m1) # 2,$%% 75 # ,%%% % 2 & $ 0years) # ,%%% x8ayments

9ring the values to 75.

5ebts"


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8ayments" x 1 &2.22$ > &=.2'

x 1 # $,2&=.


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Example &" :hat t!o equal payments at the end of 2 years and $ years !ilequitably replace the follo!ing interestBbearing debtsA a) # 2%%% due in & years !ith accumulated interest from today at %/ compounded quarterly.

b) # %%% due in ( years !ith accumulated interest from today at =/ compounded semiBannually.3oney is !orth '/ eective rate.

Solution" Cet x be the equal payment on the 2ndand $thyears. f a debt bears intere

*rst compute for the maturity value on the due date.

+he maturity value of the *rst debt is" 2%%% 1 2'=.(((

+he maturity value of the second debt is" %%% 1 '$.==7hoose $ years as 75

5ebts 6ate" 0'/, m1) 2'=.((( 75 '$.==

% 2 & $ ( ' 0years) x x8ayments


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9ring the values to 75 using '/ eective rate.

5ebts" ,'$.==

x > .2$=(2x 1 &&(.&$ > $'(.(& 2.2$=(2x 1 (2$.%%=

x 1 2,%=.%2


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vi) Varying nterest"

f the interest rate changes during an investment term, the amount at the

previous rate is *rst obtained before applying the ne! rate.

Example " Find the amount in $ years if # $%% is invested at '/ compoun semiBannually in the *rst $ years, $/ compounded semiBannually in the n years and '/ compounded quarterly in the last years.

Solution"

# $%%

% $ = $

0'/, m12) 0$/, m12) 0'/, m1)

% periods ' periods 2 periods


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+he principal of # $%% at an interest rate of '/ 0m12) !ill on the $thye amount to"


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Example 2" # 2,%%% is invested for % years in a savings ban4 that gives =compounded quarterly in the *rst years and %/ compounded quarterly

last years. :hat is the amount at the end of % yearsA

Example &" Dn 3arch , ===, ?yang deposited # $,&2% in a ban4. -odid she have after % years if the =/ interest payable quarterly !as in euntil September , 2%%2, and if %/ !as payable semiBannually thereaft

3 equation of value

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