(2) Given: equal P; equal t
8% compounded semi-annually i^(2) =0.08 m=2 p t
___ compounded i^(4) =? m=4 p t
Let F1 be the future value when interest is compounded quarterly, and f2 be the future value when
interest is 8% compounded semi-annually.
F1=F2
P (1 + i^4/4) ^(4)t = P(1 + i^(2)/2^(2)t
(1 + i^ (4)/4)^4= (1 + 0.08/2)^2
(1 + i^ (4)/4)64 = (1.04)^2
1 + i^ (4)/4) = [(1.04)^2]^(1/4)
1 + i^ (4)/ (4) = (1.04)^1/2
1 + I ^ (4)/ (4) = 1.019804
i(4)/(4) = 1.019804 -1
i(4)/(4) = 0.019804
I(4) = (0.019804)(4)
i(4) = 0.079216 or 7.9216%
Answer: 7.9216% compounded quarterly
(3) Given: equal P; equal t
12% compounded monthly i^(2)= 0.12 m= 12 p t
____ compounded semi-annually i^(2)= ? m= 2 p t
Let F1 be the future value when interest is compounded semi-annually, and F2 be the future value when
interest is 12% compounded monthly.
F1=F2
P(1+i(2)/2(2)t = p(1+i(12)/12)(12)t
(1+i(2)/2)2 = (1+0.12/12)12
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Solution
F = P(1+j)^n
15,500 = 12,000(1+0.005)^n
log (15,500/12,000) = n log(1.005)
n=log(12,917)/log(1.005)=51.3145 periods
Answer: t=n/m=51.3145/12=4.28 years
3. Shirl is planning to invest P20,000. At what rate compounded semi-annually will accumulate her
money to P25,000 in 3 years?
Given: P=20,000 F=25,000 t=3 years m=2 n=mt=(2)(3)=6
Find: i^(2)
Solution. F=P(1+j)^n
25,000=(20,000)(1+j)^n
1.25=(1+j)^6
1.25^1/6=1+j
(1.25)^1/6-1=j
j=0.0379 or 3.79%
The interest rate in each conversion period is 3.79%
The nominal rate can be computed by
j=i^(12)/m
0.0379=i^(2)/2
i^(2)=(0.0370)(2)
Answer: ^(2)=0.0758 or 7.58%
4. What nominal rate compounded monthly is equivalent to 12% compounded annually? Round off your
answer to six decimal places.
Given: i(1)=0.12 m=1
Find i^(12)
Solution. F1=F2
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Example 3. At what nominal rate compounded semi-annually will P10,000 accumulate to P15,000 in 10
years?
Given: F=15,000 p=10,000 t=10 m=2 n=mt=(2)(10)=20
Find: i^(2)
Solution.
F=P(1+j)^n
15,000=10,000(1+j)^20
15,000/10,000=(1+j)^20
1.5=(1+j)^20
(1.5)^1/20=1+j
(1.5)^1/20-1=j
J=0.0205
The interest rate per conversion period is 2.05%.
The nominal rate (annual rate of interest) can be computed by
j=i^(m)/m
0.0205=i^(2)/2
i^(2)=(0.0205)(2)
i^(2)=0.0410 or 4.10%
Hence, the nominal rate is 4.10%
Example 4. At what interest rate compounded quarterly will money double itself in 10 years?
Given: F=2P t=10 years m=4 n=mt=(4)(10)=40
Find: i^(4)
Solution.
F=P(1+j)^n
2P=P(1+j)^n
2=(1+i)^40
(2)^1/40=1+j
(2)^1/40-1=j
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J=0.0175 or 1.75%
The interest rate in each conversion period is 1.75%
The nominal rate can be computed by
j=i^(4)/m
0.0175=i(4)/4
i^(4)=(0.0175)(4)
i^(4)=0.070 or 7.00%
Therefore, the nominal rate that will double an amount of the money compounded quarterly in
10 years is 7.0%
Example 5. What effective rate is equivalent 10% compounded quarterly?
Given: i^(4)=0.10
m=4
Find: effective rate i^(1)
Solution
Since the equivalent rates yield the same maturity value, then
F1=F2
P(1+i^(1))^t= P[1+1^(4)/m]^ml
Dividing both sides by P results to
` (1+i^(1))^t= [1+1^(4)/m]^ml
Raise both sides to 1/t to obrtain
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P[1+i^(12)/m]mf = P(1+j)^1
[1+i^(12)/12]^12t= (1+i^(1))^t
[1+i^(12)/12]^12 =(1+0.12)
i^(12)/12=(1.12)^(1/12)-1=0.009489
i^(12)=12(0.009489)=0.113868 or 11.3868%
Answer; The nominal rate compounded monthly equivalent to 12% compounded annually is 11.3868%
5. What simple interest rate is equivalent to 10% compounded semi-annually at the end of year 1?
Given: i^(2) =0.10 m=2 t=1
Find:r
Solution.
Simple Interest Compound Interest
Fs = Fc
P(1+RsT) = P[1+i^(2)/m]
(1+Rst) =
[1+i^(2)/2]^2r
Substitute t=1 to obtain:
(1+Rs) = [1+i^(2)/2]^2
(1+Rs) = [1+0.10/2]^2
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