3. Find Interest Rate (i) Example
1. Find Interest Rate i = ? Regular Deposit (PMT Amount) C = 1000, Time n = 5 Year, Present value PV = 4169.87, Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year) for Present value of Annuity Due method
Solution: C=1000 (Cash flow per year)
n=5 years (Number of periods)
PV=4169.87 (Present value)
Now, Present value (Annuity Due) formula is PV_("Annuity Due")=C*[(1-(1+i)^(-n))/(i)]*(1+i)
:.4169.87=1000*[(1-(1+i)^-5)/(i)]*(1+i)
:.(4169.87)/(1000)=[(1-(1+i)^-5)/(i)]*(1+i)
:.[(1-(1+i)^-5)/(i)]*(1+i)=4.17
Now, find one solution using Newton Raphson methodHere ((1-(1+x)^-5)/x)*(1+x)=4.17:.((1-1/((1+x)^5))/x)(1+x)-4.17=0Let f(x) = ((1-1/((1+x)^5))/x)(1+x)-4.17d/(dx)(((1-1/((1+x)^5))/x)(1+x)-4.17)=((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2)d/(dx)(((1-1/((1+x)^5))(1+x))/x-4.17)=d/(dx)(((1-1/((1+x)^5))(1+x))/x)-d/(dx)(4.17)d/(dx)(((1-1/((1+x)^5))(1+x))/x)=((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2)d/(dx)(((1-1/((1+x)^5))(1+x))/x)=((x) * d/(dx)((1-1/((1+x)^5))(1+x))-((1-1/((1+x)^5))(1+x)) * d/(dx)(x))/(x)^2d/(dx)((1-1/((1+x)^5))(1+x))=5/((1+x)^5)+(1-1/((1+x)^5))d/(dx)((1-1/((1+x)^5))(1+x))=(d/(dx)(1-1/((1+x)^5)))(1+x)+(1-1/((1+x)^5))(d/(dx)(1+x))d/(dx)(1-1/((1+x)^5))=5/((1+x)^6)d/(dx)(1-1/((1+x)^5))=d/(dx)(1)-d/(dx)(1/((1+x)^5))d/(dx)(1/((1+x)^5))=-5/((1+x)^6)d/(dx)(1/((1+x)^5))=-5/((1+x)^6)*d/(dx)(1+x)d/(dx)(1+x)=1d/(dx)(1+x)
=d/(dx)(1)+d/(dx)(x)
=0+1
=1 =-5/((1+x)^6)*1=-5/((1+x)^6) =0-(-5/((1+x)^6))=5/((1+x)^6) d/(dx)(1+x)=1d/(dx)(1+x)
=d/(dx)(1)+d/(dx)(x)
=0+1
=1 =(5/((1+x)^6))(1+x)+(1-1/((1+x)^5))*1=5/((1+x)^5)+(1-1/((1+x)^5)) =((x) * (5/((1+x)^5)+(1-1/((1+x)^5)))-((1-1/((1+x)^5))(1+x)) * (1))/(x^2)=((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2) =(((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2))-0=((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2)-0=((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2) :. f'(x) = ((5x)/((1+x)^5)+x(1-1/((1+x)^5))-(1-1/((1+x)^5))(1+x))/(x^2)x_0 = 0.11^(st) iteration :f(x_0)=f(0.1)=((1-1/((1+0.1)^5))/0.1)(1+0.1)-4.17=-0.000135f'(x_0)=f'(0.1)=((5*0.1)/((1+0.1)^5)+0.1(1-1/((1+0.1)^5))-(1-1/((1+0.1)^5))(1+0.1))/(0.1^2)=-6.861802x_1 = x_0 - f(x_0)/(f'(x_0))x_1=0.1 - (-0.000135)/(-6.861802)x_1=0.09998Approximate root of the equation ((1-1/((1+x)^5))/x)(1+x)-4.17=0 using Newton Raphson method is 0.09998 (After 1 iterations) n | x_0 | f(x_0) | f'(x_0) | x_1 | Update | 1 | 0.1 | -0.000135 | -6.861802 | 0.09998 | x_0 = x_1 |
:.i=0.09998
:.i=10 % per year
2. Find Interest Rate i = ? Regular Deposit (PMT Amount) C = 5000, Time n = 3 Year, Present value PV = 13677.69, Deposit Frequency = at the beginning (Annuity Due) of every Year (1/year) for Present value of Annuity Due method
Solution: C=5000 (Cash flow per year)
n=3 years (Number of periods)
PV=13677.69 (Present value)
Now, Present value (Annuity Due) formula is PV_("Annuity Due")=C*[(1-(1+i)^(-n))/(i)]*(1+i)
:.13677.69=5000*[(1-(1+i)^-3)/(i)]*(1+i)
:.(13677.69)/(5000)=[(1-(1+i)^-3)/(i)]*(1+i)
:.[(1-(1+i)^-3)/(i)]*(1+i)=2.74
Now, find one solution using Newton Raphson methodHere ((1-(1+x)^-3)/x)*(1+x)=2.74:.((1-1/((1+x)^3))/x)(1+x)-2.74=0Let f(x) = ((1-1/((1+x)^3))/x)(1+x)-2.74d/(dx)(((1-1/((1+x)^3))/x)(1+x)-2.74)=((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2)d/(dx)(((1-1/((1+x)^3))(1+x))/x-2.74)=d/(dx)(((1-1/((1+x)^3))(1+x))/x)-d/(dx)(2.74)d/(dx)(((1-1/((1+x)^3))(1+x))/x)=((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2)d/(dx)(((1-1/((1+x)^3))(1+x))/x)=((x) * d/(dx)((1-1/((1+x)^3))(1+x))-((1-1/((1+x)^3))(1+x)) * d/(dx)(x))/(x)^2d/(dx)((1-1/((1+x)^3))(1+x))=3/((1+x)^3)+(1-1/((1+x)^3))d/(dx)((1-1/((1+x)^3))(1+x))=(d/(dx)(1-1/((1+x)^3)))(1+x)+(1-1/((1+x)^3))(d/(dx)(1+x))d/(dx)(1-1/((1+x)^3))=3/((1+x)^4)d/(dx)(1-1/((1+x)^3))=d/(dx)(1)-d/(dx)(1/((1+x)^3))d/(dx)(1/((1+x)^3))=-3/((1+x)^4)d/(dx)(1/((1+x)^3))=-3/((1+x)^4)*d/(dx)(1+x)d/(dx)(1+x)=1d/(dx)(1+x)
=d/(dx)(1)+d/(dx)(x)
=0+1
=1 =-3/((1+x)^4)*1=-3/((1+x)^4) =0-(-3/((1+x)^4))=3/((1+x)^4) d/(dx)(1+x)=1d/(dx)(1+x)
=d/(dx)(1)+d/(dx)(x)
=0+1
=1 =(3/((1+x)^4))(1+x)+(1-1/((1+x)^3))*1=3/((1+x)^3)+(1-1/((1+x)^3)) =((x) * (3/((1+x)^3)+(1-1/((1+x)^3)))-((1-1/((1+x)^3))(1+x)) * (1))/(x^2)=((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2) =(((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2))-0=((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2)-0=((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2) :. f'(x) = ((3x)/((1+x)^3)+x(1-1/((1+x)^3))-(1-1/((1+x)^3))(1+x))/(x^2)x_0 = 0.11^(st) iteration :f(x_0)=f(0.1)=((1-1/((1+0.1)^3))/0.1)(1+0.1)-2.74=-0.004463f'(x_0)=f'(0.1)=((3*0.1)/((1+0.1)^3)+0.1(1-1/((1+0.1)^3))-(1-1/((1+0.1)^3))(1+0.1))/(0.1^2)=-2.329076x_1 = x_0 - f(x_0)/(f'(x_0))x_1=0.1 - (-0.004463)/(-2.329076)x_1=0.0980842^(nd) iteration :f(x_1)=f(0.098084)=((1-1/((1+0.098084)^3))/0.098084)(1+0.098084)-2.74=0.00001f'(x_1)=f'(0.098084)=((3*0.098084)/((1+0.098084)^3)+0.098084(1-1/((1+0.098084)^3))-(1-1/((1+0.098084)^3))(1+0.098084))/(0.098084^2)=-2.339843x_2 = x_1 - f(x_1)/(f'(x_1))x_2=0.098084 - (0.00001)/(-2.339843)x_2=0.098088Approximate root of the equation ((1-1/((1+x)^3))/x)(1+x)-2.74=0 using Newton Raphson method is 0.098088 (After 2 iterations) n | x_0 | f(x_0) | f'(x_0) | x_1 | Update | 1 | 0.1 | -0.004463 | -2.329076 | 0.098084 | x_0 = x_1 | 2 | 0.098084 | 0.00001 | -2.339843 | 0.098088 | x_0 = x_1 |
:.i=0.098088
:.i=9.81 % per year
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