Question

Investment X for 100, 000 is invested at a nominal rate of interest, j, convertible semiannually. After four years, it accumulates to 214, 358.88.
Investment Y for 100, 000 is invested at a nominal rate of discount, k, convertible quarterly. After two years, it accumulates to 232, 305.73.
Investment Z for 100, 000 is invested at an annual effective rate of interest equal to j in year one and an annual effective effective rate of discount equal to k in year two. Calculate the value of investment Z at the end of two years.
I know the answer is 200,000 but Im not sure how to get it

Answer 1

X:
Future Value=Present Value*(1+r/m)^(m*n)=100000*(1+j/2)^(2*4)

=>100000*(1+j/2)^(2*4)=214358.88
=>j=((214358.88/100000)^(1/8)-1)*2=20.0000%

Y:
Future Value=Present Value*(1+r/m)^(m*n)=100000*(1+r/4)^(4*2)

=>100000*(1+r/4)^(4*2)=232305.73
=>r=((232305.73/100000)^(1/8)-1)*4
=>r=44.4444%

where r is nominal rate convertible quarterly

annual effective interest rate i=(1+44.4444%/4)^4-1=52.4157%

annual effective discount rate=i/(1+i)=52.4157%/(1+52.4157%)=34.3900%

discount rate convertible quarterly k=(1-(1-d)^(1/4))*4=(1-(1-34.3900%)^(1/4))*4=40.0000%

Z:
Future Value=100000*(1+20.0000%)*(1+40%/(1-40%))=200000