at the he end of each eight K Tund paying interest at an annual effecnive Intere

Question

at the he end of each eight K Tund paying interest at an annual effecnive Interest is paid out at the end c nest at an onth for eight of each year to a fund with an (eye "of 7.2' of 4%. Both funds are liquidated at each year to anual effective itof Melanie's liquidation from the two fnd of ten years o funds. an d. Darlene deposits $1.000 at the beginning of each interest at a nominal interest rate of i(12)-6% und is paid out monthly and is reinvested at a quarterly liquidates her money at the end of twelve years, 2 a ten-year period, Dute into a fund earning ring interest fron r her last deposit of $1,000. Find an ex years after yers of Darlern the accumulate rate j. [HINT: Begin by making a any an t having the quarterly interest rate of annuity symbols used are at value of Darlene's investme aINT: Begin by making a timeline showing the deposits i inter- coun n interest periods has a payment at the end of each m-th ot unt Q more than their predecessor. Show that this annuity has mulated value rn Ps ) + rn2 r)GH) _ n) at the end of the n-th interest 8) An annuity lasting interest od. The first payment is for an amount P and further payments an interest period period ty lasting n interest periods has a payment at the end of each m-th of riod. The first payment is for an amount P. Payments are level an interest within each interest period, and the the individual payments increases by an

Explanation / Answer

Soln : As given here P = first payment and Q is the increment every subsequent payment in the predecessor

Now, we can write the annuity value upto nth period, A = P/(1+r)m + (P+Q)/(1+r)m+1 + (P+2Q)/(1+r)m+2 +.......+ (P+nQ)/(1+r)m+n

If you will see here there are 2 GP series:

A1 = P/(1+r)m  +.......(P)/(1+r)m+n , A1 = P*(1-(1/1+r)n+m)/r............(I)

A2 = Q*(1/(1+r)m+1 + 2/(1+r)m+2 +..........n/(1+r)m+n )..........................(II)

Divide eqn (II) from (1+r) and subtract it from eqn II

We will get (1-1/(1+r))*A2 = Q*(1/(1+r)m+1 + 1/(1+r)m+2 +..........1/(1+r)m+n ) + Q*n/( (1+r)m+n+1

A2 = (Q*(1+r)n+1 +nrQ-Q) /r2(1+r)m+n

A = A1 + A2 = P*(1-(1/1+r)n+m)/r + Q*((1+r)n+1 +nr -1)/r2(1+r)m+n

here r = i , interest rate

A = i*P*(1-(1/1+i)n+m) + Q*((1+i)n+1 +ni -1)/i2(1+i)m+n

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