Question

Jeff deposits $100 into a fund today and $200 into the same fund 15 years later. Interest for the first 10 years is credited at a nominal discount rate of d compounded quarterly, and thereafter at a nominal interest rate of 6% compounded semiannually. The accumulated value in the fund at the end of 30 years is $1000. Calculate d.

Answer 1

Value of “d” is 4.5837% or say 0.0458

Detailed Solution

In this question, since we have been given the accumulate value at the end of year 30, and we have to calculate the nominal interest rate for first 10 years, we need to first calculate the amount accumulated at the end of year 15 and thereafter the amount accumulated at the end of year 10 and then nominal interest rate.

Accumulated amount with given interest rate

Computation of Amount at year 15 from Accumulated amount at the end of Year 30

A=P*(1+r)t

Where A = Accumulated Amount at the end of the period (i.e. end of Year 30) = $ 1000

P = Principal Amount at the beginning of the period (i.e. end of Year 15)

r = Nominal Interest Rate for each compounding period = since the rate is compounded semi annually, the compounding period is 6 months and therefore 6%*6/12= 3% i.e. 0.03

t = No. of Compounding periods (since the rate is compounded semi annually, the number of compounding will be 2 in a year) =

Year 30 – Year 15 = 15 years*2 compounding per year = 30

Therefore

A= P*(1+r)t

Or P= 1000/(1+0.03)30

Or P = 1000/(1.03)30

Or P = 1000/2.42726

Or P= $ 411.987

The Accumulated amount at the end of year 15 is $ 411.987. Now, similar to the above calculation, we have to calculate the amount accumulated at the end of year 10.

Computation of Amount at year end 10 from Accumulated amount at the end of Year 15

A=P*(1+r)t

Where A = Accumulated Amount at the end of the period (i.e. end of Year 15) = $ $ 411.987-$200 (Since Jeff deposited $ 200 at year 15 end, $ 200 will be reduced from the amount of $ $ 411.987) =   $ 211.987

P = Principal Amount at the beginning of the period (i.e. end of Year 10)

r = Nominal Interest Rate for each compounding period = since the rate is compounded semi annually, the compounding period is 6 months and therefore 6%*6/12= 3% i.e. 0.03

t = No. of Compounding periods (since the rate is compounded semi annually, the number of compounding will be 2 in a year) =

Year 15 – Year 10 = 5 years*2 compounding per year = 10

Therefore

A= P*(1+r)t

Or P= 211.987/(1+0.03)10

Or P = 211.987/(1.03)10

Or P = 211.987/1.34392

Or P= $ 157.738

The Accumulated amount at the end of year 10 is $ 157.738. Now, given the Initial deposit of $ 100 at today (year 0), we have to calculate the interest rate for first 10 years.

Computation of interest rate from Accumulated amount at the end of Year 10 and Initial investment

A=P*(1+r)t

Or A/P = (1+r)t

Or (A/P)1/t = 1+r

Or r = (A/P)1/t-1

Where A = Accumulated Amount at the end of the period (i.e. end of Year 10) = $ 157.738

P = Principal Amount at the beginning of the period (i.e. Year 0) = $ 100

r = Nominal Interest Rate for each compounding period = since the rate is compounded quarterly, the compounding period is 3 months (We have to compute the rate)

t = No. of Compounding periods (since the rate is compounded quarterly, the number of compounding will be 4 in a year) =

Year 10 – Year 0 = 10 years*4 compounding per year = 40

r = (A/P)1/t-1

Or   r = (157.738/100)1/40-1

Or   r = (1.57738)0.025-1

Or r= 1.011459-1

Or r = 0.011459 or say 1.1459%

Since this rate is quarterly compounded, the nominal rate of interest will be r*4 i.e. 1.1459%*4 = 4.5837%