We will use the TVM function on a financial calculator to get the answer of question e by inserting the following parameters.
N(No. of periods) = 4 (given in question)
I/Y(Rate of interest) = ?
PV = 0 (No initial deposit is made)
PMT(Reccuring deposits made) = -200 (Given in question, The negative sign is for an outflow).
FV(Future Value) = 1000 (Amount needed on 1 January 2016).
And then we will press the CPT that is compute button and get the annually compounded interest rate which will be equals to 15.091108.

Although the method shown above is prefered
we can also use the calculation method written below.

Calculation method takes a hit and trial method to solve.

First we will assume the rate to be 13% and then solve for rate using the below equation.

Let the rate of interest required be 'r'.

-200*(1+0.13) + {-200*(1+0.13)^2} + {-200*(1+0.13)^3} + {-200*(1+00.13)^4}

-200*(1.13) + {-200*(1.13)^2} + {-200*(1.13)^3} + {-200*(1.13)^4}

= $969.9594

Now as the value we get using the rate of 13% is less than the amount of $1000 required at the emd of 4 years therefore now we will try the same calculation using a rate of 15%.
-200*(1+0.15) + {-200*(1+0.15)^2} + {-200*(1+0.15)^3} + {-200*(1+00.15)^4}

-200*(1.15) + {-200*(1.15)^2} + {-200*(1.15)^3} + {-200*(1.15)^4}

=$998.675
Now we get more close to the $1000 amount required at the end of 4 years.
Now we will try using a rate of 15.1%

-200*(1+0.151) + {-200*(1+0.151)^2} + {-200*(1+0.151)^3} + {-200*(1+00.151)^4}

-200*(1.151) + {-200*(1.151)^2} + {-200*(1.151)^3} + {-200*(1.151)^4}

=$1000.12939
Which is more than $1000 amount required therefore the rate would be somewhere between 15% to 15.1%.
Which is calculated to be 15.09%.

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