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QUESTION

The rate of interest is 8\%. What will \pounds 100 be worth in
three years' time using

\begin{description}

\item[(a)]
simple interest

\item[(b)]
annual compound interest.

\end{description}
You may assume that the UK will not join EMU in that time.


ANSWER

\begin{description}

\item[(a)]
Simple interest pays only on the original amount $M_0$. Thus if
the rate is $r$,

$$M_1=M_0+M_0r;\ M_2=M_1+M_0r;\ M_3=M_2+M_0r$$

or after $T$ periods of interest

$$M_T=M_{T-1}+rM_0\Rightarrow M_T=M_0(1+rT)$$

Thus if $r=0.08 (8\%),\ T=3,\ M_0=100$

$$M_3=100\times(1+0.08\times3)=\pounds124$$

\item[(b)]
Compound interest pays on the original investment $M_0$ plus any
interest so far. Thus if the quoted annual rate is $r$,

$$M_1=M_0(1+r);\ M_2=M_1(1+r);\ldots;M_n=M_{n-1}(1+r)\Rightarrow
M_t=M_0(1+r)^T$$

Clearly compound interest pays more than simple interest, since

$$\underbrace{(1+r)^T}_{\textrm{compound}}
=\underbrace{1+rT}_{\textrm{simple}}+\underbrace{\frac{T(T-1)T^2}{2}}_{\textrm{other
extra stuff}}+\ldots$$

Therefore

$$M_3=100(1+0.08)^3=\pounds125.97\ (>\pounds124)$$

\end{description}



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