What is the difference between simple interest and simple discounting?

Question

I have been given the following statements:

"Simple interest: $C$ now $\equiv (1+in)C$ in $n$ years; $C$ in $n$ years $\equiv \frac{C}{1+in}$ now.

Simple discounting: $C$ in $n$ years $\equiv (1−dn)C$ now; $C$ now $\equiv \frac{C}{1-dn}$ in $n$ years.

Where $d=\frac{i}{1+i}$"

These statements imply that simple interest and simple discounting are not equivalent, since the statements do not equate to one another. Why is this, and what is the difference between simple interest and simple discounting?

Thanks

Answer 1

I think you have the wrong formula for d.

$C_{0c}$: present value (simple compound)

$C_{0d}$: present value (simple discount)

$C_{nd}$: future value (simple discount)

$C_{nc}$: future value (simple compound)

$C_{nc}=C_{0c}\cdot (1+in)$

$C_{0d}=C_{nd}\cdot \frac{1}{1-d\cdot n}=C_{nd}\cdot \frac{1}{1-\frac{i\cdot n}{1+i\cdot n}}$

with $\boxed{d=\frac{i}{1+i\cdot n}}$

Here I have a different expression for d.

The future value of the simple compound has to be equal to the present value of the simple discount. This is your asked connection.

$C_{0c}\cdot (1+in)=C_{nd}\cdot \frac{1}{1-\frac{i\cdot n}{1+i\cdot n}}$

$C_{0c}$ and $C_{nd}$ can be cancelled, because they are equal, too.

$(1+in)= \frac{1}{1-\frac{i\cdot n}{1+i\cdot n}}$

Multiplying the equation by the denominator of the RHS.

$(1+in)\cdot \left( 1-\frac{i\cdot n}{1+i\cdot n} \right)=1$

Now you can proof, if the equation is true.