Question

Suppose that the interest rate (r) is such that the present value of receiving $V2 in t2 years from

now is the same as the present value of receiving $V1 in t1 years from now, t2 > t1. Assume
that interest is compounded annually.
a. Show that V2 > V

Answer 1

a. It is shown that
`\(V2\)` is greater than
`\(V1\)`.

b. The present value of receiving
`\(V₂\)` at
`\((2+k)\)` years from now is equal to the present value of receiving
`\(V₁\)` at
`\((t+k)\)` years from now for any value of
`\(k\)`.

How did we arrive at this assertion?

The formula for the present value (PV) of a future amount (FV) compounded annually is given by:

`\[PV = (FV)/((1 + r)^n)\]`

where:

-
`\(PV\)` is the present value,

-
`\(FV\)` is the future value,

-
`\(r\)` is the interest rate per period, and

-
`\(n\)` is the number of periods.

To show that
`\(V_2 > V_1\)`

We are given that the present value of receiving
`\(V_2\)` in 12 years is the same as the present value of receiving
`\(V_1\)` in 4 years.

`\[ (V_2)/((1 + r)^(12)) = (V_1)/((1 + r)^4) \]`

Cancel out the denominators and rearrange the equation:

`\[ V_2 = V_1 \cdot (1 + r)^(12-4) \]`

`\[ V_2 = V_1 \cdot (1 + r)^8 \]`

Since
`\( (1 + r)^8 > 1 \)` (because
`\( r > 0 \))`, it follows that
`\( V_2 > V_1 \)`.

(b) Show that the present value of receiving
`\(V_2\)`,
`\((2+k)\)` years from now, is also equal to the present value of receiving
`\(V_1\)`,
`\((t+k)\)` years from now, for any value of
`\(k\)`.

We are asked to show that:

`\[ (V_2)/((1 + r)^(2+k)) = (V_1)/((1 + r)^(t+k)) \]`

Cancel out the denominators and rearrange the equation:

`\[ V_2 = V_1 \cdot \left( ((1 + r)^(t+k))/((1 + r)^(2+k)) \right) \]`

Simplify the expression:

`\[ V_2 = V_1 \cdot (1 + r)^(t-k) \]`

This shows that the present value of receiving
`\(V_2\)`
`\((2+k)\)` years from now is equal to the present value of receiving
`\(V_1\) \((t+k)\)` years from now for any value of
`\(k\)`.

Therefore, the present value of receiving
`\(V₂\)` at
`\((2+k)\)` years from now is equal to the present value of receiving
`\(V₁\)` at
`\((t+k)\)` years from now for any value of
`\(k\)`.

Complete question:

3. Suppose that the interest rate (r) is such that the present value of receiving $V2 in 12 years from now is the same as the present value of receiving $
`V_1` in four years from now, 12>4. Assume that interest is compounded annually.

(a) Show that V₂ >
`V_1`.

(b) Show that the present value of receiving $V2, (2+k) years from now is also equal to the present value of receiving $V₁, (t+k) years from now for any value of k. (That is, it is the absolute difference between time periods that matter.)