An investment offers to pay $100 a year forever starting at the end of year 6. If the interest rate is 8%, what is the investment's value today?

Para calcular el valor presente de una inversión que paga $100 al año para siempre, comenzando al final del año 6, con una tasa de interés del 8%, debemos seguir estos pasos:

1. Calcular el valor presente de una perpetuidad que comienza en el año 6: La fórmula para el valor presente de una perpetuidad que paga $C$ cada año con una tasa de interés $r$ es: $PV = \frac{C}{r}$ En este caso, $C = 100$ y $r = 0.08$: $PV = \frac{100}{0.08} = 1250$

2. Descontar el valor presente de la perpetuidad al año 5: Dado que los pagos comienzan al final del año 6, el valor presente calculado en el paso anterior es el valor al final del año 5. Para encontrar el valor presente hoy (año 0), debemos descontar este valor 5 años hacia atrás: $PV_{\text{hoy}} = \frac{1250}{(1 + 0.08)^5}$

3. Calcular el valor presente hoy: $PV_{\text{hoy}} = \frac{1250}{(1.08)^5}$ Calculamos $(1.08)^5$: $(1.08)^5 \approx 1.4693$ Entonces: $PV_{\text{hoy}} = \frac{1250}{1.4693} \approx 850.34$

Por lo tanto, el valor de la inversión hoy es aproximadamente $850.34.

To determine the present value of an investment that pays $100 a year forever starting at the end of year 6, with an interest rate of 8%, we need to follow these steps:

Step 1: Calculate the Present Value of the Perpetuity at Year 5

A perpetuity is a series of equal payments that continue indefinitely. The formula to calculate the present value of a perpetuity is:

$PV_{\text{perpetuity}} = \frac{C}{r}$

where:

• $C$ is the annual payment ($100 in this case),
• $r$ is the interest rate (8% or 0.08 in this case).

So, the present value of the perpetuity at the end of year 5 is:

$PV_{\text{perpetuity at year 5}} = \frac{100}{0.08} = 1250$

Step 2: Discount the Present Value of the Perpetuity Back to Today

Now, we need to discount this value back to today (i.e., the present value at year 0). The formula to discount a future value is:

$PV = \frac{FV}{(1 + r)^n}$

where:

• $FV$ is the future value ($1250 in this case),
• $r$ is the interest rate (8% or 0.08 in this case),
• $n$ is the number of years (5 years in this case).

So, the present value today is:

$PV_{\text{today}} = \frac{1250}{(1 + 0.08)^5}$

Step 3: Perform the Calculation

First, calculate $(1 + 0.08)^5$:

$(1 + 0.08)^5 = 1.08^5 \approx 1.4693$

Now, divide $1250 by 1.4693:

$PV_{\text{today}} = \frac{1250}{1.4693} \approx 850.34$

Conclusion

The present value of the investment today is approximately $850.34.