Present values with denser discounting

Question

With an annual interest rate of 10%, the present value of 100 dollars received one year from now is $PV = \frac{100}{1.1}=90.91$.

If instead the 100 dollars is received in two installments of 50 dollars and the first payment is made 6 months from now, the present value is $PV = \frac{50}{1.05}+\frac{50}{1.05^2}=92.97$.

So the denser compounding increases the present value. However, if you will receive 100 dollars each year till eternity, the present value will be $PV = \sum_{t=1}^\infty \frac{100}{1.1^t}=1000$. If you instead receive 50 dollars every half year, the present value will be $PV = \sum_{t=1}^\infty \frac{50}{1.05^t}=1000$.

What's going on here? Why isn't the latter PV higher? To add to the confusion, I have a finance textbook here saying that the latter should be higher because you receive the first 50 already after 6 months, and after a year you will have accumulated 102.5, whereas in the first case you only have 100 after the first year.

Answer 1

the half yearly rate is not 0.05. It is

$$(1+r)^2 = 1.1$$

$$r = 0.048809$$

Now compute it using the formula $\frac{CF}{r}$ for a stream to perpetuity

In the first case it is $\frac{100}{0.1} = 1000$. In the second case it is $\frac{50}{0.048809} = 1024.04$. Yes, the PV is higher indeed.

Answer 2

The six month calculation assumes compounding at six month intervals. The fair way to compare it with the year calculation would be to divide by $\sqrt {1.1}$ for the payment at six months, so the present value would be $\frac {50}{\sqrt {1.1}} + \frac {50}{1.1}\approx 93.13$ This reflects the fact that the proper interest for six months to equal $10\%$ per year is not $5\%$ but $\sqrt {1.1}-1 \approx 4.88\%$

For the permanent annuity, the six month value should be higher as you get some of the money earlier. It will be $WPV = \sum_{t=1}^\infty \frac{50}{\sqrt{1.1}^t}\approx 1024.4$