Interest involving different years, but same or lower 2nd value?

Question

When selling your property you receive the following offers :

(a) € 100 000 on 1.1.18 and € 100 000 on 1.1.20

(b) € 150 000 on 1.1.18 and € 50 000 on 1.1.19

Assume interest is 3% Which offer is better?

---

Now this is how I believe this should be done. Is my understanding correct or am i completely off?

a) FV= 100 000(1+0.03*2)=106 000+ 100 000=206 000

b) FV= 150 000(1+0.03*1)=154 000+ 50 000= 204 500

Answer 1

I think you should not calculate the end value of both payments, as it implies comparing (using your numbers) $\$206 000$ at 1.1.2020 to $\$204 500$ at 1.1.2019. It's better to compare the net present value of both proposals, which are

$$NPV_1=100 000+\frac{100 000}{(1+0.03)^2}=194259.59$$

and

$$NPV_2=150 000+\frac{50 000}{(1+0.03)}=198543.69$$

As you can see, it's better to choose the second option. This is not surprising, as the total $\$200000$ is received earlier than in the first.

Edit: The net present value is just the value today. That's why we discount future payments, i.e., we divide by $(1+r)^t$. The future value is the value at some point in the future. And this is the reason for capitalization, i.e., multiplying by $(1+r)^t$. In the first case, we want to determine how much is worth today a given payment schedule, while in the second we obtain the value at some point in the future. Both the future value and the present value should give you the same answer provided the future values are obtained for the same (future) date.

For example, if you computed the future value of the schedules in your problem at 1.1.2020, you'd obtain

$$ FV_1=100000(1+0.03)^2+100000=206090$$ $$ FV_2=150000(1+0.03)^2+50000(1+0.03)=210635.$$

This is due to the fact that the future value is just the net present value capitalized. In your case, it is easy to see that

$$ FV_i=NPV_i(1+0.03)^2.$$

Answer 2

Not quite. The second offer should look better because you receive money earlier

Your error is to calculate the values at the point the last payments are received, but these are different times, making them incomparable. You should choose a particular date for the comparison

• If you are measuring the value at 1.1.20 then your calculations should be $$€100000(1+0.03)^2+€100000 = €206090$$ $$€150000(1+0.03)^2+€50000(1+0.03) = €210635$$

• If you are measuring the value at 1.1.19 then your calculations should be $$€100000(1+0.03)+€100000/(1+0.03) \approx €200087$$ $$€150000(1+0.03)+€50000 = €204500$$

• If you are measuring the value at 1.1.18 then your calculations should be $$€100000+€100000/(1+0.03)^2 \approx €194260$$ $$€150000+€50000/(1+0.03) \approx €198544 $$