ECON: Confusing math regarding bonds?

theBEAST

Here is a screenshot of a slide that the professor went over in class (PV = Present Value):


I think the reason why I don't understand what's going on in the math is because I don't understand the question at all. So the bond pays $100 each year and on the final year pays $1000. Then there is an interest rate of 7% per year (which means you would gain 7%. Am I right so far? Could someone please explain the intuition behind the math?

Edit: Since it pays a total of $1300 would that mean you would get a profit of $1300-$1078.73?

tiny-tim

hi theBEAST!

7% is not the interest the bond is giving

7% is the interest you're losing by not having the money in the bank (which would give you 7% interest)

so the bond-issuer needs to put $100/1.07 in the bank now to have enough money to pay the first $100 at the end of the first year (etc)

theBEAST

Thanks for clearing things up!

But if the bank gives 7% interest, wouldn't it be better to put the money in the bank? With this bond you get $1300 at the end but if you put it in the bank for 3 years at 7% per year you get:
1078.73*1.07^3 = $1321 which is greater than $1300

tiny-tim

ah, but then you wouldn't be able to pay your annual $100 subscriptions to Economics Weekly for the first two years, because the money would be stuck in the bank for three years!

theBEAST

One last question, I know that if you deposit money into the bank, the bank pays you interest. But if you borrow money from the bank you pay interest. Are the two interest rates always the same or do we just assume they are the same in simple macroeconomics models?

tiny-tim

he he

we assume they are!

Matterwave

Actually if you put 1000 bucks in the bank at 7% interest, you'd make 1000*(1.07)^3=1225 dollars, so you are better off with the bond. The bond must pay a higher interest rate, or else no one would buy it.

alan2

You have to be careful with your interest rates. The bond pays 10% and the risk free rate, to which everything is compared, is 7%. The calculation given is the amount that you would pay for the bond today in order to receive the rate of 7% for the life of the bond. To see this, you pay $1,078.73 today for the bond. In one year it is worth (1078.73)(1.07)=$1,154.24. They pay you $100 and the bond is now worth $1,054.24. One year later it is worth (1054.24)(1.07)=$1,128.04. They pay $100 and it is worth $1,028.04. One year later it is worth (1028.04)(1.07)=$1,100, exactly the last payment.
Secondly, you can't just deposit the money into the bank and compare the accumulated value after 3 years to the receipts from the bond. You have to compare the same cash flows, in this case removing $100 at the end of year 1 and year 2. This is, in fact, the calculation above. The two investments are exactly equal, assuming there is no risk in the bond. The reason that there is a difference in the interest rates is because the risk free rate changes with time. This is why government bonds (which are risk free) never sell at face value. If I bought a 10% bond several years ago but the risk free rate has fallen to 7%, then I hold something which is worth more than its face value as in the example that your professor gave you.