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Now, I wanted to find a discount bond.

Â Bonds typically pay coupons.

Â This is an old word but they still say coupons.

Â It used to be, that if you invested in

Â a corporate bond or

Â a government bond it would be on a piece of paper they would give you.

Â This is like for hundreds of years.

Â And around the exterior of the pieces of paper,

Â were little coupons that you would clip every six months typically.

Â And you would clip your coupons every six months and take them to a bank,

Â and the bank would then give you the money.

Â So, each coupon would be so much money.

Â And then, at the end of the maturity of the bond,

Â you could take the whole thing back and you get your amount back.

Â So a typical bond back then,

Â in 1900, and 1800,

Â and 1700. Going way back.

Â If you bought a 100-dollar bond,

Â and it was issued for $100,

Â and had say as a three-dollar coupon.

Â Then, you would pay $100.

Â You'd wait six months,

Â you'd clip a coupon,

Â and it would say, pay to the bearer one dollar and fifty cents.

Â You'd go to the bank and get your one dollar and fifty cents.

Â And you can see some of these bonds,

Â they're framed and on display.

Â Ones that defaulted, otherwise,

Â the coupons would be already clipped and gone.

Â You can see them, I think they're on displays on the fourth floor of this building.

Â A discount bond is a bond that carries no coupon.

Â Now, why would you buy a bond that carries no coupon?

Â How do you get interest from it?

Â This is also time in memorial.

Â People have traded discount bonds for a long time,

Â and the answer is,

Â because you buy it for less than $100.

Â You buy it at a discount.

Â So, they tend to be two different kinds of bonds,

Â the coupon bonds which are more common,

Â tend to be sold at par.

Â You buy it initially for $100,

Â and you sell it for a 100,

Â you get back at the end when it matures after so many predefined years.

Â With a discount bond,

Â there are no coupons but of course,

Â you buy it at a discount.

Â There would be no other reason to buy it.

Â Maybe today they're selling not at a discount with a negative interest rates.

Â But normally, they're sold at a discount.

Â We still call them discount bonds even there's

Â a negative interest rate than they're selling for more than $100 initially.

Â So, if we look at the price of the discount bond,

Â we can infer the yield to maturity from that bond.

Â So, if someone says, I have a bond that will pay let's say,

Â one dollar in T years,

Â and it's compounding once a year,

Â and the price I want is P,

Â I can compute using this formula.

Â What the yield to maturity is.

Â So, I would basically take one over P,

Â if solving this equation,

Â I take one over P to the one over T power,

Â and that's the yield to maturity.

Â In a sense paying an interest rate of r, every year.

Â Compounding once a year if I call it that.

Â But typically, bonds pay interest rate every six months, that's the tradition.

Â So, you might use this formula instead which has the bond compounding twice a year.

Â If T is a number of years to maturity,

Â and P is the price,

Â then we will take P as

Â the present value of the principle which I have as one dollar here,

Â divided by one plus or over two,

Â to the two T. So,

Â the price today of the bond is called, the present value.

Â If it's a one dollar principle of one dollar at time capital T. And as a general rule,

Â P is going to be less than one.

Â I say is a general rule because it might not hold right now which is a little puzzling.

Â But over most of history, it's a discount.

Â P is less than one.

Â So, is that clear?

Â Any questions about that?

Â You have to specify the compounding interval.

Â But normally, for pedagogical purposes,

Â it's convenient to take the compounding as once a year.

Â And we just use this formula.

Â By the way, you could do it continuously till you could say,

Â what does that continuously compounded yield to maturity.

Â And that we have P equals e to the minus r times capital T. Okay.

Â Now, we can define the present discounted value of any cash flow,

Â not just a coupon flow which would be the case for a coupon bond,

Â or the principal after T years for discount bond.

Â Because we know that implicit in market prices for discount bonds,

Â we can calculate the present discounted value of a dollar in any number of years.

Â So, what is a dollar one year today?

Â What is the present discounted value of that?

Â It's one over one plus r,

Â where r is the yield to maturity on a one year discount bond.

Â And what is the present discounted value of a dollar in n years?

Â It's one over one plus r to the nth power.

Â Now, this is obvious to a banker who always thinks.

Â When you talk money with a banker,

Â and you're talking about money in future years,

Â there's a little calculator going in his head,

Â he has memorized the prices of all these discount bonds going out.

Â And he's translating it into present value or present discounted value, PDV.

Â Amateur's mess this up.

Â Instead, they become vulnerable to fishes.

Â Lots of people make mistakes.

Â Maybe you should develop the habit of always computing the present discounted value.

Â On the other hand, you live in a very good time for ignoring this.

Â Because right now, interest rates are virtually zero.

Â But it will come back.

Â I think Nick you are right,

Â we are going to have two percent and higher interest rates at some date in the future.

Â I just don't know when.

Â If you have a cash flow,

Â x sub t, where x sub one is the money coming in in one year.

Â x sub two is the money coming in in two years.

Â And let's assume that the discount rate is the same for all these different maturity.

Â This is a simplification.

Â Then the present discounted value of

Â the cash flow is the summation t equals one of the capital T,

Â of the cash flow x sub t,

Â divided by one plus r over the t. That's one of the most famous formulas in finance.

Â So, now let's look at a conventional coupon varying bond which is issued at par.

Â How do they issue them at par by the way?

Â It's tradition to issue at par

Â because you're getting your interest in the form of coupons.

Â What they have to do is judge the market.

Â If I want to issue with par,

Â what coupon is the market demanding on a $100?

Â And once I know that,

Â I'll just pick that coupon,

Â and I can be pretty sure that my bond will be picked up

Â for $100 because I've got the market coupon.

Â So, I'm going to use c for the amount of coupon.

Â Now, this is measured in currency,

Â if we're dealing in dollars,

Â c is so many dollars and the prices in so many dollars.

Â And what it is now?

Â Now, here I have two versions.

Â This is compounded annually,

Â and this is the more realistic compound in every six months where t measures years.

Â So, what you get in a simple case when you buy a coupon bond,

Â is you get after one year,

Â this is the annual compound after one year,

Â I can clip a coupon for c dollars.

Â And then, I have to wait another year,

Â and then I can clip a coupon for c dollars again,

Â and clip another coupon in three years for c dollars.

Â And then at the end, I get my last coupon of c dollars,

Â plus the principal which is a 100.

Â So, I get 100 plus c dollars at the end.

Â What is the present value of that if it's discounted at rate r?

Â It turns out that's the formula for the present value.

Â Now it's interesting to take the limit of this as t goes to infinity.

Â If t goes to infinity,

Â this term goes to zero, right?

Â And this term goes to zero.

Â So, we're left with c times one over r. So, that's the console.

Â I think I have another slide for that.

Â This is the more complicated formula for six-month compounding.

Â This formula was sufficiently difficult that in the old days,

Â and people didn't have calculators.

Â Bankers would carry around the table, bond yield table.

Â Also you can't solve this back.

Â If I'm told the price of a bond,

Â and I want to compute the yield to maturity r,

Â I've got to solve this equation for r. And you can't.

Â It's not algebraically possible.

Â Unless, t is very small.

Â So, you need a book.

Â But now, it's probably already on your mobile phone.

Â I think it is. I know it is.

Â Go to Wolfram Alpha on your mobile phone and you'll get,

Â or there must be other places.

Â Must be hundreds of places that will solve this equation for you because it's standard.

Â So many people think in terms of present values.

Â And they want to know what the yield to maturity.

Â What's the interest rate on a bond given its price.

Â