[MUSIC] Welcome, everyone. Now that you've attempted the quiz on bonds, I'm going to walk you through to what the correct answers are and how to arrive at them. So let's start with Question 1. And think through each of these answers to the question of what motivates negative interest rates. Choice a is the least acceptable answer because negative interest rates actually make money cheaper. So banks lend more and consumers borrow more which can help to kickstart a sluggish economy. Choice b is counter intuitive, since the countries that adapt negative interest rates, were trying to weaken their currencies, vis-a-vis the Euro to make their exports more competitive, and their imports more expensive. Choice c is the best answer because negative interest rates signal to commercial banks that they will pay a penalty to keep money with the central banks, which is what hoarding refers to. And this should lead them to lend more money out to businesses, again, to boost the economy. As mentioned in the video, negative interest rates are trying to do the opposite of choice d, to avoid deflation when prices fall, since this will just reduce national income and cause an economic contraction. So the correct answer is c, negative interest rates discourage banks from hoarding money. Let's look at Question 2 now, which asks what a typical corporate bond's coupon rate is quoted as. Choice a and b speak to effective annual rates or EARs. EARs convert the annual percentage rate, that is, the APR, the rate which is advertised in the bond issue into a rate that takes into account the effect of compounding. This conversion is actually not done when the coupon rate is quoted. The correct choice is d, and not d, since the coupon rate simply advertise the APR, or the stated rate, typically on semiannual basis. So if the coupon rate, let's say is 6% compounded semiannually, this means that keeping in mind the face value of a bond is $1,000. The semiannual rate of 6% divided by 2, which is 3%, will be $30 that will be paid out every 6 months. Okay, let's look at Question 3 which asks whether the following statements are true or false. Statement a is false because, although it defines the current yield correctly, the yield to maturity is not the same thing. The yield to maturity is a market-determined rate that takes into account both the coupon, the current price, as well as the maturity of the bond which includes the time value element. Statement b is also incorrect because discount bonds sell below $100. Statement c however is true because premium bonds sell above $100. Now it's important to note that the yield to maturity on discount bonds exceeds the coupon rates, while the yield to maturity on premium bonds is lower than the coupon rate. Choice d is false because the call feature allows the issuer to call the bonds back and reissue another bond at a different rate, plus a penalty called the call premium, and this is not always true for all bonds. Okay, let's look at Question 4. Now this is an easy one to explain because all of these statements are correct, and were explained in the video. And the choice of course is d, which is all of the above. Question number 5, this requires us to pick out the relevant information in the problem and then maybe depict it on a timeline, and then we can work through the calculations. So let's isolate the important information. First of all, we have the face value also known as the par value of the bond with is $1,000. We also have the coupon rate, and the coupon rate is stated to be 10%. And 10% of course of the par value which gives us $100. The maturity of the bond, which is how long it is outstanding, is given to be 8 years, whereas the yield to maturity is stated as 12%. Notice the difference between the coupon rate and the yield to maturity. And very important to note that there is a frequency of compounding, which as you remember from previous videos, is denoted by m. In this case it is semi annual, so it is twice a year. Now right away when we see the frequency is greater than one, we should make adjustments to the coupon rate, to the majority and to the yield. What will happen is this is paid annually, so if we're paying this semi-annually, we must divide this by 2, which will give us $50 every 6 months. Where as the maturity will now be 2 periods per year, we have 8 years, so we multiply this by 2. This gives us 16 periods and, of course, the yield again stated annually must be divided by 2 which gives us a semi-annual rate of 6%. So, why don't we put this up on a timeline? And that we'll help us to visualize what exactly we're trying to do. So what we can see on the timeline is we start at some point, let's say time 0. And we have 16 periods to depict on this timeline. So we can do that. All the way to period 16. And then we want to put these cash flows on this timeline so we understand the valuation process. The cash flows as you can see here, include, first off all we have the $50 coupon payments every 6 months, so we can put that over here. The $50 that we can expect to receive form the issuer all the way to the last period. And then we also get back the phase value which is 1000, again in the last period. And now our task is to simply discount these values back at the very important market rate, known as the yield to maturity. So here we can state now the equations to value the bond. This is what we were after in the first place. The bond price today, so this is the value we're after, right now, today, is going to equal to the present value of these coupon payments. So we can just write that. Present value of the coupon payments. Plus, we have to add to this the present value of the face amount of the loan which we can call the principle, present value of the principle amount. So the equation as you can see is quite simple. All we have to do is plug some numbers in and off we go to the races. So what are the numbers? Well, we have here an annuity which we've already learned is a series of equal numbers made in equal time intervals. We have a discount rate, we have a time period, so this is easy. So the market value is going to be the present value of this annuity, which is $50 multiplied by the present value annuity factor, which will be based on 6% interest for, as you can see, 16 remaining periods. And to this, we will add the second half, which is a lump sum, 1 number, we have to zap back, which is simply the present value of 1000 multiplied by the present value factor 4 at 6% for 16 periods. So, if you crunch the numbers, what do we get? We get a value equal to $899, and $899 corresponds to the correct answer. All right, let's look at Question number 6. Now given the information that these bonds are identical, and we need the yield to maturity as we just saw to calculate the price. What we can do is if first find the yield to maturity for bond A. Now if we set the information up just like we did, where we are given the bond price and that bond price is $885.30. That's the price which we know is equal to the present value of the coupon. We know the coupon is 50, and so the present value annuity factor for again, this time we have 20 periods, but we don't know the yield. So that is the missing variable that we're looking for. Plus, we're going to have the $1,000 principal returned to us. Again, present value 20 periods at the yield to maturity which is what we're looking for. So we have an equation with one unknown. Now, mind you, the mathematics of this is a little bit tedious, because the formula that we use actually to compute these values. If I isolate that formula for you, which is this annuity formula, the annuity formula is 1- 1 over 1 + r raised to the power t over r. That's this annuity formula. Whereas the lump sum formula is easier, which is simply 1 over 1 + r raised to the power t. So we don't want to get into the mathematics, and we just use our trusted calculator which gives us the value. Keep in mind that again we're looking for a semi-annual yield because that's how the problem was set up. And if you solve for that semi-annual yield, what you are going to get, in this example. The answer is going to be the yield to maturity, on a semi annual basis, is equal to 6%. Now, that's what we were looking for to be able to solve the bond price for the next bond b. And so all we have to do now is plug the numbers in for this kind of equation. And this is going to be fairy straightforward. So what do we know about Bond B? We know the coupon instead of 50, for Bond B it is $70. And again, we'll multiply this now with the present value annuity factor at 6%, which we've figured out now, for the 20 periods that it's going to be outstanding for. Plus we're going to add again the face value component, the $1000, for a lump sum factor, present value 6%, 20 periods, okay? These, of course, can be calculated here, but If you have a financial calculator, you're just laughing because you're punching in these values. And you've seen that in the financial calculator tutorial, you work this out and you get your answer which is $1,114.70. Notice that this bond is selling at a premium Okay, whereas we had the situation of the first bond, which was selling less than 1,000, which is selling at a discount.
