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I am learning for a class that is partly about finance and I don't have any background in finance at all.

I am struggling with a question that was asked in last years exam:

A bond trades at £1015, has a duration of 5 and yields 4.69%.
If yields increase to 4.87%, what will the new price be?

There is no other information. (The only type of yield we had is Yield to Maturity, if that is relevant.)

I am currently trying out some variations (moving terms around ...) of the formula for the present value of money, but I can't come up with anything that behaves in a sensible way.

I also tried to read this article and followed some of the links, but I am still lost. :(

  • Is it current yield or yield to maturity? – JoeTaxpayer Apr 6 '16 at 2:18
  • What is the coupon rate? Is it a Zero coupon bond? – RajSharma Apr 6 '16 at 14:42
  • There is a formula to calculate the theoretical price of the bond i,e Price = PV of Coupons + PV of redemption payment where PV= present value. – RajSharma Apr 6 '16 at 14:43
  • I'd strongly recommend checking how duration is defined in your class notes (as it must have been if it's being used in exams). Note that duration is not simply the time to maturity. – atkins Apr 8 '16 at 14:17
  • Also, this might be a better fit for quant.stackexchange.com. – atkins Apr 8 '16 at 15:06
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I am currently trying out some variations (moving terms around ...) of the formula for the present value of money

The relationship between yield and price is much simpler than that.

If you pay £1015 for a bond and its current yield is 4.69%, that means you will receive in income each year:

4.69% * £1015 = £47.60

The income from the bond is defined by its coupon rate and its face value, not the market value. So that bond will continue to pay £47.60 each year, regardless of the market price. The market price will go up or down according to the market as a whole, and the credit rating of the issuer.

If the issuer is likely to default, the market price goes down and the yield goes up. If similar companies start offering bonds with higher yields, the market price goes down to make the bond competitive in the market, again raising yield.

So if the yield goes up to 4.87%, what is the price such that 4.87% of that price is £47.60?

£47.60 / 4.87% = £977.48


Another way to think of it: if the yield goes up from 4.69% to 4.87%, then yield has increased by a factor of:

4.87% / 4.69% = 1.0384

Consequently, market price must decrease by the same factor:

£1015 / 1.0384 = £977.48

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    What irritates me in this answer is that the duration of the bond is not taken into account – icehawk Apr 6 '16 at 10:05
  • @icehawk So you are looking for yield to maturity, not current yield? In that case, I think you'd need more information, at least one of the bond's par value, or its coupon rate. See investopedia.com/terms/y/yieldtomaturity.asp for some example calculations. The formula is non-trivial and they solve it by iteratively guessing. – Phil Frost Apr 6 '16 at 11:41
  • I am actually not sure. As said, its a question from a past exam and there is no additional information given. I don't think they would ask a non-solvable question. Also that investopedia looks much more complicated than could be justified for the 3 marks assigned to the question. So I would think I am looking for current yield, yes, but if so, why would they provide the duration? – icehawk Apr 6 '16 at 13:04
  • @icehawk I'm as confused as you are. Maybe it was meant to be about current yield, and the duration is intentionally irrelevant information thrown in there to see if you are paying attention. Or maybe it was meant to be about yield to maturity, and whoever wrote the question wasn't thinking hard enough about how hard it would be to answer, or if it's even possible with the information given. I know in my education I encountered more than once questions which were "accidentally" unanswerable, or turned out to be more difficult than intended, as a result of teacher error. – Phil Frost Apr 6 '16 at 13:29
  • Well I guess current yield is the best possible solution then. :/ – icehawk Apr 6 '16 at 15:36
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The duration of a bond tells you the sensitivity of its price to its yield. There are various ways of defining it (see here for example), and it would have been preferable to have a more precise statement of the type of duration we should assume in answering this question.

However, my best guess (given that the duration is stated without units) is that this is a modified duration. This is defined as the percentage decrease in the bond price for a 1% increase in the yield. So,

change in price = -price x duration (as %) x change in yield (in %)

For your duration of 5, this means that the bond price decreases by a relative 5% for every 1% absolute increase in its yield. Using the actual yield change in your question, 0.18%, we find:

change in price = -1015 x 5% x (4.87 - 4.69) = -9.135

So the new price will be 1015 - 9.135 = £1005.865

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Edited to incorporate the comments elsewhere of @Atkins

Assuming, (apparently incorrectly) that duration is time to maturity:

First, note that the question does not mention the coupon rate, the size of the regular payments that the bond holder will get each year. So let's calculate that.

Consider the cash flow described. You pay out 1015 at the start of Year #1, to buy the bond. At the end of Years #1 to #5, you receive a coupon payment of X. Also at the end of Year #5, you receive the face value of the bond, 1000. And you are told that the pay out equals the money received, using a time value of money of 4.69%

So, if we use the date of maturity of the bond as our valuation date, we have the equation:

Maturity + Future Value of coupons = Future value of Bond Purchase price

1000 + X *( (1 + .0469)^5-1)/0.0469 = 1015 * 1.0469^5

Solving this for X, we obtain 50.33; the coupon rate is 5.033%. You will receive 50.33 at the end of each of the five years.

Now, we can take this fixed schedule of payments, and apply the new yield rate to the same formula above; only now, the unknown is the price paid for the bond, Y.

1000 + 50.33 * ((1 + 0.0487)^5 - 1) / .0487 = Y * 1.0487^5

Solving this equation for Y, we obtain: Y = 1007.08

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    How do you know the face value? Its not given in the question and I don't think there is a legal reason that bonds need to come in 1000 Pound increments? – icehawk Apr 6 '16 at 9:15
  • Bond prices and yields in the secondary market are usually quoted for a fixed amount. Actual purchases can be for any amount desired from a pool of the specific bond... And you must make some assumptions to solve this problem; there simply isn't enough information explicitly stated... – DJohnM Apr 6 '16 at 18:32
  • @DJohnM No, there is enough information in the question without making assumptions about face value or time to maturity - the duration encapsulates all of that information. – atkins Apr 8 '16 at 14:11
  • @atkins What about the coupon rate, then? – DJohnM Apr 8 '16 at 15:28
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    @DJohnM You don't need to know it to answer the question: the duration is by definition the sensitivity of the price to the yield. To calculate the duration you need to know all these details (time to maturity, coupon rate, face value, price, etc.), but if we are given the duration, the details become irrelevant. The purpose of the duration is to allow us to quickly calculate hedge ratios: if all yields move up by 1% how much of bond A would hedge the price move of bond B? The ratio of their durations tells us instantly. – atkins Apr 8 '16 at 15:35

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