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I am studying about bonds and I am a bit puzzled. So far my understanding is that bonds are loans issued by the government or a company and it works very similarly to loans.

However, I have a question regarding a premium bond and a discount bond. I am not quite understanding who is making money and who is losing. Can someone explain to me using the following example?

Smith purchases a bond of face value $100,000$ at a rate of $10\%$ for $10$ years. The yield rate is $12\%$.

hyg17
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  • Hallo! At what price did he buy it? At 100000? This is price stated by the issuer not necessarily the current market value. Perhaps your example has that current market value is 83.33 or something similar – Jimmy R. Nov 13 '14 at 00:45
  • He gets $10000$ of interest each year and $100000$ repayment at the end of ten years. But as he buys the bond for less than $100000$ at the start (i.e. at a discount), as far as he is concerned his yield from the bond is more than the nominal $10%$ – Henry Nov 13 '14 at 00:49

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Bonds have an inverse relationship with yield. So if Smith purchases a $10\%$ bond and now the interest rate is $12\%$, the face value of the bound has gone down. Therefore, the bond is trading at a discount at present.

If at some time later, the interest rate was $8\%$, the bond would be worth more and trading at a premium. Assuming there is still some time value left on the bond.

As $t\to 10$, the bond converges to its par value regardless if it was trading at a discount or premium and the entity that issued it isn't going bankrupt.

Suppose Smith buys this bond when it was issued new. At the time of the bond inception, the interest rate was determined to be $10\%$. At the time the bond came to the market (let's say today), the interest is $12\%$.

The present value of an ordinary annuity is $$ PVOA = C\Bigl(\frac{1 - (1 + i)^{-n}}{i}\Bigr) $$ where $C$ is cash flow, $i$ is interest rate, and $n$ is the number of payments. We will assume semi-annual payments so the cash flow is $$ C = 1000*0.1*100 = 10000 $$ or $5000$ semi-annually. $$ PVOA = 5000\Bigl(\frac{1 - (1 + .06)^{-20}}{.06}\Bigr) = 57349.6 $$ The present value of the bond is $$ PV_{\text{bond}}= \frac{F_v}{(1+i)^{-n}} $$ where $F_v$ is the face value at maturity. $$ PV_{\text{bond}}= \frac{100000}{(1+.06)^{-20}} = 31180.5 $$ The bonds total value is $PVOA + PV_{\text{bond}}$. The bond is trading at the discounted rate of $\$ 88530.1$ Since we assumed semi-annually payments, the interest rate was divided by $2$ and $n$ was multipled $2$.

Is buying a premium bond ever worth it? Yes.

Consider two $10$ year bonds where one is trading at a premium and the other a discount.

Let the coupon on the premium bond be $10\%$ and the coupon on the discount bond be $4\%$.

Premium Bond:

Initial cost $\$120,000$, annual cash flow $\$10,000$, and 10 year cash flow $\$100,000$.

Premium Net Cash Flow: $100000 - (120000 - 100000) = \$80000$

Discount Bond:

Initial cost $\$90,000$, annual cash flow $\$4,000$, and 10 year cash flow $\$40,000$.

Discount Net Cash Flow: $40000 - (90000 - 100000) = \$50000$

Therefore, we can't say that buying a premium bond is a bad thing. It depends on the other bonds available to the investor at the time. A premium could be a better bet. In the toy problem above, we could either raise the interest rate of the discounted bond by another $3\%$ or increase the discounted price by $\$30,000$ before they two bonds break even or we could do a combination of change the interest rate and the discount. That is, a discounted bond can be structured to look more attractive then a premium which may conjure a negative connotation since we are paying more than par but unless the situations are analyzed we don't know what the better deal is.

dustin
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  • I'm sorry but I need a bit more explanation. Is having a discount a good thing or a bad thing for the purchaser? – hyg17 Nov 13 '14 at 02:43
  • @hyg17 it can be a good thing if the purchaser holds the bond to maturity since the bond holder collects the coupon and the par value. Therefore, the purchaser will get asset appreciation too. – dustin Nov 13 '14 at 02:47
  • So, does that mean that if the bond is premium it's a bad thing? – hyg17 Nov 13 '14 at 03:42
  • I'm so sorry but I still don't understand the situation about bonds. The more I read about it the more confusing it gets. So, I would like to ask you this question first. Why would someone want to buy a bond at premium? I want to say this because the present value of the bond is more expensive than buying it at a discount. My understanding is, the higher the yield rate, the more money you get. So if the yield rate is less than the coupon rate, your bond decreases in price as you go on. Right? – hyg17 Nov 13 '14 at 22:57
  • What the hell... you are right. Is there a way to decide which one is better without actually calculating the yield? What I mean is, is it possible to decide which is better simply by looking at the coupon and/or yield rates? – hyg17 Nov 13 '14 at 23:56
  • @hyg17 in order to determine which one is better, you need to know what the bonds are currently trading at, time value, and coupon. In the example above, I made the time value the same to simplify things. To determined how much the investment yields, it is: total cash flow - (cost paid - par value). A 5 year 10% on 100k has cash flow of 50k whereas our 10 year 4% was only 40k. So you have to pay attention to time to maturity too. – dustin Nov 14 '14 at 00:01
  • I see... one last question. When one buys a bond, the person will always earn money? – hyg17 Nov 14 '14 at 04:49
  • @hyg17 are we excluding bankruptcy, coup d'etat (for government bonds), nationalization of assets (which could be a plus look up bond holders Chavez nationaliztion), and theoretical events such as bond whose premium is so high the coupon wont cover the excess cost (no one would ever buy this but it could theoretically exist)? – dustin Nov 14 '14 at 19:22
  • Hmm... I do not know enough to even know what all of those mean, but sure. I am thinking that those events are rather uncommon and I would like to know generally, or perhaps naturally, what would happen. – hyg17 Nov 14 '14 at 22:47
  • @hyg17 bankruptcy bond holders can get screwed and less than par value as well as no more coupon payments, government instability can especially via a coup could cause the government bonds not to be paid, nationalize is when a foreign country takes over assets of a corporation can be good or bad, and someone could theoretically by a bond where the premium is so high it isnt worth it. – dustin Nov 14 '14 at 22:51