Bond prices and how to compare

Question

I have a couple of basic questions regarding bonds that I would like to ask and the following problem is what I used.

Find the "price" of the following bonds "redeemable at par". Let $F=100$ be the face price. $r$ be the coupon rate and $j$ be the yielding rate. Let $t$ be the amount of years. And

a), $t=10, r^{(2)}=5\%, j^{(2)}=7.2\%$

b), $t=10, r^{(2)}=5.5\%, j^{(2)}=7.7\%$

c), $t=12, r^{(2)}=5\%, j^{(2)}=7.2\%$

d), $t=12, r^{(2)}=5.5\%, j^{(2)}=7.7\%$

Also, show how to compare their prices without actually calculating the numerical values?

So far I have learned amortization and a bit about cash flow, so I understand that

$$P=Cv_j^n+Fra_{\overline {n}\rceil j}$$

Question 1:

I'm a bit iffy about the vocab. Is the "price" of the bond always the present value ? Is there a special word for the future value?

Question 2:

I assumed that in this problem, $F=C$ because of the word "redeemable at par". Am I right? Or was there another reason why this would be true? Either way I simplified the equation as

$$P=F(1+(r-j)a_{\overline {n}\rceil j})$$

Question 3:

I was able to get the numbers correct to the answer, but I am conceptually still not sure how things are going.

$$P \approx 84.52, \quad 84.85, \quad 82.52, \quad \text{and} \quad 82.96$$

for a), b), c) and d) respectively. I also understand that the smaller the present value, the larger the future value due to the effect of the yield rate.

I can at least say that the more time there is the more money one will make, so c>a and d>b in terms of yield. The yield rate directly affects how the money grows, so I can also say that this determines the future value drastically, so b>a and d>c. However, I am not confident to compare b and c. I can see that the value $r-j$ are the same for both so it might do something with it, but is there a way to compare these two without actual calculation?

Answer 1

1. In general, every security's price is equal to the present value of its cash flows.

2. You have interpreted redeemable at par correctly.

3. c can be seen to have a greater value than a by the reason you gave and similarly d can be seen to have a greater value than b. This can be seen by examining the present value equation and seeing that it increases with maturity.

Comparing b and a is not so easy because although the maturity is the same and b has a higher coupon rate, b also has a higher yield rate, and when the yield increases the price decreases. As you can see from the calculation, the prices are very similar. We can get a comparison from your second pricing equation by noting that $5-7.2=5.5-7.7$ so the higher yield means the value of the actuarial symbol will be higher, so b>a. For the same reason, d>c.

For comparing b and c, the question reduces to comparing $a_{10\rceil 7.7}$ and $a_{12\rceil 7.2}$. Heuristically the extra years for the 12 year bond happen far in the future, so that seems to suggest that the higher yield might outweigh the longer maturity. I know of no good way to compare them without calculating it, though.