1. Exam FM Problem 1054 Learning Objective: “Bonds”
Wisconsin Center of Actuarial Excellence Technology Enhanced Learning Project
Exam FM Problem 1054
Learning Objective: “Bonds”
Welcome to the tutorial on Exam FM. Today we are going to go over question 1054, which is part of the learning objective “Bonds.” Here is a view of the problem.

2. Please choose one of the answers and submit it
Please choose one of the answers and submit it. Once submitted, you will not be able to take the quiz again.
Let us clearly define the multiple givens in the question. First, we are given that the nominal annual coupon rate is 8%; however, since coupons are paid semiannually, we can translate this rate into a semiannual rate of 4%. From here let’s establish that the notation “i” will refer to the yield earned per payment period, which in this case the payment period is six months. So the given nominal annual rate of interest of 6% really means a semiannual yield of 3%. Next, we see that the bond’s term is 20 years. We are also given that the bond is callable at par value X – this means the redemption value X is equal to the face value of the bond. Continuing, we also know the bond is callable starting at the end of year 15. Lastly, the price of the bond is
We are looking to solve for X, the redemption value of the bond, which is the principal amount returned to us at term end.

3. Let’s build a strategy to solve this problem.
First, we will take the givens and draw a timeline of cash flows to understand what this bond looks like.
Next, we want to figure out whether the bond is a premium or discount bond. This fact will then tell us when the bond will likely be called by the issuer of the bond.
Finally, after establishing when/if the bond is called before the 20 year term, we can go ahead and set up an equation for the bond price and solve for X, the redemption value.

4. Let’s draw out a timeline and figure out what this bond looks like
Let’s draw out a timeline and figure out what this bond looks like. It is easiest to draw out cash flows in terms of payment period, so for this problem I will draw cash flows per semiannual period. We know that every six months we will receive a coupon payment equal to .04X, the semiannual coupon rate times the face amount, X. These coupons are paid out starting six months after today, or time zero.
The bond’s term can go up to 20 years if not called by the bond issuer, but the issuer holds the right to call back the bond early at the end of year 15.
Finally, the price paid for the bond today is $

5. When the bond is redeemed at the end of its life, the bondholder is paid the final coupon payment plus the redemption value. As mentioned before, this bond’s redemption value is the par value. Now let’s figure out if this bond is a premium or discount bond – this will help us figure out whether or not the bond issuer will find it advantageous to call the bond back earlier than the full 20 years.
Compare the semiannual coupon rate to the semiannual yield rate. The coupon rate is 4% whereas the yield is 3%. This means bondholders can earn a higher rate than the yield rate ,so to compensate for this gain the bondholder will pay more initially for it. Premium bond prices are higher than the face value of the bond.
As a general rule, for bonds that sell at a premium, it is advantageous for the bond issuer to call back the bond early so always assume that the redemption date will be at the earliest date possible. Bond issuers may call back bonds early because they can refinance their debt at a lower rate than what they are currently paying. Thus, we establish that this bond will be called back at the end of year 15.

6. Now that we know that the bond will be paid off in full at the end of year 15, we revise our earlier timeline. As before, the bondholder receives coupon payments of .04X every 6 months, up to the end of year 15 for a total of 30 coupon payments. Also at the end of year 15, the bondholder receives the redemption value, X.
Next, let’s set up an equation for the price of the bond. The price of the bond must equal the present value of all future payments to the bondholder. Here is the equation. For the 30 coupon payments, we use the equation for an annuity with N=30, I/Y=3%, and PMT=.04 for those of you using a financial calculator. The redemption value payment is a one-time payment so we just have to discount it back 30 terms using the semiannual rate, hence the use of v^30. Here’s a review of the annuity and v equations.
The rest is simple – just plug in the given values and solve for X. We get the final answer of $1440, which is answer choice C.

7. Thanks for watching. Questions. Comments
Thanks for watching! Questions? Comments? Please us at: Funding provided by the Society of Actuaries and the Wisconsin School of Business Voice: Alyssa Webb Faculty Supervisor: Jed Frees
That is the end of this tutorial. Thanks for watching!