Question

Suppose that Dr. Kim has an obligation to pay back $100,000 in 2.5 years. To immunise this obligation, Dr. Kim plans to invest in 3-year x% coupon bond at a yield of 25.73% pa. Coupon rate is an annualized percentage rate and coupons are paid annually. Yield to maturity is an annualized simple interest rate compounded annually. (a) Please find coupon rate x% that will immunize Dr. Kim's obligation. (b) How much Dr.Kim should invest today in this coupon bond for immunization? (c) Using the coupon rate you computed in part (a), please find the total dollar amount of par value of the coupon bond that Dr. Kim should invest in. (In case you didn't get an answer for part (a), please assume the coupon rate is 10%. I will still give you full mark for part (c) if the rest of steps are all correct.)

Answer 1

To immunize Dr. Kim's obligation of $100,000 in 2.5 years, the coupon rate should be 8.58%. Dr. Kim should invest $74,574.82 today in the coupon bond for immunization. The total dollar amount of the coupon bond that Dr. Kim should invest in is $6,390.74.

Step-by-step explanation:

To immunize Dr. Kim's obligation to pay back $100,000 in 2.5 years, we need to find the coupon rate that will match the yield to maturity. The yield to maturity is 25.73% pa, so the coupon rate should be the same to ensure that the investment's returns cover the obligation. Given that the coupon rate is paid annually and the bond is for 3 years, the coupon rate should be x = 25.73% / 3 = 8.58%.

To find the amount Dr. Kim should invest today in this coupon bond for immunization, we can use the present value formula. The present value can be calculated as:

Present Value = Face Value / (1 + Yield to Maturity)^n

Using the given information, the present value of the bond should be:

Present Value = $100,000 / (1 + 0.2573)^2.5 = $74,574.82.

Using the coupon rate of 8.58% calculated in part (a), we can find the total dollar amount of par value of the coupon bond that Dr. Kim should invest in. The amount can be calculated as:

Total Dollar Amount = Coupon Rate * Face Value = 0.0858 * $74,574.82 = $6,390.74.

Answer 2

Immunizing Dr. Kim's Obligation

Part (a): Coupon Rate

To immunize Dr. Kim's obligation, the present value of the bond's future cash flows (coupons and principal) must equal the obligation amount. We can use the following equation to solve for the coupon rate (x):

How to solve

PV(Coupons) + PV(Principal) = $100,000

Step 1: Calculate Present Value of Coupons

Coupon amount = x% * Par value

Since coupons are paid annually for 2.5 years, the total number of coupons = 2.5

Present value of each coupon = Coupon amount / (1 + Yield)^t

Present value of all coupons = Sum of present values of individual coupons

Step 2: Calculate Present Value of Principal

Present value of principal = Par value / (1 + Yield)^t

t = 3 years (maturity of the bond)

Step 3: Solve for Coupon Rate (x)

Substitute the above expressions into the initial equation and solve for x:

x * Par value * [(1 - (1 + Yield)^(-2.5)) / Yield] + Par value / (1 + Yield)^3 = $100,000

This equation cannot be solved explicitly for x. We need to use numerical methods like trial and error or iterative methods to find the coupon rate that satisfies the equation.

Part (b): Investment Amount

Once you have the coupon rate (x), you can calculate the investment amount needed by Dr. Kim using the following equation:

Investment Amount = $100,000 / (1 + Yield)^2.5

Part (c): Par Value

The par value of the bond is the face value that Dr. Kim will receive at maturity. We can use the following equation to calculate the par value:

Par Value = $100,000 / [(1 - (1 + Yield)^(-2.5)) / Yield]