A zero-coupon bond is a bond that pays no interest and only returns the face value at maturity. A zero-coupon yield is the annualized rate of return for investing in a zero-coupon bond. A zero-coupon yield curve is a plot of zero-coupon yields for different maturities.
A par bond is a bond that pays a fixed coupon rate and is priced at its face value. A par yield is the coupon rate of a par bond for a given maturity. A par yield curve is a plot of par yields for different maturities.
To construct a par yield curve from a zero-coupon yield curve, we need to find the coupon rates that make the present value of the bond’s cash flows equal to its face value.
Basic Theory:
Zero-coupon yields represent the yield of a bond that does not pay periodic interest but instead is issued at a
discount and redeemed at face value. Par yields, on the other hand, are the interest rates at which the bond is
priced at par. The relationship between these two is based on the concept of present value, where the sum of
the present values of future cash flows equals the bond’s face value.
Procedures:
- Determine the Cash Flow Periods: Identify the time periods for which zero-coupon yields are
available. These periods will serve as the time to maturity for the zero-coupon bonds. - Calculate Present Values: Use the formula for present value to calculate the present value
of each zero-coupon bond’s future cash flow. - Iterate to Find Par Yield: Iterate over different interest rates until the sum of present
values equals the face value of the bond. This is the par yield.
Scenario:
Let’s consider a scenario with three zero-coupon bonds with face values of $1000 each and corresponding yields
of 3%, 4%, and 5% for maturities of 1 year, 2 years, and 3 years, respectively.
Excel Table:
| Time to Maturity (Years) | Yield (%) | Future Value (Face Value) | Present Value |
|---|---|---|---|
| 1 | 3 | 1000 | =1000/(1+3%)^1 |
| 2 | 4 | 1000 | =1000/(1+4%)^2 |
| 3 | 5 | 1000 | =1000/(1+5%)^3 |
| Total | =SUM(D2:D4) |
Calculation:
1. For the 3% bond: 
2. For the 4% bond: 
3. For the 5% bond: 
4. Total Present Value = 
Now, iterate over different interest rates until the Total Present Value equals the total face value of the bonds
($3000). The result is the par yield.
Result:
After iterating, the par yield is found to be approximately 4.5%.
Other Approaches:
- Solver Function: Use Excel’s Solver function to find the interest rate that minimizes the
difference between the total present value and the total face value. - Curve Fitting: Utilize Excel’s curve fitting functions to find a polynomial function that
best fits the zero-coupon yields. The derivative of this polynomial at time
gives the par yield.
