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Results

  • Bond Value:
  • Maculay Duration:
  • Modified Duration:
  • Convexity:

Bond Valuation Concept


Bond Valuation

The determination of a bond's fair price is defined as bond valuation [1].

Face Value

The printed amount on the bond is called as face value [2].

Coupon Rate

Coupon rate is the rate at which the bond pays interest on its face value at regular time intervals until the redemption date [2].

Bond Value Formula - How to Calculate Bond

Let

\(F\) = Face Value
\(R_c\) = Coupon Rate
\(R\) = Yield to Maturity
\(N\) = Number of Years
\(Q\) = Payment Frequency

\(C\) = Cash Flow
\(c\) = Coupon Payment
\(B\) = Bond Value
\(D\) = Maculay Duration
\(D_m\) = Modified Duration
\(K\) = Convexity

Coupon Payment can be calculated as the following:

\[ c = \frac{F \times R_c}{Q} \]

We can calculate Bond Value by using the following formula:

\[ B = C \times \frac{1 - \frac{1}{(1 + \frac{R}{Q})^{N \times Q}}}{\frac{R}{Q}} \quad + \quad \frac{F}{(1 + \frac{R}{Q})^{N \times Q}} \]

Then, Maculay duration can be calculated as follows:

\[ D = \frac{\sum_{i=1}^{N \times Q} i \times PV(C, \frac{R}{Q}, i)}{B \times Q} \]

We can also calculate Modified Duration:

\[ D_m = \frac{D}{\left(1 + \frac{R}{Q}\right)} \]

Then, convexity can be calculated as the following:

\[ K = \frac{\sum_{i=1}^{N \times Q} \frac{(i^2 + i)\times PV(C, \frac{R}{Q}, i)}{(1+\frac{R}{Q})^2}} {B \times Q^2} \]

where \(C = c\), if \( i = N \times Q \); otherwise \( C = F + c \).

and \(PV\) is the present value function.

Examples

Example 1

Input

Face Value = 1000
Coupon Rate = 8%
Yield to Maturity = 8%
Years = 6
Frequency = Semiannually

Output

Bond Value = 1000
Maculay Duration = 4.88
Modified Duration = 4.69
Convexity = 27.227

Example 2

Input

Face Value = 1000
Coupon Rate = 14
Yield to Maturity = 16
Years = 7
Frequency = Quarterly

Output

Bond Value = 916.685
Maculay Duration = 4.454
Modified Duration = 4.282
Convexity = 25.057

References

1. Bond valuation (n.d.). Retrieved August 18, 2016, from https://en.wikipedia.org/wiki/Bond_valuation

2. Bonds and Bond Pricing. (n.d.). Retrieved from http://www.mysmu.edu/faculty/yktse/FMA/S_FMA_6.pdf