Discuss on Bond Duration and Bond Volatility with Example.
A bond will face interest rate risk if the holding period differs from duration. For any bond, there is a holding period at which the effect of reinvestment risk and price risk balances each other. The lose of reinvestment of interest can be compensated by a capital gain on the sale of the bond and vice versa. For this holding period, there is no interest rate risk. The holding period at which interest rate risk disappears is called the duration of the bond.
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Thus, the duration of a bond is a required time period at which the price risk and the reinvestment risk of a bond are of equal magnitude but opposite in direction. Therefore, the bond duration is the weighted average life of the bond. The various time periods at which the bond generates returns are weighted according to the respective size of the present value of these returns.
The formula for calculating bond duration can be expressed as:
d = [1×{I1/(1+k)1}+2×{I2/(1+k)2}+3×{I3/(1+k)3}+ – – – – – – – + n ×(In+MV)/(1+k)n] / P0
This equation requires discounting the series of cash flows, which are multiplied by the time period in which they occur. The sum of these cash flows is divided by the price of the bond obtained by using the present value model. However, the above formula can be expressed in a more general format as:
n n
d = ∑ [(t)(Ct)/(1+k)t] / ∑ Ct/(1+k)t]
t = 1 t = 1
where,
d = bond duration,
Ct = annual cash flow both interest and principal amount,
k = discount rate which is the proxy of market interest rater,
t = time period of each cash flow and
n = number of periods.
A 5-year bond having 15 percent coupon rate was issued at a premium of 5 percent 3 years ago. The prevailing interest rate in the market is 18 percent. The calculation of the bond duration can be summarized as:
Assume that the face value of the bond is Tk. 100.
| Year | Cash flow
(Ct) |
PV factor
@ 18 % |
Present value (PV) | PV multiplied by years |
| 1
2 3 4 5 Advertisement
5 |
15
15 15 15 15 105 |
0.8475
0.7182 0.6086 Advertisement
0.5158 0.4371 0.4371 |
12.7125
10.7730 9.1290 7.7370 6.5565 45.8955 |
12.7125 Advertisement
21.5460 27.3870 30.9480 32.7825 229.4775 |
| 92.8035 | 354.8535 |
n n
Bond duration (d) d = ∑ [(t)(Ct)/(1+k)t] / ∑ Ct/(1+k)t]
t = 1 t = 1
= 354.8535/92.8035 = 3.82 years
The duration of the 5-year maturity bond is 3.82 years. If the bond is held for 3.82 years, the interest rate risk can be eliminated. The impact of reinvestment risk and price risk would offset each other to reduce the interest rate risk to zero.
Bond volatility is the absolute value of the percentage change in the bond price for a given change in yield to maturity. If we divide the percentage change in price by the percentage change in yield to maturity, we simply get bond volatility.
Therefore, bond volatility can be expressed by applying the following formula:
Bond volatility = [∆P/P]/ ∆r
where,
∆P/P = percentage change in price
∆r = percentage change in yield to maturity
To understand bond volatility, let us take an example. Suppose the interest rate increases from 10 percent to 12 percent and price changes from Tk. 100 to Tk. 90. Thus, bond volatility can be expressed as:
Bond volatility = [∆P/P]/ ∆r
= [100-90/100]/(12-10)
= 10 %/ 2% = 5
Read More: Bond Pricing Theorems
Discuss on Bond Duration and Bond Volatility with Example