Question

Using excel, illustrate and check the following observations on the duration of bonds.

1. The duration of a zero-coupon bond equals its time to maturity

2. With time to maturity and yield to maturity held constant, a bond's duration and interest rate sensitivity are higher when the coupon rate is lower

3. With the coupon rate held constant, a bond's duration and interest rate sensitivity generally increase with time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par.

4. With other factors held constant, the duration and interest rate sensitivity of a coupon bond are higher when the bond's yield to maturity is lower.

5. The duration of a level perpetuity is: (1+y)/y

You can make up the excel macros

Answer 1

SOLUTION TO PART (I)          
          
Let us assume a Rs 1000 8% 4 year ZCB           
          
Year   Cash Flow   PV @ YTM 10%   PV x Year
1   0   0   0
2   0   0   0
3   0   0   0
4   1000   735   2,940
TOTAL       735   2,940
          
MD = (PV X Time to Cash flow) / PV of All the cash flow          
MD = 2940/735.02 = 4          
          
SOLUTION TO PART (II)          
          
Let us take 2 bond           
1). Rs. 1,000 5 year 10% annual coupan bond trading at a YTM of 10%            
1). Rs. 1,000 5 year 2% annual coupan bond trading at a YTM of 10%            
          
Now let us Calculate duration of these 2 bonds          
          
BOND 1          
          
Year   Cash Flow   PV @ YTM 10%   PV X Year
1   100   91   91
2   100   83   165
3   100   75   225
4   100   68   273
5   1,100   683   3,415
TOTAL       1,000   4,170
          
MD = 4169.87/1000 = 4.16          
          
BOND 2          
          
Year   Cash Flow   PV @ YTM 10%   PV X Year
1   20   18   18
2   20   17   33
3   20   15   45
4   20   14   55
5   1,020   633   3,167
TOTAL       697   3,318
          
MD = 3317.66/696.74 = 4.76          
          
Since duration is the sensitivity of bond price with respect to change in interest rate, change in price of "BOND 2" will be higher because it has higher duration          
          
          
SOLUTION TO PART (III)          
          
Let us take 2 bond           
1). Rs. 1,000 5 year 10% annual coupan bond trading at a YTM of 10%          
1). Rs. 1,000 10 year 10% annual coupan bond trading at a YTM of 10%            
          
Now let us calculate MD of these bonds          
          
Year   Cash Flow   PV @ YTM 10%   PV X Year
1   100   91   91
2   100   83   165
3   100   75   225
4   100   68   273
5   1,100   683   3,415
TOTAL       1,000   4,170
          
MD = 4169.87/1000 = 4.16          
          
          
Year   Cash Flow   PV @ YTM 10%   PV X Year
1   100   91   91
2   100   83   165
3   100   75   225
4   100   68   273
5   100   62   310
6   100   56   339
7   100   51   359
8   100   47   373
9   100   42   382
10   1,100   424   4,241
TOTAL       1,000   6,759
          
MD = 6759.02/1000 = 6.75          
          
BOND II with higher time to maturity of 10 years has higher MD and therefore higher sensitivity to interest rate change          
          
          
          
SOLUTION TO PART (IV)          
          
Let us take 2 bond           
1). Rs. 1,000 5 year 10% annual coupan bond trading at a YTM of 10%          
2). Rs. 1,000 5 year 10% annual coupan bond trading at a YTM of 5%            
          
Now let us calculate MD of these bonds          
          
Year   Cash Flow   PV @ YTM 10%   PV X Year
1   100   91   91
2   100   83   165
3   100   75   225
4   100   68   273
5   1,100   683   3,415
TOTAL       1,000   4,170
          
MD = 4169.87/1000 = 4.16          
          
Year   Cash Flow   PV @ YTM 5%   PV X Year
1   100   95   95
2   100   91   181
3   100   86   259
4   100   82   329
5   1,100   862   4,309
TOTAL       1,216   5,174
          
MD = 5174.27/1216.47 = 4.25          
          
BOND II with lower YTM of 5% has higher MD and higher sensitivty to change in interest rate          
          
          
SOLUTION TO PART (V)          
          
let us take a bond           
Rs. 1,000 perpetual 10% annual coupan bond trading at a YTM of 10%          
          
Year    CF   PV   PV X Year
1   100   90.91   90.91
2   100   82.64   165.29
3   100   75.13   225.39
4   100   68.30   273.21
5   100   62.09   310.46
6   100   56.45   338.68
7   100   51.32   359.21
.   .   .   .
.   .   .   .
.   .   .   .
95   100   0.01   1.11
96   100   0.01   1.02
97   100   0.01   0.94
98   100   0.01   0.86
99   100   0.01   0.79
100   1100   0.08   7.98
TOTAL       1,000.00   10,999.20
          
MD = 10999/1000 = 11          
which can be directly calculated using formulate (1+y)/y          
ie 1.1/0.1 = 11