# Should you sell your bond and buy one at a higher interest rate when interest rate increases?

Suppose you buy a bond 4% coupon 3 years at market rate with 4%.. In the first year you collected your coupon and interest rate rises to 5%.

Value of your bond after 1st year: (40/1.05^2) + (1040/1.05^3)= 945.12

Should you hold to maturity , and earn the cashflow in total (40+40+1040=1120)

Or should you sell your bond and buy a new one at 5% interest rate ? i.e., Total amount earned (40 + 945.12 *0.05 + 945.12*1.05) =1079.64

What puzzles me is why doesnt the 2 strategies equal ?

Assume no transaction cost to buy a new bond...

You are only taking time value of money into consideration in one part of your example and neglects it in the other?

But let's start at the beginning, by building up the cashflow of your original scenario:

``````|                 |           t |     t + 1 |     t + 2 |        t + 3 |
|-----------------|-------------|-----------|-----------|--------------|
| Cashflow        |             |    40.00  |    40.00  |    1,040.00  |
| Discount factor |             |   1.04^1  |   1.04^2  |      1.04^3  |
|-----------------|-------------|-----------|-----------|--------------|
| PV              |   1,000.00  |    38.46  |    36.98  |      924.56  |
``````

Present value of your original scenario, with 4% interest is 1,000 dollars, which is fortunate, because this is what you have paid for the bond.

After we have received the first payment of 40 dollars, the Present Value of the bond (at t + 1) is:

``````|                 |           t |     t + 1 |     t + 2 |        t + 3 |
|-----------------|-------------|-----------|-----------|--------------|
| Cashflow        |             |           |    40.00  |    1,040.00  |
| Discount factor |             |           |   1.04^1  |      1.04^2  |
|-----------------|-------------|-----------|-----------|--------------|
| PV              |             | 1,000.00  |    38.46  |      961.54  |
``````

The price of the bond at t + 1 is still 1,000 because nohting has changed.

Now we introduce a change in interest rate going from 4% to 5%. Now the price of our bond changes:

``````|                 |           t |     t + 1 |     t + 2 |        t + 3 |
|-----------------|-------------|-----------|-----------|--------------|
| Cashflow        |             |           |    40.00  |    1,040.00  |
| Discount factor |             |           |   1.05^1  |      1.05^2  |
|-----------------|-------------|-----------|-----------|--------------|
| PV              |             |   981.41  |    38.10  |      943.31  |
``````

The price of our bond is now 981.41 (at t + 1).

Let's say we are to sell our bond in exchange for the new and improved 5% bond (with 2 years to maturity).

We now get the following cashflow:

``````|                 |           t |     t + 1 |     t + 2 |        t + 3 |
|-----------------|-------------|-----------|-----------|--------------|
| Cashflow        |             |           |    49.73  |    1,030.48  |
| Discount factor |             |           |   1.05^1  |      1.05^2  |
|-----------------|-------------|-----------|-----------|--------------|
| PV              |             |   981.41  |    46.73  |      934.67  |

Note: 981.41 x 0.05 = 49.73
``````

See how our PV of our bonds at t + 1 are the same in both scenarios?

value of your bond after 1st year: `(40/1.05^2) + (1040/1.05^3)= 945.12`

This is where your mistake is - the value of the 3-year bond after one year would be

``````(40/1.05^1) + (1040/1.05^2)= 981.41
``````

With that correction, the cash flow of the 2-year bond would be:

``````(40 + 981.41 *0.05 + 981.41*1.05) =
40 + 49.07        + 1030.48      = 1,119.55
``````

And the only difference in PV is a slight difference in discounting (~\$9 is received one year sooner)