Time Value of Money
The first thing to understand, is this concept called the Time Value of Money. In other words, $1 today is worth more than $1 in the future. Or, $1 in the future is worth less than $1 now.
So, how much is $1 now worth in the future? To answer this question, we will need to know how much can this $1 grow riskfree. In other words, without any risk, how much interest can this $1 gain and grow into.
This interest rate that in question is called the riskfree rate. We usually use the central bank's prime lending rate for this. The prime lending rate or prime rate, is the rate at which the country's bank charges the rest of the bank for lending it money. Singapore's central bank is the Monetary Authority of Singapore (MAS).
For example, if the prime rate is 2%, $1 today will be worth $1.02 next year, and $1.0404 in the subsequent year (1.02 x 1.02 = 1.0404). Similarly, $1 in the next year is worth $0.98 today (1 / 1.02 = 0.98). This idea is key.
An explanation into bonds
Bonds are a form of borrowing. When you purchase a bond, you are lending money to the other party. In return, you will get back regular payments of interest, and at the end, you will also get back the money you lent at the start. This is the simple form of bonds. There are also types of bonds such as zero coupon bond which do not pay any regular payments but pay you back more money in the end than what you have initially lent.
Calculating bond price
Let's use this example for illustration. Suppose you purchase a bond at $1,000. In return, you get back $100 each year for 5 years. At the end of 5 years, you will get back $1,000. The prime rate is 2%.
This point is important. The bond price at any point, will be equal to the value of the payments in the future.
Assume we have just lent out $1,000. For the first year's interest, it will be worth $980 today ($1,000 / 1.02 = $980). For the second year's interest, it will be worth $961 ($1,000 / 1.02 / 1.02 = $961). This goes on till the last year. For the last year, you will be receiving $1,100 due to both interest and principal repayment. This $1,100 is worth $996 today ($1,100 / 1.02 / 1.02 / 1.02 / 1.02 / 1.02 = $996). The total value of all the future payments is worth $1,377, which is also the bond price.
Year

Receive

Divided
by

Value

1

$100

1.02

$98

2

$100

1.02^2

$96

3

$100

1.02^3

$94

4

$100

1.02^4

$92

5

$1,100

1.02^5

$996

Total:

$1,377

When interest rates fall, bond price increases
Let's run through the same example with prime rate being lowered to 1% instead of 2%. In this case, the first year's receipt of $100 will be worth $99 ($100 / 1.01 = $99), and the second year's receipt will be worth $98. The last year's receipt of $1,100 is worth $1,045 ($1,100 / 1.02 / 1.02 / 1.02 / 1.02 / 1.02 = $1,045). The full table is replicated below. Calculations are similar to the previous example.
Year

Receive

Divided
by

Value

1

$100

1.01

$99

2

$100

1.01^2

$98

3

$100

1.01^3

$97

4

$100

1.01^4

$96

5

$1,100

1.01^5

$1,045

Total:

$1,435

As you can now see, the bond price has increased to $1,435, due to the drop in prime rate.
When interest rates rise, bond price falls
We will now change the prime rate to 3% instead of 2%. The table is replicated below. The bond price has dropped to $1,321 when the interest rate rose.
Year

Receive

Divided
by

Value

1

$100

1.03

$97

2

$100

1.03^2

$94

3

$100

1.03^3

$92

4

$100

1.03^4

$89

5

$1,100

1.03^5

$949

Total:

$1,321

Conclusion
Hope this can give a better picture of why bond prices and interest rates move in opposite direction. I will be doing another post on why bonds and stock prices move in opposite directions, and how bonds can be a good hedge in any portfolio.
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