1. Time value of money
When calculating bond valuations, we need to understand the time value of money (or TVM).
2. Time value of money (TVM)
Specifically, the value of an asset is the present value of the expected future cash flows generated by that asset. The intuition behind this is that the value of $1 today is worth more to you than $1 tomorrow because you can invest the $1 you have today at some return so you get more money in the future.
Another way to look at it is as follows. Suppose you won $10,000 in a game show, which of the following would you choose? Receive $10,000 today or one year from now?
If you have the money today, you can invest it and get more than $10,000 next year. So the game show host needs to offer you more than $10,000 to entice you to receive the $10,000 one year from now.
3. Future value
What is future value? The future value is how much $1 today is worth next year, two years from now, and so on. Since we need to be compensated to agree to forego receiving that $1 today, we need some interest rate "r" that will make you accept giving up that $1 today.
We can calculate this as follows. The future value one year from now "fv1" is equal to the present value "pv" multiplied by one plus the interest rate "r." Similarly, the future value two years from now "fv2" is simply the "fv1" grown at one plus the interest rate "r."
So, the further into the future you receive the cash flow, the more you have to be compensated.
4. Present value
The reverse of future value is 'present value'. Present value tells us the value today of $1 received one year from now, two years from now, and so on. Since you prefer to receive money today, you will be willing to receive less than $1 today to avoid having to wait one year or two years to receive that same $1.
We can calculate this as follows. The present value "pv" is equal to the future value one year from now "fv1" discounted one period using one plus the interest rate "r." Similarly, the present value of a cash flow received two years from now is discounted two periods.
5. TVM applied to bonds
We can apply these Time Value of Money (or TVM) concepts to bonds.
Let's use the following example. Consider a bond with a par value (or face value) of $100. The bond pays a fixed 5% coupon rate, and matures in 5 years. Its price today is $100.
Should you buy this bond?
6. Bond investor's trade-off
To determine the value of the bond, we can show graphically the cash outflow and inflow over time.
Today is Year 0 and you would have to pay $100 to purchase the bond. Then, you receive $5 in Years 1 to 4. In Year 5, you get the last $5 coupon payment plus the $100 principal payment. After that, the bond matures and is no longer outstanding.
7. Comparing cash flows
To know if it is worth giving up the $100 today, you would need to know whether the present value of the coupons and principal payments is greater than $100.
To do this, you take the $5 in Year 1 and calculate its present value.
You do the same for the $5 in Year 2, Year 3, and Year 4. In Year 5, you calculate the present value of the $105. The sum of those present values is equal to the value of the bond.
If the sum of the present values exceeds $100, you buy the bond. If it's lower than $100, you don't buy the bond.
8. Let's practice!
You will now apply the time value of money concept in the following exercises.