Previously, we have discussed the discounting of one future cashflow to its present value.

In this blog post, we will be going into:

- How the discounting of future cashflows of a bond’s coupon determine its price
- How the change of the bond yield (risk-free rate) affect bond price

## Discounting Coupon Payout

For bonds, the future cashflow are known beforehand and are dictated by the coupon rate (you could think of this as the bond yield although there is a slight difference).

In this example, the bond we would be considering would be a 5-year bond with 10% coupon rate and a face value (price that the coupon payout is based on) of $1,000. It would give have a coupon payout of $100 for the next 5 years and for the last coupon payout, it would include the $1000 face value.

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | |
---|---|---|---|---|---|

Coupon Payout | $100 | $100 | $100 | $100 | $1,100 |

Present Value | $90.91 | $82.64 | $75.13 | $68.30 | 683.01 |

*Coupon payout is taken as annually for simplicity, but it is usually semi-annually

The risk-free rate (discount rate) is taken to be the same as the bond’s *yield-to-maturity (YTM – *same as coupon rate in this case).

If we summed up the present value of all the discounted future cashflows, we would have the present value of the bond. Which is $1,000, the same as the bond’s face value.

This is only so because we assumed that the bond yield is constant at 10%, the same as the coupon rate. However, the bond yield is not constant and changes like everything else in the market, controlled by supply and demand.

## Change in Bond Yield

As the risk-free rate is based upon the bond yield, when the bond yield changes so does the risk-free rate. And since the discounting of the future cashflow is based upon this risk-free rate, the present value of the cashflows would change too.

Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | |
---|---|---|---|---|---|

Coupon Payout | $100 | $100 | $100 | $100 | $1,100 |

Present Value (0% risk-free rate) | $100 | $100 | $100 | $100 | $1,100 |

Present Value (10% risk-free rate) | $90.91 | $82.64 | $75.13 | $68.30 | 683.01 |

Present Value (20% risk-free rate) | $83.33 | $69.44 | $57.87 | $48.23 | $442.07 |

Despite the coupon payout being the same $100 per year, the present value of the bond changed greatly depending on the bond yield.

When the YTM is 0%, the price of the bond (total present value of all future cashflows) is $1,500. You would pay $1,500 to buy the bond today and receive $1,500 at the end of the 5 years, making 0% return.

When the bond YTM is 20%, the price of the bond is $700.94. More on YTM would be elaborated later on.

This is to point out the risk when investing in an instrument that is priced heavily with respect to future cashflow, it will be very sensitive to the change in interest rate (same meaning as bond yield, usage is based on context). When yield increases, the price of the bond you are holding would fall.

## Supply and Demand Controls Bond Price/Yield

This section is written so that it could perhaps clear up some common misinterpretation on bonds.

Firstly, bond price is based on the market’s supply and demand of the bond. When demand increases, the bond price increase. It is this increase in bond price that decreases the YTM.

Although from this article, it might seem that it is the decrease in the YTM (risk-free rate) that increases bond price, the relationship is opposite. **It is the increase in bond price that drive yield down.**

Secondly, YTM is the hypothetical annual return of the bond if **all coupons are reinvested at the same yield**.

\[ \text{First coupon: }$100*1.2^4=$207.36 \\ \text{Second coupon: }$100*1.2^3=$172.8 \\ \text{Third coupon: }$100*1.2^2=$144 \\ \text{Forth coupon: }$100*1.2=$120 \\ \text{Last coupon: }$1,100 \\ \text{Total sum: }$1744.16\]

We would also arrive at the same total sum if we compound price of the bond $700.94 by 20% annually for 5 years.

\[ $700.94*1.2^5=$1744.16 \]