# How to calculate the value of a $bDUET

<mark style="background-color:yellow;">Before we calculate the value of a bond, we need to understand the basics of a bond</mark>

1. **Face Value(F)**: The principal amount owed by the bond issuer.&#x20;
2. **Epoch**: The time interval between every payment, Duet defines an Epoch to be 1-month equivalent amount of blocks, depending on different block times of different blockchains. All pending payments of all bonds are claimable on the same block every month.&#x20;
3. **Coupon Rate(r)**: The Coupon interest rate is used to calculate interest payments each epoch, for example, a bond with face value of 100 Duet and a coupon rate of 10% per epoch will be paid 10%\*100=10 Duet tokens each epoch as interest payments&#x20;
4. **Duration(n)**: The amount of epochs left to be paid&#x20;
5. **Discount Rate(R)**: Discount rate is derived using market price of a bond, duration of the bond and expected payments of the bond.&#x20;
6. **Net Present Value(NPV)**: The NPV is arrived by discounting all expected payments of a bond with respective discount rates, NPV is the fair value of a bond.&#x20;
7. **Current Price（P)**: P is the market price at which a bond is being traded at&#x20;
8. **Yield to Maturity(YTM)**: YTM is annualized return on principal of a bond

**Now Consider a simplest bond,** a bond with a face value of 100, coupon rate of 10% and duration of 1 epoch.

**Coupon payments=F\*r=100\*10%=10 Duet**

**Principal Payment= F =100**

The bond holder is expected to receive 110 Duets in 1 epoch time. On the date of issuance, coupon rate is always equal to discount rate, which is 10%.

**NPV=Payment/(1+R)n=110/(1+10%)1=100 Duets**

When a bond’s current price is equal to its face value, the bond is trading at par, when it is trading above the face value, it is trading at a premium and when it is trading at lower than the face value, it is trading at a discount.

**Now consider after the issuance of the bond,** discount rate drops to 5%, the value of the bond would be

**NPV=Payment/(1+R)n=110/(1+5%)1=104.76 Duets**

This example shows how the value of a bond can appreciate when its coupon rate is higher than the discount rate (which is typically coupon rate of a newly issued bond)

**Now back to the more complicated bond that is mentioned in the overview,** a bond with face value of 100 Duets, coupon rate of 10% and a duration of 6 epochs. The payment structure of the bond is shown below

<table data-header-hidden><thead><tr><th>Date</th><th data-type="number">Payments</th></tr></thead><tbody><tr><td>Month 1</td><td>10</td></tr><tr><td>Month 2</td><td>10</td></tr><tr><td>Month 3</td><td>10</td></tr><tr><td>Month 4</td><td>10</td></tr><tr><td>Month 5</td><td>10</td></tr><tr><td>Month 6</td><td>110</td></tr></tbody></table>

**The above payment structure can be replicated with 6 bonds like follows**

<table><thead><tr><th width="150" data-type="number">Bond No.</th><th width="185" data-type="number">Face Value</th><th width="193.86830680173662">Coupon Rate</th><th data-type="number">Duration</th></tr></thead><tbody><tr><td>1</td><td>10</td><td>0%</td><td>1</td></tr><tr><td>2</td><td>10</td><td>0%</td><td>2</td></tr><tr><td>3</td><td>10</td><td>0%</td><td>3</td></tr><tr><td>4</td><td>10</td><td>0%</td><td>4</td></tr><tr><td>5</td><td>10</td><td>0%</td><td>5</td></tr><tr><td>6</td><td>110</td><td>0%</td><td>6</td></tr></tbody></table>

Calculating the NPV of the bond now is just as easy as adding NPVs of these 6 bonds.

$$
NPV=Payment/(1+r)^1+Payment2/(1+r)^2+....+Payment6/(1+r)^6
$$

Assuming the discount rate is as follows such that it increases as durations lengthen, the NPVs are calculated as

<table><thead><tr><th width="150">Bond No.</th><th width="150" data-type="number">Face Value</th><th width="150">Coupon Rate</th><th width="150" data-type="number">Duration</th><th>Discount Rate</th><th data-type="number">NPV</th></tr></thead><tbody><tr><td>1</td><td>10</td><td>0%</td><td>1</td><td>1%</td><td>9.90099</td></tr><tr><td>2</td><td>10</td><td>0%</td><td>2</td><td>2%</td><td>9.61169</td></tr><tr><td>3</td><td>10</td><td>0%</td><td>3</td><td>3%</td><td>9.15147</td></tr><tr><td>4</td><td>10</td><td>0%</td><td>4</td><td>4%</td><td>8.54804</td></tr><tr><td>5</td><td>10</td><td>0%</td><td>5</td><td>5%</td><td>7.83526</td></tr><tr><td>6</td><td>110</td><td>0%</td><td>6</td><td>6%</td><td>77.5457</td></tr><tr><td>Total</td><td>160</td><td></td><td>null</td><td></td><td>122.59</td></tr></tbody></table>

**The bond’s fair value or NPV is 122.59 Duets, thus the bond is trading at a premium, selling immediately will amount to a return of 22.59 Duets. This is the result from a lower coupon rate after the issuance of the original bond. However, by selling the bond now, the holder will have to accept a discount of 1-(122.59/160)=23.38%.**