**Wikipedia**- "The YTM calculation formulates certain stability conditions of the security, its owner, and the market going forward: The owner reinvests all interest payments rather than spending them, to gain the benefit of compounded returns."

**Investopedia**- "Calculating the yield to maturity can be a complicated process, and it assumes all coupon or interest payments can be reinvested at the same rate of return as the bond."

**Annette Thau**(author of The Bond Book): "YTM calculations are based on the assumption that coupons are never spent; they are always reinvested. [...] coupons are reinvested at an interest rate equal to the yield-to-maturity."

However, the following paper by 3 researchers states:

I am trying to verify their claim, but I am unable to do so. Let's take the following coupon bond:This note addresses a common misconception, found in investment texts and popular investment education literature,

that in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments. We identify a

sample of text and professional sources making this claim, demonstrate that yield to maturity entails no assumption of

coupon reinvestment, discuss a cause for this confusion and offer a possible remedy

Code: Select all

```
Maturity: 3 years
Principal: $1,000
Current Price: $1,000
Coupon Payment: $50
```

Let's verify whether we can get a 5% Annualized Return (as predicted by the 5% YTM).

So if I purchase the bond for $1,000, then I will get the following cashflows: $50, $50, $50, $1,000 = $1,150. This means I will earn a Total Return of $150, expressed as a percentage: $ 150 / $1,000 = 15%. The Annualized Return would be (1 + 15%) ^ (1 / 3) - 1 ≃ 4.77%.

The Annualized Return of 4.77% does not match the stated YTM of 5%.

Investopedia, Wikipedia and Annette Thau are saying that in order to earn the 5% YTM, we actually need to reinvest the coupons also at the 5% rate. The researchers say that they are wrong. Let's try reinvesting the coupons and see if we get to 5%.

Here is the table showing the interest payments we get when we reinvest the interest payments at 5%

Code: Select all

```
Interest payments End of year 1 End of year 2 End of year 3
Bond 1 50 50 50
Bond 2 (from interest of bond 1) 2.5 2.5
Bond 3 (from interest of bond 1) 2.5
Bond 4 (from interest of bond 2) 0.125
```

This would mean that if we reinvest the coupon payments at 5%, we end up with $157.625 in interest income, instead of only $150. What's the Total Return? $157.625 / $1,000 = 15.7625%. The Annualized Return would then be (1 + 15.7625%) ^ (1 / 3) - 1 = 5%.

This matches the 5% YTM.

So now I am confused. The paper from the researchers says that "it is an erroneous assumption", that "in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments".

How is it erroneous? If I do not reinvest the coupon payments, my Annualized Return will be ~4.77%. If I do reinvest the coupons, my Annualized Return will be 5.00%, exactly equal to YTM.

**Question:**If the research paper is correct and achieving an Annualized Return of 5% does not require reinvesting coupons, then I should get a 5% Annualized Return when not reinvesting the coupons, but I only get 4.77%. Why is that?

**Question:**Perhaps the research paper is not correct, because the only way to get a 5% Annualized Return is to reinvest the coupons, contrary to what the research paper says. Is the research paper incorrect?