✏️ More example YTM questions

As you know, calculating $YTM$ is just calculating the $IRR$ for the process of purchasing a bond.

The $PV$ of the outflows is just the price of buying the bond, $P_B$.

The $PV$ of the inflows is just the $PV$ of the cash flows you get from the bond.

Therefore, when you write down the equation to solve for $IRR$, you just get the bond pricing formula‼ You just solve this equation for $i$, the discount rate. The answer you get is the $YTM$, which is:

1. The effective “interest rate” on the bond
2. The $IRR$ of purchasing the bond

Here are our three bond pricing formulas:

$$\begin{aligned} &P_{CouponBond} = \frac{Fc}{(1+i)^1} + \frac{Fc}{(1+i)^2} + \frac{Fc}{(1+i)^3} + ... + \frac{Fc + F}{(1+i)^T} \\ &P_{ZCB} = \frac{F}{(1+i)^T} \\ &P_{Consol} = \frac{Fc}{i} \end{aligned}$$

The good news is that there are only three types of bonds, so there are only three types of $YTM$ problems.

Yield to Maturity (YTM) Examples

✏️ Bond #1: Consol
$c = 6\%$
$F = \$1000$
$T = ∞$
$P_B = \$982$

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Write down the bond pricing formula and solve for $i$

$$\begin{aligned} P_{Consol} &= \frac{Fc}{i} \\ \$982 &= \frac{\$1000 \times 6\%}{i} \end{aligned}$$

Switcheroo:

$$\begin{aligned} i &= \frac{\$1000 \times 6\%}{\$982} \\ &= \frac{\$60}{\$982} \\ &= 6.11\% \end{aligned}$$

✏️ Bond #2: 1 year Coupon Bond
$c = 5\%$
$F = \$1000$
$T = 1$
$P_B = \$982$

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The general formula is:

$$P_{CouponBond} = \frac{Fc}{(1+i)^1} + \frac{Fc}{(1+i)^2} + \frac{Fc}{(1+i)^3} + \cdots + \frac{Fc}{(1+i)^T} + \frac{F}{(1+i)^T}$$

For a one period bond, this becomes:

$$P_{CouponBond} = \frac{Fc}{(1+i)^1} + \frac{F}{(1+i)^1} = \frac{Fc + F}{(1+i)^1}$$

Plugging in the numbers for this problem, we get:

$$\$982 = \frac{\$1050}{(1+i)^1}$$

Next we do the Switcheroo:

$$\begin{aligned} (1+i)^1 &= \frac{\$1050}{\$982} = 1.0692 \\ (1+i)^1 &= 1.0692 \\ 1+i &= 1.0692 \\ i &= 6.92\% \end{aligned}$$

✏️ Bond #3: Zero Coupon Bond
$c = 0\%$
$F = \$1000$
$T = 4$
$P_B = \$749$

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Write down the bond pricing formula and solve for $i$

$$\begin{aligned} P_{ZCB} &= \frac{F}{(1+i)^T} \\ \$749 &= \frac{1000}{(1+i)^4} \end{aligned}$$

Switcheroo:
$(1+i)^4 = \frac{\$1000}{\$749} = 1.3351$

Move the $4$ over to the other side and turn it into a $\frac{1}{4}$.
$(1+i) = 1.3351^{1/4}$
(Mathematically speaking, you are taking the $\left(\frac{1}{4}\right)$ power of both sides, but practically, you’re moving the exponent over to the other side and “flipping it.”)

Tip: Be careful how you enter this into a spreadsheet or the Google search box:

• Correct: 1.3351^(1/4) = 1.0749

• Incorrect: 1.3351^1/4 = .333775 (don’t forget the parentheses!)

• Correct: 1.3351^.25 = 1.0749 (it’s okay to calculate 1/4 in advance)

  $1+i = 1.3351^{(1/4)} = 1.074928$
  $i = 1.074928 - 1$
  $i = 7.4928\%$