🔎 PDV vs PB vs IRR vs YTM

There are four concepts that it is easy to mix up:

In Lecture 6:

• PDV - Present Discounted Value of an Investment
  • “The PDV of investing in that apartment projects is $\$21.64 \;million$” means that the fair value of the apartments is $\$21.64$. (fair value = the most you should pay for it)
• NPV - Net Present Value of an Investment
  • “The present value of the cash generated by the apartment complex is $\$21.64 \;million$, but we only have to pay $\$10 \;million$ for it, so our NPV is $\$11.64 \;million$” means that we have a $\$11.64$ profit on the investment.
  • $NPV = PDV \;of \;cash \;Inflows - PDV \;of \;cash \;outflows$.
• $IRR$ - Internal Rate of Return for an Investment
  • “Our investment in that apartment project had an $IRR$ of $21\%$” means that we had a $21\%$ return on our investment. $IRR$ tells you the percent return on an investment.

In Lecture 7:

• $P_B$ - The price of a bond
  • To calculate the price of a bond, you calculate the $PDV$ of the cash flows from the bond. This tells you the fair value of the bond - ie it tells you the most you should pay for the bond.
• $YTM$ - Yield to Maturity of a Bond
  • To calculate the percent return on the bond, you just calculate the $IRR$ of purchasing the bond. Practically, this means that you write down the relevant bond pricing formula and solve for $i$.

Conclusion: Bond Pricing and $YTM$ are just applications of $PDV$ and $IRR$ .

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

Suppose you have an investment that you can put any amount of money into at any time. Every year, your investment grows by $i\%$. This is a reasonable assumption, as many of us can invest extra money into bank accounts, mutual funds, or other investments.

Suppose you are evaluating a different project.

The present value of a future payment is the amount of money that would need to be invested today to produce a future payment (or multiple future payments).

It tells you the fair value that someone should pay for those future cash flows.

✏️$T=3$, $c=6\%$, $i=8\%$, $F=\$1000$, $P_B = \;?$

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Cash Flows

$T$	$CF$
$1$	$6\% \times \$1000=\$60$
$2$	$\$60$
$3$	$\$60 + \$1000$

Because $i>c$, this is a discount bond and $P_B<F$.
$P_B = \frac{\$60}{(1+8\%)^1} + \frac{\$60}{(1+8\%)^2} + \frac{\$1060}{(1+8\%)^3} = \$948.46$
More concisely: $P_B = \frac{60}{1.08} + \frac{60}{1.08^2} + \frac{1060}{1.08^3} = \$948.46$

✏️$T=4$, $c=9\%$, $i=7\%$, $F=\$1000$, $P_B=\;?$

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Because $i<c$, this is a premium bond, and $P_B>F$
The coupon payment is $c \times F = 9\% \times \$1000 = \$90$
$P_B = \frac{90}{1.07} + \frac{90}{1.07^2} + \frac{90}{1.07^3} + \frac{1090}{1.07^4} = 1067.74$

✏️ $T=10$, $c=0\%$, $i=7\%$, $F=\$1000$, $P_B=\;?$

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$P_B = \frac{\$1000}{(1+7\%)^{10}} = \$508.35$

✏️ $T=∞$ (consol), $c=9\%$, $i=7\%$, $F=\$1000$, $P_B=\;?$

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The cash flows are:

$T$	$cF$
$1$	$9\% \times \$1000=\$90$
$2$	$\$90$
$3$	$\$90$
$4$	$\$90$
$5$	$\$90$
$6$	$\$90$
$P_B = \frac{\$90}{7\%} = \$1,285.71$

Now let’s try a $YTM$ problem.

✏️$T=∞$, $c=9\%$, $F=\$1000$, $P_B=\$1100$, what is $YTM$?

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Every year, you get $cF = 9\% \times \$1000 = \$90$ Approaching a $YTM$ problem is just like doing an $IRR$ problem. You write down a formula, and solve for $i$.
$\$1100$ = $\frac{\$90}{i}$
$i = \frac{\$90}{\$1100} = 8.18\%$

✏️$T=1$, $c=9\%$, $F=\$1000$, $P_B=\$1010$, what is $YTM$??

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$P_B = \frac{cF}{(1+i)^1} + \frac{F}{(1+i)^1}$
The bond pricing formula for a $T=1$ bond: $P_B = \frac{(cF+F)}{(1+i)}$
Plugging in:
$\$1010 = \frac{(\$90+\$1000)}{(1+i)}$
$\$1010 = \frac{\$1090}{(1+i)}$
Switcheroo:
$1+i = \frac{\$1090}{\$1010} = 1.0792$
$i =7.92\%$

✏️$T=5$, $c=0\%$, $F=\$1000$, $P_B=\$720$, what is $YTM$?

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The bond pricing formula for a $ZCB$ is $P_B = F/(1+i)^T$
Plugging in:
$\$720 = \frac{\$1000}{(1+i)^5}$
Switcheroo:
$(1+i)^5 = \frac{\$1000}{\$720} = 1.388889$
$(1+i)^5 = 1.388889$
We need one more trick:
$((1+i)^5)^{\frac{1}{5}} = 1.388889^{\frac{1}{5}}$
$1+i = 1.388889^{\frac{1}{5}} = 1.067907$
In a spreadsheet or with a calculator you would enter this as 1.388889^(1/5). (Don’t omit the parentheses.)
$i=6.79\%$