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Consider a property developer who buys a property at time $0$ for $\$90,000$. He also spends $\$10,000$ at time $0$ to buy some materials he will use to develop the property. Ignoring Inflation , the investor thinks that the property will be worth $\$110,000$ today if some improvements to the property are made. The developer expects to make the improvements over the next year and sell the property at time $1$. Inflation is $5\%$.

In the above question Real and Nominal yield received by the investor are asked.

Now for real yield , we settle all the payments according to the common purchasing power , so we write the payments received at $t=1$ in terms of $t=0$ , which is : $\dfrac{110,000}{1.05}$.

Thus real yield is i : $90,000 + 10,000 = \dfrac{110,000}{1.05} (1+i)^{-1}$ { That's what I think .}

In the solution the real yield is given as : $100,000 = \dfrac{110,000\times1.05}{1.05} (1+i)^{-1}$.

Can anyone explain ? And I have no idea about the Nominal yield. Is that the 'money' yield ? { Money Yield - Ignoring Inflation }.

User9523
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1 Answers1

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The relation between real and nominal interest rates and the expected inflation rate is given by the Fisher equation $$1+i = (1+r) (1+\pi_e)\tag 1$$ where

  • $i =$ nominal interest rate;
  • $r=$ real interest rate;
  • $\pi_e =$ expected inflation rate.

$110,000$ represents the undiscounted expected net benefit at the end of year 1, evaluated at constant prices; so you have $$ 100,000=\frac{110,000}{1+r}$$ where no inflation components are included in either prices or the interest rate. That is a real yield rate $r=10\%$.

Alternatively, considering the inflation rate, $$ 100,000=\frac{110,000\times (1+\pi_e)}{1+i}=\frac{110,000\times (1+\pi_e)}{(1+r)(1+\pi_e)}=\frac{110,000}{1+r} $$

This means a nominal rate $i=15.5\%$ (as you can check also from the Fisher equation).

In general considering constant prices $C_k^{(0)}$ related to the base year $t=0$, we have that the present value is $$ PV=\frac{C_1^{(0)}}{1+r_1} + \frac{C_2^{(0)}}{(1+r_1)(1+r_2)} + \cdots + \frac{C_n^{(0)}}{(1+r_1) \cdots (1+r_n)}\tag 2 $$ Otherwise, considering nominal prices $C_k$ related to the current year, we have that the present value is $$ \text{PV}=\frac{C_1}{1+i_1} + \frac{C_2}{(1+i_1)(1+i_2)} + \cdots + \frac{C_n}{(1+i_1) \cdots (1+i_n)}\tag 3 $$ Through the Fischer equation (1) and considering that $C_k=C_k^{(0)}(1+\pi_{e,1})\cdots(1+\pi_{e,k})$ $$ \begin{align*} \text{PV} & = \frac{C_1^{(0)}(1+\pi_{e,1})}{(1+r_1)(1+\pi_{e,1})} + \frac{C_2^{(0)}(1+\pi_{e,1})(1+\pi_{e,2})}{(1+r_1)(1+r_2)(1+\pi_{e,1})(1+\pi_{e,2})} + \cdots \\ & \qquad \cdots + \frac{C_n^{(0)}(1+\pi_{e,1})\cdots(1+\pi_{e,n})}{(1+r_1)(1+r_2)\cdots(1+r_n)(1+\pi_{e,1})(1+\pi_{e,2})\cdots(1+\pi_{e,n})} \end{align*} $$ So we have that (2)=(3), that is, the present value derived by either equation will be identical. This alleviates any question concerning the analysis in terms of constant or nominal prices and in terms of real or nominal rate of interests.

alexjo
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  • In your first relation $100,000 = \dfrac{110,000}{1+r}$ , you are saying no inflation components are included , where $r$ is the real yield , but to obtain real yield we have to incorporate inflation , don't we ?@alexjo – User9523 Mar 06 '16 at 06:49
  • @User9523 I've added some explanations. If you consider constant prices, you must consider real interest rate. If you consider nominal prices, you must consider nominal interest. $110,000$ is the worth today, i.e. at constant price. So the value at $t=1$ is $110,000(1.05)$ and, according to the common purchasing power, the payment received at $t=1$ in terms of $t=0$ is $\frac{110,000(1.05)}{1.05}$ and then you can evaluate the real interest from $100,000=\frac{110,000(1.05)}{1.05}(1+r)^{-1}$. – alexjo Mar 06 '16 at 11:41