Calculating the present value of a payment after n days (APY)

Question

I am currently trying to fully understand the APY concept in finance (I know very basic). I would be very greatful, if you could help me with my problem.

The following example is given: The current 1 month market rate is 5%. The term-structure is flat. Calculate the present value of a 10$ payment in 30 days. The year has 360 days.

Now there should be 2 ways to tackle this problem, right?

1. Transforming the monthly (compounded rate) into the APY:

I would do this by the following formula:

$ (1+\frac{0.05}{12})^{12} -1 = 0.0512 $

In order to discount with the correct rate I would calculate the following:

$ \frac{10}{1.0512^{\frac{30}{360}}} = 9.9585 $

2.Now there is also a second method that I do not fully understand:

If I calculate the following I get the same result as in 1:

$ \frac{10}{1+0.05*\frac{30}{360}} = 9.9585 $

Can anyone explain to me, why both methods yield the same result? I understand that the payment occurs in 30 days and hence multiplying the rate by $\frac{30}{360}$ makes sense somehow, however I do not understand why this works with the monthly rate. Can anyone explain this to me? I am somehow struggling to understand the concept of the monthly compounded rate (in this case the rate of 5%).

Answer 1

The results are close, but they are not exactly the same. The results are $ \frac{10}{1+0.05\cdot \frac{1}{12}} = 9.958506224066... $ and $\frac{10}{1.0512^{\frac{1}{12}}}=9.9584761436...$ I have cancelled the fractions to make clear that we are referring to monthly periods.

To evaluate the monthly interest rate we start with the yearly interest r. Let's say we have an amount of $C_0$. Then after one year we gain $C_0\cdot r$ interest. And in total we have $C_0+C_0\cdot r=C_0\cdot (1+r)$. So the factor is $(1+r)$. To get an equivalent factor for a monthly compounding or discounting we take the $12$-th root: $(1+r)^{\frac1{12}} \qquad (*)$

We can also make it more simple. If we gain $C_0\cdot r$ interest in one year, we gain $C_0 \cdot \frac{1\cdot r}{12}=C_0 \cdot \frac{ r}{12}$ interest in one month. And in two month $C_0 \cdot \frac{2\cdot r}{12}$ interest. And so on. We just calculate the proportion. Now we take the simple interest rate $\frac{r}{12}$ and add 1 to get the interest factor for one month: $1+\frac{r}{12}$. We raise it to power 12 to get the yearly interest rate factor: $\left(1+\frac{r}{12}\right)^{12}$. The binomial theorem states that $(x+y)^n = \sum\limits_{k=0}^n {n \choose k}\cdot x^{n-k}\cdot y^k $. For $n=12, x=1$ and and $y=\frac{r}{12}$ we get

$$\left(1+\frac{r}{12}\right)^{12}=\sum\limits_{k=0}^{12} {12 \choose k}\cdot \left(\frac{r}{12}\right)^k$$

The first four summands of the sum are $\color{blue}{1+r}+\frac{11}{24}r^2+\frac{55}{432}r^3$. Usually interest rates are close to $0$, especially smaller than 1, i.e. $6\%=0.06$. The higher the exponent, the closer is the summand to zero. For $r=0.06$ we have $\frac{55}{432}r^3=0.0000275$. So we can take the first two summands only for a good approximation.

$$\left(1+\frac{r}{12}\right)^{12}\approx 1+r$$

Taking the 12-th root

$$1+\frac{r}{12}\approx (1+r)^{\frac1{12}}$$

The right hand side is equal to $(*)$. We see that the fractional discount factor $1+\frac{r}{12}$ is a good a approximation for the equivalent discount factor. You example confirms that fact.