Purchase A $1000 Face Value Bond With 6 Years To Maturity

You Purchase A 1000 Face Value Bond With 6 Years To Maturity For 850

You purchase a $1000 face value bond with 6 years to maturity for $850. One year later, you sell the bond to a bond investor for $714. Suppose that the bond investor holds the bond to maturity. Calculate the annualized rate of return that the bond investor will receive. Enter your answer into blackboard as a percentage rounding to two decimal places. For example if your answer is 1.25 percent then you should enter "1.25".

Paper For Above instruction

The process of determining the annualized rate of return on a bond that is sold before maturity involves understanding the concept of yield to maturity (YTM) and the effects of holding period returns. In this scenario, an investor initially purchases a bond at a discount, and then sells it after one year at a lower price, with the remaining maturity being five years. The goal is to calculate the bond investor's annualized rate of return based on this transaction.

Initially, the bond was purchased for $850 with a face value of $1000, which implies a purchase yield that accounts for the difference between the purchase price and face value over the remaining period to maturity. After holding the bond for one year, the sale price decreases to $714, and since the bond is held to maturity, the investor will receive the face value of $1000 at maturity. The return on investment involves two components: the income received from the coupon payments (if any, but unspecified in this case), and the capital gain or loss resulting from the sale before maturity combined with the face value at maturity.

Assuming the bond pays no coupons or that coupons are rolled into the calculations equally over the period, the cash flows for the investor include the sale price after one year ($714), and the face value received at maturity ($1000). Specifically, the investor pays $850 initially, holds the bond for one year, and then sells for $714, but will receive $1000 at maturity. To find the annualized rate of return, we analyze the entire process from the initial purchase to the eventual maturity cash flow, considering the effective annual return over the duration of ownership.

Mathematically, the total return over the one-year period can be formulated as the ratio of the proceeds at maturity over the initial purchase price, compounded over the holding period. The total amount the investor effectively receives at maturity includes the face value ($1000) plus any coupons, but since none are specified, we consider only the face value and sale price in calculations. The key step involves solving the following equation for the annualized return r:

\[

(850 \times (1 + r)) = 714 \times (1 + r)^{0} + \text{future cash flows}

\]

However, in this case, because the bond is sold at a lower price than the original purchase, and the face value is $1000, the primary calculation involves solving the internal rate of return (IRR) based on the initial purchase, sale price, and face value at maturity.

Specifically, the overall rate of return considers the initial investment of $850, the sale price after one year ($714), and the face value at maturity ($1000). The cash flow timeline is as such: initial outlay of $850, after one year, the bond is sold for $714, and at maturity in five years, the bondholder receives $1000. For the purpose of this calculation, since the bond is sold after one year, the annualized return is primarily dictated by the change in bond price and the face value at maturity, compounded over the remaining period.

To accurately find the bond investor's annualized rate of return, we apply the IRR formula to the cash flows: -$850 at time 0, $714 at time 1 (sale price), and $1000 at time 6 (maturity date). But since the bond is sold after one year, and we're asked for the annualized rate of return over the one-year holding period, we focus on the cash flows within this period, specifically the initial investment and the sale proceeds, taking into account the expected payout at maturity.

In practice, the precise calculation involves solving the equation for r where:

\[

850 \times (1 + r) = 714 + \frac{1000}{(1 + r)^5}

\]

or, more straightforwardly, considering the return from the initial purchase to the sales price and subsequent face value at maturity, using the following approximation:

\[

(1 + r)^{6} = \frac{1000}{850}

\]

Extracting a 6-year annualized return gives:

\[\

\( (1 + r)^6 = \frac{1000}{850} \Rightarrow 1 + r = \left(\frac{1000}{850}\right)^{1/6} \)

Calculating, we find:

\( 1 + r = \left(\frac{1000}{850}\right)^{1/6} \approx (1.17647)^{1/6} \approx 1.0274 \)

Thus, the annualized rate of return is approximately:

\( r \approx 0.0274 \) or 2.74%.

Since the question asks for the annualized rate of return the bond investor will receive assuming they hold the bond to maturity, considering the sale price and the face value at maturity, the approximate annualized yield is around 2.74%. This calculation aligns with typical bond valuation principles and yield computations for bonds bought at a discount and held to maturity.

References

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  • Investopedia. (2021). Yield to Maturity (YTM) Explained. Retrieved from https://www.investopedia.com/terms/y/ytm.asp
  • SEC. (2020). Understanding Bonds and Bond Valuation. U.S. Securities and Exchange Commission.
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