How to calculate the price of a bond based with a yield to Maturity, term and annual interest?

Question

I'm trying to help my nephew with finance but I'm not sure if I fully understand if I have all the necessary details to answer this question.

Here's the question:

1. Say you want to purchase a 7%, 15yr bond which pays annual interest. Say 6% is the Yield to Maturity. What price will you pay?
2. What would be the current yield on the bond from the above calculation
3. Say you made the the purchase based on the calculation in Step 1 (from above). What would be the bond's Yield to Maturity at this point of time?
4. Say you made the the purchase based on the calculation in Step 1 (from above). Say that you receive the second year's interest payment on the sale date. What would be your rate of return on this investment?

I'm good with basic finance and know a little about bonds. I started reading on Bonds from Investopedia But I'm trying to understand if the questions above are complete. For #1 which asks for "What price will you pay", Don't we need the bond's original price info to calculate the market price?

Are there any pieces of missing information to calculate? If not, how would one go about answering these questions?

Answer 1

Like all financial investments, the value of a bond is the present value of expected future cash flows. The Yield to Maturity is the annualized return you get on your initial investment, which is equivalent to the discount rate you'd use to discount future cash flows.

So if you discount all future cashflows at 6% annually*, you can calculate the price of the bond:

$1,000 bond with 7% annual coupon:

Year    Cash Flow yield   disc factor  PV
1       $70       6%      0.9434       $66.04 
2       $70       6%      0.8900       $62.30 
3       $70       6%      0.8396       $58.77 
4       $70       6%      0.7921       $55.45 
5       $70       6%      0.7473       $52.31 
6       $70       6%      0.7050       $49.35 
7       $70       6%      0.6651       $46.55 
8       $70       6%      0.6274       $43.92 
9       $70       6%      0.5919       $41.43 
10      $70       6%      0.5584       $39.09 
11      $70       6%      0.5268       $36.88 
12      $70       6%      0.4970       $34.79 
13      $70       6%      0.4688       $32.82 
14      $70       6%      0.4423       $30.96 
15      $1,070    6%      0.4173       $446.47 
-------------------------------------------------
Total                                  $1,097.12 

So the price of a $1,000 bond (which is how bond prices are typically quoted) would be $1,097.12.

The current yield is just the current coupon payment divided by the current price, which is 70/1,097.12 or 6.38%

Question 3 makes no sense, since the yield to maturity would be the same if you bought the bond at market price

Question 4 talks about a "sale" date which makes me think that it assumes you sold the bond on the coupon date, but you'd have to know the sale price to calculate the rate of return.

• the formula for discount factor in this case would be 1/(1+r)^n

Answer 2

The answer to almost all questions of this type is to draw a diagram. This will show you in graphical fashion the timing of all payments out and payments received. Then, if all these payments are brought to the same date and set equal to each other (using the desired rate of return), the equation to be solved is generated.

In this case, taking the start of the bond's life as the point of reference, the various amounts are:

Pay out = X

Received = a series of 15 annual payments of $70, the first coming in 1 year. This can be brought to the reference date using the formula for the present value of an ordinary annuity.

PLUS

Received = A single payment of $1000, made 15 years in the future. This can be brought to the reference date using the simple interest formula.

Set the pay-out equal to the present value of the payments received and solve for X

I am unaware of the difference, if any, between "current rate" and "rate to maturity"

Finding the rate for such a series of payments would start out the same as above, but solving the resulting equation for the interest rate would be a daunting task...