Question

A corporation made a coupon payment yesterday on its 9.8​%-coupon, ​$1000 par value bonds that make​ semi-annual coupon​ payments, and mature in 14.5 years. You purchased one of these bonds 6.5 years ago​ and, at the​ time, the yield to maturity on these bonds was 4.99​% ​(APR). If you sold your bond today for ​$1856.81​, what​ APY% did you earn on your investment in the​ bond? ​ (In percent with 3​ decimals.) Please state in N , I/Y , PV, PMT, FV calculator form . TI BA ll Plus

Answer 1

There are 3 steps to arrive at APY:
Step 1: Calculate the purchase price of bond
Step 2: Calculate the holding period return
Step 3: Calculate the annual return of the bond

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Step 1:

As per Time Value of Money calculations, we need below values to calculate the purchase price:
N = Period till maturity
FV= Value at maturity
Coupon = Periodic coupon amount
R = Rate of interest /YTM (Yield to Maturity)
PV= Present value or Purchase price at the beginning

Values given are as below:

1) N = Holding period

Time of Purchase : 6.5 years ago
Maturity from today: 14.5 years
Therefore at the time of purchase bond, maturity of bond was 6.5+14.5 = 21 years.
This is N in the TVM calculation. However as the coupon payment is semi-annual, N will be 21*2 = 42.

2) FV= Value at maturity

Par value of Bond which is $ 1,000.

3) Coupon = Periodic coupon amount

Coupon rate is 9.80% annually.
Coupon amount is $98 annually.
However, since coupon payments are semiannual, coupon amount will be $98/2 = $49.

4) R = Rate of interest /YTM (Yield to Maturity)

YTM at the time of purchase was 4.99%.
Therefore, semiannual YTM = 4.99 /2 = 2.495 %        

Input below values in Excel PV function or TI BA II Plus:
FV = $1,000; Coupon = $49; I/Y = 2.495%, N = 42.
This will give the PV as = ($1,621.529)       |

Please note, PV is negative because it indicates Purchase price, outflow of money whereas Coupon and FV are positive as these indicate inflow of money. Incorrect use of signage will lead to wrong PV value.

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Step 2:    

Purchase Price = ($1,621.529)
Selling Price = $1,856.810

Return = (Selling Price / Purchase Price)-1
= (1,856.810 /1,621.529) - 1
= 1.145098 - 1
= 0.145098
= 14.510 %

This is return of the total period - which is 6.5 years.

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Step 3:    

The formula for calculating average annual interest rate is:
Annualized Rate = (1 + ROI over N months)^{12 / N}
where,
ROI = Return on Investment

Substituting the values are below,

Annualized rate = [(1 + 14.5%) ^ 1/6.5 years (or alternatively 12months/78 months)]-1
= [(1.0145)^1/6.5]-1
= 1.021063 - 1
= 0.021063
= 2.106 %

This is the annual percentage yield on the investment in this bond.

I hope this explanation and additional notes were helpful. Please share you feedback. Thank you in advance!