BOND VALUATION IN A NUTSHELL

 Hello everyone! I am back again to give a good gist to my readers about Bond valuation techniques. This will help anyone attain a holistic view of how debt securities are valued like below:

Bond coupon = 100 or 10%

Discount rate = 8%

Redeemed = 10 years

Current Market Index= 800

 

Traditional Bond Valuation using Time value of money

100/1.08 + 100/1.08² + ……. + 1100/1.08¹⁰ = PV of bond

 

Zero Coupon Bond valuation

If it is semi-annual

1000/ (1+0.08/2)^10x2 = 1000/1.04²⁰ = PV of future cash flows

 

Current Bond Yield

This is the same for semi-annual and annual

1000 x 10% / 800 = Current yield

 

Yield-To-Maturity

It is the annualized IRR of the bond, interest rate or APR

(100 + (1000 – PV of bond/10 years))/ (1000 + PV of bond/ 2)

or

800 = 100(PVIFA15%, 10 years) + 1000(PVIF15%, 10 years) 

800 = 750

800 = 100(PVIFA 8%, 10 years) + 1000(PVIF 8%, 10 years)

800 = 100*6.710 + 1000*0.4632

800 = 1134.2

When we equate the both, if 800 > 800, lower the discount rate to 8% so that 800 < 800.

To find the exact Kd = YTM between 8-15%

8% + (15%-8%) * 1134 - 800 MV / 1134 - 750

YTM = 8.060%

or

(100 + (1000 - 800 MV)/10 years) / (0.4*1000 + 0.6*800 MV) = Approximation YTM

 

 Calculate bond value using Spot rates using Bond Equivalent Yield Method

Here, annual coupon is one half of the annual coupon cash flow 100. If the Treasury bond spot rates for 1, 2, 10 years is 4%, 3%,…, 7%

50/ (1+0.04/2)¹ + 50/ (1+0.03)² + 1 + 1050/ (1+0.07)¹⁰ = PV of Bond

(Note: The spot rates can be derived from forward rates, say, for two periods & 10 periods by using this formula Spot _{10th year rate} = (1+f₀ x 1+f₁ X 1+f₂ x ….. x 1+f₁₀)^1/10 – 1. Where f₀ is the today’s bond forward rate, f_1-10 are yearly forward rates. Similarly, Forward rates can be derived from Spot rates by using this formula (1+S₂)² / (1+S₁) – 1 = gives the current forward rate for next year.                    

Next, (1+S₁₀)²/ (1+S₉) – 1 = gives the current forward rate for 10^th year)

 

Calculate bond value using Forward rates

Since it is lengthy to calculate for 10 years, let’s compute for 3 years

100/ (1+f₀) + 100/ (1+f₀)(1+f₁) + 100/ (1+f₀)(1+f₁)(1+f₂) = PV of bond. Here, ‘f’ denotes Forward rate and ‘s’ denotes Spot rate. 

These are some of the valuations pertinent to Bonds and please dont hesitate to drop in a comment if you know of other techniques.

 

 Intrinsic value of bond

Vo = 100 (PVIFA 8%, 3 years) + 1000 (PVIF 8%, 3 years) 

8% is the expected rate of return on bond, discount rate and we have to search for values from PVIF tables under 8% & 3 years. 

Semi-annual interest rate

I/2 = 100/2 = 50 (Semi-annual interest)

50(PVIFA (8/2)%, 3 years) + 1000(PVIF (8/2)%, 3 years)

Notes:

when 8% discount rate = 10% coupon rate, Bond value = Face value

When 8% discount rate > 10% coupon, then Bond value < Par value

Bond's price moves inversely proportional to YTM