Fixed income analysis with matlab

An investor initially buys a 5-year, 8% annual coupon payment bond at the price of 85.00 per 100 of par value.

 syms x

 eqn = 8/(1+x)^1 + 8/(1+x)^2 +8/(1+x)^3 +8/(1+x)^4 +108/(1+x)^5==85

 S = solve(eqn,x,'Real',true) 

0.1218

or we know discount rate as 0.12 (%12) and we calculate price of bond

eqn = 8/(1+0.12)^1 + 8/(1+0.12)^2 +8/(1+0.12)^3 +8/(1+0.12)^4 +108/(1+0.12)^5==x 

S = solve(eqn,x,'Real',true) 

85


So the investor receives the series of 5 coupon payments of 8 (per 100 of par value) for a total of 40, plus the redemption of principal (100) at maturity. On top of collecting the coupon interest and the principal, there’s an opportunity to reinvest the cash flows.

If the coupon payments are reinvested at 12.18% immediately they are received, the future value of the coupons on maturity date will amount to 51.004 per 100 par value.

End of Year 1	End of Year 2	End of Year 3	End of Year 4	End of Year 5
$8×1.12184	$8×1.12183	$8×1.12182	$8×1.12181	$8×1.12180

  

The 1^st coupon payment of $8 is reinvested at 12.18% for 4 years until the end of 5^th year, 

the 2^nd is invested for 3 years,

the 3^nd is invested for 2 years,

the 4^nd is invested for 1 years,


The amount in excess of the coupons, 11.004 (= 51.004 – (5 × 8)), is called “interest-on-interest” gain from compounding.

 8 * (1.1218)^4+ 8 * (1.1218)^3 + 8 * (1.1218)^2+ 8 * (1.1218)^1+ 8

51.0049

The investor’s total return is 151.004, the sum of reinvested coupons (51.004) and the redemption of principal at maturity (100). The realized rate of return is 12.18%, as we have calculated previously.

$$85=\frac{151.004}{{/(1+r)}^5}$$

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The yield-to-maturity at the time of purchase equals the investor’s rate of return under three assumptions: (1) The investor holds the bond to maturity, (2) there is no default by the issuer, and (3) the coupon interest payments are reinvested at that same rate of interest.

Constant Yield Price Trajectory

 8/(1+0.12)^1 + 8/(1+0.12)^2 +8/(1+0.12)^3 +8/(1+0.12)^4 +108/(1+0.12)^5

ans =

   85.5809 [First day]

>> 8/(1+0.12)^1 + 8/(1+0.12)^2 +8/(1+0.12)^3 +108/(1+0.12)^4

ans =

   87.8506 [First year]

>> 8/(1+0.12)^1 + 8/(1+0.12)^2 +108/(1+0.12)^3

ans =

   90.3927  [Second year]


>> 8/(1+0.12)^1 + 108/(1+0.12)^2

ans =

   93.2398 [Third year]

>> 108/(1+0.12)^1

ans =

   96.4286 [Forth year]

last day 100

Source: https://analystprep.com/cfa-level-1-exam/fixed-income/understanding-fixed-income-risk-return/