Question

must be completed by hand

A $1,000 par value bond with seven years left to maturity has a 9 percent coupon rate (paid semiannually) and is selling for $945.80. What is its yield to maturity? (An equation is sufficient.)

Answer 1

Yield to Maturity is the rate at which Present of value of all cash inflows bond equal to Bond value at present i.e sale price of bond.

We have following information-

Par value (FV) ) = $ 1,000

Coupon rate = 9 %

Coupon rate semi annual = 0.09/2 = 0.045

Semi annual coupon amount = 1000*0.045 = $ 45

Maturity = 7 years

Semi annual period (n) = 7*2 = 14

Selling Price of Bond (BV) = $ 945.80

YTM = Y

To calculate YTM we will use following equation

$BV = C \sum_{n=1}^{n}\frac{1}{(1+Y)^{n}} + \frac{FV}{(1+Y)^{n}}$

where,

C = semi annual coupon

Y = YTM

n= no. of semi annual period

BV = Bond value today

FV = Par value of Bond

To calculate YTM manually, we need to refer Present value Interest Annuity Table (PVIAF(%,n) and Present value Interest factor table (PVIF(%,N)

$\sum_{n=1}^{n}\frac{1}{(1+Y)^{n}} = PVIAF(Y,n)$

$\frac{1}{(1+Y)^{n}} = PVIF(Y,n)$

Thus, we can write above equation in following manner-

$BV = C\times PVIAF(Y,n) + FV\times PVIF(Y,n)$

Now, to solve this we need to choose two random rates in such manner that PV of cash inflows with higher rate is less than Bond selling price and with Lower rate PV of cash inflows is greater than selling price.

Tips for choosing random rate-

The Bond Price is equal to its Par value when coupon rate is equal to yield to maturity rate

The Bond Price is less than its par value when coupon rate is less than YTM and vice versa.

In our case, Selling Price Bond is less than the Par value. Therefore, two random rates higher than coupon rate.

Let's select 6% and 5% -

using below interest factor we calculate the Bond value.

PVAIF(6%,14) = 9.2950

PVAIF(5%,14) = 9.8986

PVIF(6%.14) = .44230

PVIF(5%,14) = .50507

Now putting the values in equation for Y = 6%

$BV = 45 \times 9.2950 + 1,000\times 0.44230$

$BV = 860.575$

Thus BV at Y=6% is $ 860.575

putting the values in equation for Y=5%

$BV = 45\times9.8986 + 1,000 \times0.50507$

$BV = 950.507$

Thus, BV at Y=5% is $ 950.507

Now to calculate YTM(Y) rate in above case, use following formula

$Y = Lower rate + \frac{BV at lower rate - Sale Price}{BV at lower rate - BV at higher rate}$

Y = 5% + (950.507 - 945.80)/(950.507 - 860.75)

Y = 5% + 4.707/89.757

Y = 5.05 % (semi annual)

Thus, Annual YTM rate would be 5.05*2 = 10.1% pa.

Please note -

we can also calculate YTM rate using excel . to calculate YTM in excel use "=irr" formula selecting all cash inflows.