Bank discount yield, holding period yield, effective annual yield, and money market yield

Example 6: Calculating bank discount yield

What is the bank discount yield for a Treasury bill trading at $98,000? The face value of the Treasury bill is $100,000, and it has 180 days left until maturity.

Solution:

Discount to the face value (D) = 100,000 - 98,000 = 2,000
Bank discount yield = (2,000/100,000)(360/180) = 4.00 percent.

Example 7: Calculating T-bill price given its bank discount yield

A Treasury bill is quoted at a bank discount yield of 3.00 percent. The bill has 120 days to maturity. Compute the market price of the Treasury bill.

Solution:

Bank discount yield = 3.00 percent
Unannualized discount for 120 days period = 0.03(120/360) = 0.01
Dollar discount on the face value = 0.01*100,000 = 1,000
Market price of the Treasury bill = Face value of bill - Dollar discount on the bill = 100,000 - 1,000 = $99,000.

Example 8: Calculating the holding period yield for a pure discount bond

A zero-coupon bond was bought at a price of $850 and is sold at a price of $984. What is the holding period yield for the bond?

Solution:

Holding period yield for a pure discount bond = (P₁ - P₀)/P₀ = (984-850)/850 = 15.76 percent.

Example 9: Calculating the effective annual yield

Mohan bought a stock for $45 and sold it after 180 days for $47.5. The stock also paid a dividend of $0.5 at the end of 180 days. What is the effective annual yield realized by Mohan from investing in the stock?

Solution:

For calculating the effective annual yield, we need to calculate the holding period yield first.

HPY = (P₁ - P₀ + D₁)/P₀ = (47.5 - 45 + 0.5)/45 = 6.67 percent
EAY = (1+HPY)^(365/180) - 1 = 1.0667^(365/180) - 1 = 13.98 percent.

Derivation of the second formula of money market yield from the first one

HPY = Dollar Discount/(Face Value – Dollar Discount) = D.D./(F.V. – D.D.)

(D.D./F.V.)*(360/t) = r_BD

D.D. = (t/360)*(F.V.)*r_BD

HPY = (t/360)*(F.V.)*r_BD/[F.V. – (t/360)*(F.V.)*r_BD] = t*r_BD/(360 – t*r_BD)

r_MM = HPY*(360/t) = [t*r_BD/(360 – t*r_BD)]*(360/t) = 360*r_BD/(360 – t*r_BD)

Example 10: Calculating money market yield

A Treasury bill is trading at $98,500, and it has 270 days to maturity. The face value of the bill is $100,000. What is the money market yield for the bill?

Solution:

Holding period yield (HPY) = (100,000-98,500)/98,500 = 1.52 percent
Money market yield = HPY(360/t) = 0.0152*(360/270) = 2.03 percent.

Example 11: Converting bank discount yield to money market yield

The bank discount yield of a Treasury bill is 2.50 percent. The bill has 180 days to maturity. What is its money market yield?

Solution:

This question can be solved in two ways. The first way is to simply remember the formula of converting bank discount yield to money market yield and apply the formula.

Money market yield, r_MM = (360*r_BD)/[360 - (t)(r_BD)] = (360*0.025)/[360 - 180*0.025] = 2.53 percent.

The other way is a little longer one, but it is advisable to solve the question using this way as you don't want to risk forgetting the formula.

Bank discount yield = 2.50 percent
Suppose the face value of the bill = 100,000
Total dollar discount = 100,000*0.025*(180/360) = 1,250.
Market price of the bond = 100,000 - 1,250 = 98,750.
Holding period yield = 1,250/98,750 = 1.266 percent
Money market yield = HPY(360/t) = 0.01266*(360/180) = 2.53 percent.

Example 12: Calculating bond equivalent yield

The holding period yield for a bond is 0.8 percent. The bond was held for one month only. What is the bond equivalent yield for the bond?

Solution:

We first need to calculate the semi-annual yield.

Semiannual yield = (1+Monthly holding period yield)⁶ - 1 = 1.008⁶ - 1 = 4.897 percent.

Bond equivalent yield = 2*Semiannual yield = 2*4.897 percent = 9.79 percent.