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# What is a discounted factor (DF) and how to calculate

- Authors
- Name
- Benton Li

A dollar today probably won’t be worth the same as a dollar 10 years later, and the future dollar probably worth less than today (Yenno, inflation)

e.g. In 2010, you can buy a dozen eggs for $2. But today, $2 can probably buy only half a dozen.

So although this green paper is the same at present and in the future, they have different values: present value (PV) and future value (FV)

When you lend money, you can compound interest over time, that is, **compound** rate maps PV to FV. The other way would be mapping FV to PV, and the rate here is called the discount factor.

- Compounding rate: PV |→ FV
- Discounting factor: FV |→ PV

DF depends on time and therefore is a function of time. So we denote $DF_t$ as the discount factor at time t.

You can think of it as the current price of a zero-coupon bond, which is a one-payment bond that doesn’t pay coupon, but $1 at time t. The price is probably less than $1 (e.g. $0.97). Mathematically,

Before you read the following content, make sure you are familiar with yield to maturity (YTM)

**Let’s look at the bond rate (for a vanilla bullet bond).**

Suppose at time t, you receive your last payment, which is the principal + the last coupon payment. Your FV at t will be

But now, this payment is the **bond price** (PV) - the **sum of discounted coupons** between now and t-1. That is,

Consequently,

See also:

## Other DF formulas

**Depo rate** and **Swap rate** for t < 1 yr($\alpha$ = accrual factor)